A recursive approach to the O(n) model on random maps via nested loops

Planar maps, which are proper embeddings of graphs in the two-dimensional sphere, are fundamental mathematical objects whose combinatorics raises many beautiful enumeration problems, first addressed by Tutte[1]. Maps are also widely used in physics as discrete models for fluctuating surfaces or interfaces in various contexts, ranging from soft matter physics (e.g. biological membranes) to high energy physics (e.g. string theory). Most problems involve 'random maps', i.e. consist in the study of some particular statistical ensemble of maps, distributed according to some prescribed law. Interesting scaling limits may be reached when one considers ensembles of large random maps, giving rise to nice universal probabilistic objects. Several map enumeration techniques were developed over the years, which we may classify in three categories: Tutte's original recursive decomposition method [2], the technique of matrix integrals [3] and, more recently, the bijective method [4], where maps are coded by tree-like objects.

So far, the most advanced results were obtained for ensembles of maps with a simple control on the degree of, say, the faces of the maps. Examples are ensembles of triangulations (maps with faces of degree 3 only) or of quadrangulations (degree 4) where the total number of faces in the map is either fixed or governed by some Boltzmann distribution. The statistics of these maps is now well understood and many exact enumeration results were obtained within the framework of each of the above three enumeration techniques. Of particular interest is the scaling limit of large maps with prescribed bounded degrees, which realizes the so-called universality class of 'pure gravity' and gives rise to the 'Brownian map', a probabilistic object with remarkable metric properties [5]. As first recognized in physics, other universality classes of maps may be reached upon equipping the maps with statistical models, such as models of spins or particles, which present a large variety of critical phenomena that modify the statistical properties of the underlying maps. This gives rise to a large variety of universality classes for maps, whose understanding is the domain of the so-called two-dimensional quantum gravity [6]. It is worth mentioning that most results in this domain rely on matrix integral formulations of the models so that the matrix integral technique appears so far as the most powerful approach to explore new universality classes.

A particular important class of models of maps equipped with statistical models are the so-called O(n) loop models, which consist in having maps endowed with configurations of closed self-avoiding and mutually avoiding loops drawn on their edges, each loop receiving the weight n. A particular class of O(n) loop models, where loops visit only vertices of degree 3 (the degree of the other vertices being bounded), was analyzed in detail by use of matrix integral techniques in [7–12]. There it was found that these models present several phases with non-trivial universal scaling limits. We will briefly recall the results of this analysis in section 1.2.

It was recently recognized in [13] that another simple way to escape from the universality class of pure gravity consists in considering 'maps with large faces', i.e. ensemble of maps where the degree of the faces is unbounded and properly controlled, so that faces with arbitrarily large degree persist in the scaling limit. Using the bijective method, it was shown that such ensembles may give rise to new probabilistic objects corresponding to maps coded by stable trees. It was then proposed that these probabilistic objects may also describe the scaling limit of O(n) loop models in some of their phases.

The purpose of this paper is to show that this is indeed the case. More precisely, we show that a number of O(n) loop models may be bijectively transformed into models of maps without loops and with a simple control on the degree of their faces. This control involves some degree-dependent face weights whose value is fixed by some appropriate consistency relation in the form of a fixed point condition depending on n. This implies that the possible scaling limits of the O(n) loop models necessarily match the scaling limits of maps controlled simply by their face degrees. These include the pure gravity universality class as well as the new universality classes of [13] for maps with large faces, which are observed in the dense and dilute phases of the O(n) loop models.

As a reminder, we briefly describe a few of the known results about O(n) models on random maps, as previously studied in [7–12] by matrix integral techniques. We will not give very precise definitions here as these models are not those that we will study in the remainder of the paper. The O(n) loop model studied in [7–12] consists in having self-avoiding and mutually avoiding loops drawn on random maps so as to visit only vertices of degree 3, the vertices not visited by loops having arbitrary (but bounded) degrees at least 3. A vertex visited by a loop receives the weight ${\tilde{h}}$, while a vertex not visited by a loop receives the weight ${\tilde{g}}_k$ if it is k-valent (k ⩾ 3). Each loop receives, in addition, the weight n. We define ${\tilde{F}}_p^{\rm loop}$ as the corresponding generating function for loop configurations on rooted maps, where the root vertex is not visited and has degree p (by convention, the root vertex is unweighted and we set ${\tilde{F}}_0^{\rm loop}=1$). This corresponds to having a dual map with a boundary of length p. We may gather the family $\big({\tilde{F}}_p^{\rm loop}\big)_{p\ge 0}$ into the resolvent

$$\begin{equation} {\tilde{W}}(\xi )=\sum _{p\ge 0} {{\tilde{F}}_p^{\rm loop}\over \xi ^{p+1}}\ \end{equation} \tag{ 1.1 }$$

well defined for ξ large enough. In the range of weights where the model is well defined, this formal series is in fact an analytic function on C∖[γ₋, γ₊] and has a discontinuity on some segment [γ₋, γ₊]. Using this information in the recursive relation for ${\tilde{F}}_p^{\rm loop}$ obtained (in the spirit of the Tutte's recursive method) by removing the root edge, it can be shown that ${\tilde{W}}(\xi )$ is a solution of a scalar non-local Riemann–Hilbert problem:

$$\begin{equation} \fl \forall\, \xi \in [\gamma _-,\gamma _+],\qquad {\tilde{W}}(\xi + {\rm i}0) + {\tilde{W}}(\xi - {\rm i}0) + n{\tilde{W}}({\tilde{h}}^{-1} - \xi ) = \xi -\sum _{k \ge 3} {\tilde{g}}_k \xi ^{k - 1}. \end{equation} \tag{ 1.2 }$$

This uniquely fixes ${\tilde{W}}(\xi )$, as well as γ₋ and γ₊, from the requirement that ${\tilde{W}}(\xi ) \sim 1/\xi$, when ξ → ∞, and that ${\tilde{W}}$ is holomorphic in C∖[γ₋, γ₊]. A non-trivial critical point is reached when $\gamma _+ \rightarrow (2{\tilde{h}})^{-1}$, which defines a non-trivial critical surface ${\tilde{h}}={\tilde{h}}((\tilde{g}_k)_{k\ge 1})$. Equation (1.2) was first solved on this critical surface by Kostov [8], who showed that ${\tilde{W}}$ may develop a singularity of the form

$$\begin{equation} {\tilde{W}}(\xi )\vert _{\rm sing.}\sim {\rm const.} \left(\xi -{1\over 2{\tilde{h}}}\right)^{a-1} \quad {\rm for} \quad \xi \rightarrow \left({1\over 2{\tilde{h}}}\right)^+ , \end{equation} \tag{ 1.3 }$$

with a = 2 ± b, $\pi b= \arccos (n/2)$. This singularity captures the large p asymptotics of ${\tilde{F}}_p^{\rm loop}$ as

$$\begin{equation} {\tilde{F}}_p^{\rm loop}\sim {\rm const.} {(2{\tilde{h}})^{-p} \over p^a}. \end{equation} \tag{ 1.4 }$$

The smallest value a = 2 − b is observed inside the non-trivial critical surface and describes the so-called dense phase of the O(n) model, while the largest value a = 2 + b is observed only at some boundary of this critical surface and describes the so-called dilute phase. Other multicritical points, with a = 2 ± b + 2m (m = 1, 2, ...), can also be observed by tuning more and more coefficients ${\tilde{g}}_k$ [9], but this requires having some of these weights negative, which prevents interpreting these multicritical points as proper probabilistic ensembles. Equation (1.2) was then solved in all generality (outside of the critical surface) in [11] in terms of elliptic functions and of the Jacobi theta function ϑ( ·, q), the non-trivial critical surface corresponding to the limit where the nome q goes to 1 (or 0 depending on the convention). In this limit, the theta function degenerates into a trigonometric function, and one can recover the critical exponents stated above. The study of the O(n) model was extended recently to maps of any topology in [14].

Let us start by giving a brief description of our results. Much of the paper deals with O(n) loop models defined on planar tetravalent maps (maps with vertices of degree 4 only). By definition, the loops are self-avoiding and mutually avoiding and each loop receives the weight n. Different models are obtained by assigning different weights to the vertices depending on whether they are visited by a loop or not. Special attention will be paid to the simplest version of the model, the rigid loop model, where loop turns are forbidden at the tetravalent vertices.

Our analysis is simplified first by reformulating our loop model on the dual maps, which are planar quadrangulations, and then by extending them to quadrangulations with a boundary of arbitrary (necessarily even) length. Indeed, a simple characterization of the universality class of the loop model at hand is via the large p asymptotics of its generating (or partition) function F^loop_p in the presence of a boundary of length 2p. It is expected to take the form

$$\begin{equation} F_p^{\rm loop}\sim {\rm const.} {{\mathcal A}^p\over p^a} \end{equation} \tag{ 1.5 }$$

with a non-universal exponential growth factor ${\mathcal A}$ and a power-law decay factor 1/p^a involving a universal exponent a in the range 3/2 ⩽ a ⩽ 5/2. Assuming this asymptotic behavior, we show that, for a given n, at most four values of the exponent a may be observed; namely a = 3/2, corresponding to a subcritical model, a = 5/2, corresponding to a generic critical model, and

$$\begin{equation} a=2\pm b, \qquad \pi b= \arccos (n/2) \end{equation} \tag{ 1.6 }$$

corresponding to a non-generic critical model. This last behavior requires that n lies in the range 0 < n < 2 (note that, for n = 0, (1.6) yields the pure gravity exponents a = 3/2 and a = 5/2). In the case of the rigid loop model, we solve the model exactly, thus establishing the validity of (1.5) and providing a precise description of the phase diagram, which specifies the domain of physical parameters where each of the above values of a is observed.

Expressions (1.5) and (1.6) are identical to those obtained for the O(n) loop models discussed in section 1.2. Here, they are obtained by different means, which rely on an equivalence of our models with models of bipartite maps. More precisely, quadrangulations (with a boundary) endowed with loop configurations may be coded bijectively by bipartite maps without loops but with faces of arbitrary degree, each containing an O(n) loop model configuration of its own. This bijective decomposition allows one, via a simple substitution procedure, to express F^loop_p in terms of the well-understood generating function for bipartite maps with a boundary. As a consequence, we may identify the possible asymptotics of F^loop_p as those of ordinary maps with possibly large faces, leading eventually to (1.5) and (1.6).

This paper is organized as follows. Section 2 is devoted to a combinatorial study of the O(n) loop model on tetravalent maps. After defining the model in section 2.1, we present in section 2.2 a bijective decomposition that allows one to code the configurations of our model in terms of their gasket, which is a bipartite map with no loops but with holes, which are faces of arbitrary even degree, together with a content for each hole. For consistency, the effective weights for the holes are shown to obey some crucial fixed point condition, which determines them uniquely. This condition involves in particular some ring generating function, which accounts for the configuration in the immediate vicinity of a loop, and which we make explicit in section 2.3. Section 3 does not treat the O(n) model itself, but discusses the possible critical behaviors of bipartite maps, which we classify into subcritical, generic critical and non-generic critical. The first two cases are discussed in section 3.2, while particular attention is paid to the non-generic critical behavior in section 3.3. Section 3.4 discusses the so-called resolvent of bipartite maps. Section 4 is devoted to the consequences of the fixed point condition on the asymptotics of F^loop_p, leading to (1.5) and (1.6). We first concentrate on the rigid loop model in section 4.1 before addressing the general case in section 4.2. Section 5 shows how to transform the fixed point condition into a linear integral equation, both in the rigid case (section 5.1) and in the non-rigid case (section 5.2). We then analyze in section 5.3 the solution of this equation as it captures all the phase diagrams of the O(n) loop model. Section 6 presents a detailed analysis of the rigid loop model. We first show in section 6.1 how to transform the fixed point condition into an equation for the resolvent of the model. This equation is then solved, first along a line of non-generic critical points in section 6.2, and then in all generality in section 6.3. The generic critical line is discussed in section 6.4 and we summarize the resulting phase diagram in section 6.5. Section 7 considers extensions of our results to other classes of O(n) loop models. We first show how relation (1.6) may be modified by considering loops with non-symmetric weights (section 7.1) or with additional constraints on their length (section 7.2). We end our study with a discussion of the wide class of O(n) loop models defined on arbitrary even-valent maps (with bounded degrees) and show that our results nicely extend to this case. We gather our conclusions in section 8.

Our model of interest is the O(n) loop model on planar tetravalent maps defined as follows. By loop we mean an undirected simple closed path on the map (visiting edges and vertices), also sometimes called an undirected cycle. A loop configuration is a set of disjoint loops. By construction, loops are both self-avoiding and mutually avoiding. Alternatively, a loop configuration may be viewed as a set of covered edges such that each vertex is incident to either 0 or 2 covered edges.

We prefer to work with the dual map, which is a planar quadrangulation. Then, the loops cross some edges of the quadrangulations. Up to a rotation, the quadrangulation has three possible types of faces, as displayed in figure 1: (a) empty squares not visited by a loop, (b) squares visited by a loop going straight (i.e. crossing opposite sides of the square) and (c) squares visited by a loop making a turn (i.e. crossing consecutive sides of the square). The length of a loop, i.e. the total number of faces that it visits, may be even or odd. However, the number of faces of type (b) visited by a given loop is necessarily even, resulting from the fact that a planar quadrangulation is bipartite (other interesting consequences of the bipartite nature of our maps will be shown below). A loop is said to be rigid if it only visits faces of type (b).

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Generally speaking, the O(n) loop model consists in attaching a non-local weight n to each loop, in addition to some local weights. Here, we consider an 'annealed' model, where the map and the loops are drawn at random altogether: a configuration of the model is the data of a planar tetravalent map endowed with a loop configuration. Local weights are attached to the vertices of the tetravalent map, or equivalently to the faces of the dual quadrangulation, and naturally, these weights depend on the face type; see again figure 1: g per square of type (a), h₁ per square of type (b), h₂ per square of type (c). The global weight of a configuration is the product of all loop and face weights, and the generating (partition) function of the model is the sum of the global weights over all configurations. Here, n, g, h₁ and h₂ are taken as non-negative real numbers. The model is said to be well defined when the generating function is finite. (This is the case for instance when g + max (n, 1)(2h₁ + 4h₂) ⩽ 1/12.) It is then possible to normalize the configuration weights and define a probability distribution. The rigid loop model corresponds to taking h₂ = 0 so that loops are obliged to go straight within each visited square.

As usual it is easier to work with rooted maps. Recall that a rooted map has a distinguished oriented edge (the root edge). From now on, the face on the right of the root edge will be called the external face and we define a quadrangulation with a boundary of length 2p (p ⩾ 1) as a rooted bipartite planar map, where the external face has degree 2p and every other face has degree 4. We shall now consider loop configurations on the dual of a quadrangulation with a boundary (see figure 2), and for simplicity, we assume that no loop visits the external face, to which we therefore decide to attach a local weight 1. Keeping the same weights n, g, h₁ and h₂ as above, we define F^loop_p(n; g, h₁, h₂) as the generating function for the O(n) loop model on duals of quadrangulations with a boundary of length 2p. As particular cases, for p = 1, we obtain (upon squeezing the bivalent external face) rooted quadrangulations, where the root edge is not crossed by a loop, while for p = 2, we obtain (directly) rooted quadrangulations, where the external face is not visited.

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The fundamental observation of this paper is that F^loop_p(n; g, h₁, h₂) is related to the generating function F_p of bipartite planar maps without loops with a boundary of length 2p, defined as follows. Let us recall that a planar map is bipartite if and only if all its faces have even degree. We say that it has a boundary of length 2p if it is rooted and the external face, defined again as the face on the right of the root edge, has degree 2p. To each internal face of degree 2k (k ⩾ 1), we attach a weight g_k, and the weight of a rooted map is the product of all its internal face weights. We then define F_p as the sum of the weights of all bipartite maps with a boundary of length 2p. Our main statement is that

$$\begin{equation} F_p^{\rm loop}(n;g,h_1,h_2)=F_p(g_1,g_2,\ldots ) \end{equation} \tag{ 2.1 }$$

for the particular sequence (g_k)_{k ⩾ 1} of face weights satisfying the fixed point condition

$$\begin{equation} g_k= g \delta _{k,2}+n \sum _{k^{\prime }\ge 0} A_{k,k^{\prime }}(h_1,h_2) F_{k^{\prime }}(g_1,g_2,\ldots ),\qquad k\ge 1 , \end{equation} \tag{ 2.2 }$$

with $A_{k,k^{\prime }}(h_1,h_2)$ being a polynomial in h₁ and h₂ whose expression will be given in the next section. For the rigid case, it reads simply $A_{k,k^{\prime }}(h_1,0)=h_1^{2k}\delta _{k,k^{\prime }}$, leading to the simpler fixed point condition

$$\begin{equation} g_k= g \delta _{k,2}+n h_1^{2k} F_{k}(g_1,g_2,\ldots ). \end{equation} \tag{ 2.3 }$$

Let us now justify our fundamental observation, by considering a quadrangulation with a boundary of length 2p endowed with a loop configuration. For conciseness, we will often, in this section, call simply 'quadrangulation' a quadrangulation with a boundary. Thanks to the external face, the notions of exterior and interior of a loop are well defined. To each loop, we may associate its outer (resp. inner) contour formed by the edges of the quadrangulation, which (i) belong to the exterior (resp. interior) of the loop and (ii) are incident to squares visited by the loop (see figure 3). Note that these contours are closed paths living on the quadrangulation itself, and therefore, each of them has an even length. Note also that the inner contour may be empty, i.e. may have length 0 if the loop encircles a single vertex.

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Using a terminology similar to [13], we call gasket the map formed by edges that are exterior to all the loops; see figure 4 for an illustration. It is a rooted bipartite planar map containing in particular all the edges incident to the external face, which therefore remains of degree 2p. Its other faces are of two types: regular faces that are the squares of the original quadrangulation that lay outside of every loop, and holes delimited by the outer contours of the former outermost loops.

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Clearly, the transformation that goes from the original quadrangulation with loops to the gasket is not reversible. To make it reversible, we must keep track of the former content of the holes. In this respect, a first remark is that the interior of a loop may itself be viewed as a quadrangulation with a boundary endowed with a loop configuration, where the boundary is nothing but the inner contour of the loop at hand (when the inner contour has length 0, we have the so-called vertex map reduced to one vertex and one face, with no edge). In order not to lose any information, we must also consider the squares covered by the loop itself, which form a ring lying in between the inner and the outer contours. As shown in figure 3, this ring is a cyclic sequence of squares glued together. To summarize, the content of a hole of degree 2k (k ⩾ 1) is described by a pair formed by a ring with the outer length 2k and inner length 2k' (for some k' ⩾ 0), and an internal quadrangulation with a boundary of length 2k' (equal to the vertex map if k' = 0) endowed with a loop configuration.

Two important remarks are in order. First, to ensure bijectivity, both the ring and the internal quadrangulation must be rooted, respectively, on its outer contour and on its boundary. The positions of their root edges is inherited from the rooting of the original quadrangulation via a somewhat irrelevant yet well-defined procedure: for instance, we may consider the leftmost shortest path that stays within the gasket, starts with the root edge and ends with an edge of the outer contour, the edge being selected as the ring root; then, using only the data of the ring, we may easily select an edge of the inner contour, which serves as root for the internal quadrangulation (see figure 5). Second, an essential assumption is that the original map, the gasket and the internal quadrangulations are all possibly separable (i.e. may contain separating vertices whose removal disconnects them). In particular, a separating vertex incident to a hole in the gasket appears whenever a multiple point is present along the outer contour of the associated loop on the original quadrangulation. Similarly, a multiple point of the inner contour of a loop results into a separating vertex on the boundary of the associated internal quadrangulation. In contrast, all the vertices of a ring are considered as distinct (since the information about contacts is recorded in the gasket and internal quadrangulation). At this stage, it should be clear that the decomposition is reversible: given a ring and a quadrangulation with a boundary with compatible lengths, there is a well-defined procedure to join them and fill a hole of the gasket.

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Globally, we have a one-to-one correspondence between, on the one hand, quadrangulations with a boundary endowed with a loop configuration and, on the other hand, gaskets endowed with hole contents. Translating this correspondence in the language of generating functions, we deduce that the wanted quantity F^loop_p(n; g, h₁, h₂) is equal to the generating function F_p for gaskets, where each regular face still receives a weight g and each hole of degree 2k receives a weight $n \sum A_{k,k^{\prime }}(h_1,h_2)F_{k^{\prime }}^{\rm loop}(n;g,h_1,h_2)$ (see figure 6). In this hole weight, n accounts for the associated loop, $F_{k^{\prime }}^{\rm loop}(n;g,h_1,h_2)$ for the internal quadrangulation, while $A_{k,k^{\prime }}(h_1,h_2)$ is the generating function for rings with sides of lengths 2k and 2k'. This establishes equations (2.1) and (2.2).

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Let us now discuss the precise form of the ring generating function $A_{k,k^{\prime }}(h_1,h_2)$. We have the explicit expression

$$\begin{equation} A_{k,k^{\prime }}(h_1,h_2)=\sum _{j=0}^{\min (k,k^{\prime })} {2k\over k+k^{\prime }}{(k+k^{\prime })!\over (2j)!(k-j)!(k^{\prime }-j)!} h_1^{2j} h_2^{k+k^{\prime }-2j} , \end{equation} \tag{ 2.4 }$$

which follows from a simple counting argument: the term with index j in the sum corresponds to rings with 2j squares of type (b), each weighted by h₁. Such rings have k − j (resp. k' − j) squares of type (c) with their two unvisited sides along the outer (resp. inner) contour of the ring, all having a weight h₂. There are ${k+k^{\prime }\choose 2j,k-j,k^{\prime }-j}$ possible sequences of such three types of squares. However, this number has to be corrected because a ring has a distinguished edge (among 2k) on its outer contour rather than a distinguished square (among k + k') to start the sequence. This explains the corrective factor 2k/(k + k').

We have the alternative expression, which will prove useful in the following:

$$\begin{eqnarray} &&\fl \sum _{k^{\prime }\ge 0} A_{k,k^{\prime }}(h_1,h_2) z^{-k^{\prime }} = (\lambda _+(z))^k + (\lambda _-(z))^k, \qquad \lambda _\pm (z) = z\left({h_1 \pm \sqrt{h_1^2 - 4 h_2^2 + 4 h_2 z} \over 2 (z-h_2)}\right)^2 \nonumber\\ \end{eqnarray} \tag{ 2.5 }$$

(note that the sum converges for |z| > h₂). This expression naturally follows from a transfer matrix argument. As seen in figure 7, we may distinguish two 'states' for the edges of the outer contour depending on whether they follow clockwise a vertex adjacent to a vertex of the inner contour (state 1) or not (state 2). We may then write

$$\begin{equation} \fl \sum _{k^{\prime }\ge 0} A_{k,k^{\prime }}(h_1,h_2) z^{-k^{\prime }} = \mathop {\mathrm{tr}}\nolimits M(z)^{2k}, \qquad M(z) = \left(\begin{array}{@{}cc@{}} \displaystyle {h_1 z^{-1/2} \over 1-h_2 z^{-1}}& \displaystyle {{1 \over 1-h_2 z^{-1}}} \\\ms h_2 & 0 \end{array}\right) , \end{equation} \tag{ 2.6 }$$

where M(z) is the transfer matrix describing the transition between two successive edge states. This establishes (2.5), where λ_±(z) are simply the eigenvalues of M²(z), solutions of

$$\begin{equation} \left( h_2 (\lambda +z) - \lambda z \right)^2 - h_1^2 \lambda z =0 . \end{equation} \tag{ 2.7 }$$

Interestingly, λ and z play a symmetric role in this equation. It follows that λ(λ(z)) = z provided we pick the correct determination of λ. More precisely, it may be checked that z↦λ₊(z) is an involution on (h₂, ∞) with a unique fixed point:

$$\begin{equation} z^*=h_1+2h_2=\lambda _+(z^*). \end{equation} \tag{ 2.8 }$$

Finally, we observe that for rigid loops, there is a unique ring with the outer length 2k, and it has the same inner length. This yields immediately $A_{k,k^{\prime }}(h_1,0)=h_1^{2k}\delta _{k,k^{\prime }}$ as announced in the previous section. This is compatible with (2.4) for h₂ = 0, and with (2.5) upon noting that λ₊(z) = h²₁/z and λ₋(z) = 0.

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As already mentioned, a planar map is bipartite iff all its faces have even degree. To each face of degree 2k (k ⩾ 1), we attach a weight g_k, here considered as an arbitrary non-negative real number. Furthermore, we will sometimes find convenient to also attach a non-negative real weight u per vertex of the map, irrespectively of its degree. The unnormalized weight of a map is then the product of all its face and vertex weights. Note that, from Euler's relation, we could set u = 1 without loss of generality upon redefining g_k → g_ku^{k − 1}.

It is well known that the simplest map enumeration formulas are for pointed rooted maps, i.e. maps with a distinguished vertex (pointed) and a distinguished oriented edge (rooted). The generating function (i.e. the sum over all maps of the unnormalized weights above) for pointed rooted bipartite planar maps is 2u(R(u) − u), where R(u) is the smallest non-negative (possibly infinite) solution of the equation [15]:

$$\begin{equation} R(u)=u+\sum _{k\ge 1} g_k {2k-1 \choose k} R(u)^k . \end{equation} \tag{ 3.1 }$$

Here, we choose to explicitly display the dependence in u but to hide that in g_k. It is easily seen that R(u) is an increasing function of u with R(0) = 0.

Other more involved generating functions can be expressed in terms of R(u). For instance, the generating function F_k for maps with a boundary of length 2k, as defined in the previous section (with u = 1), reads [16]

$$\begin{equation} F_k={2k \choose k} \int _0^1 R(u)^k {\rm d}u . \end{equation} \tag{ 3.2 }$$

Consistently, we set F₀ = 1. Furthermore, since rooted maps are obviously in bijection with maps with a boundary of length 2 and not reduced to a single edge, we may interpret F₁ − 1 as the generating function for rooted maps. Upon integrating (3.1) and noting that ${2k \choose k}=2{2k-1 \choose k}$, we immediately deduce

$$\begin{equation} F_1-1= \sum _{k\ge 1} g_k F_k \end{equation} \tag{ 3.3 }$$

which amounts to decomposing the generating function of rooted maps (lhs) according to the root degree of the maps (rhs).

We say that the sequence (g_k)_{k ⩾ 1} of face weights is admissible if R(1) is finite. In this case, taking u = 1 and dividing the unnormalized weight of a pointed rooted map by 2(R(1) − 1), we obtain its probability in the Boltzmann ensemble of pointed rooted maps associated with the sequence (g_k)_{k ⩾ 1}. For example, random quadrangulations are obtained by choosing the sequence g_k = g δ_{k, 2} (g > 0), which is admissible for g ⩽ 1/12.

A number of properties of the Boltzmann ensemble are encoded in the expansion of R(u) around u = 1. This expansion is obtained upon inverting the expansion around R = R(1) of the inverse function u(R) that, from (3.1), reads explicitly

$$\begin{equation} u(R)= R-\varphi (R), \quad \varphi (R)=\sum _{k\ge 1} g_k {2k-1 \choose k} R^k . \end{equation} \tag{ 3.4 }$$

Here, the function φ(R) is defined through its series expansion at R = 0, whose coefficients are all non-negative. Its radius of convergence is $R_c=1/(4 \lim \sup _{k\rightarrow \infty } (g_k)^{1/k})$ since ${2k \choose k}\sim 4^k/\sqrt{\pi k}$ at large k. If the sequence is admissible, R_c is necessarily non-zero (possibly infinite), with R(1) ⩽ R_c, and it is easily seen that u'(R(1)) ⩾ 0, i.e. φ'(R(1)) ⩽ 1. The sequence is said to be critical if φ'(R(1)) = 1 (hence u'(R(1)) = 0 and R'(u) → ∞ as u → 1) and subcritical otherwise (hence R'(1) is finite). This change of behavior is visible in the large k asymptotics of F_k. Indeed, the integral (3.2) is asymptotically dominated by the vicinity of u = 1 and its behavior may be computed via Laplace's method [17]. For a subcritical sequence, we have at large k

$$\begin{equation} F_k \sim {R(1)(1-\varphi ^{\prime }(R(1))\over \sqrt{\pi }}{(4 R(1))^k \over k^{3/2}} . \end{equation} \tag{ 3.5 }$$

This form of the asymptotics is observed, for instance, when all g_k are zero, in which case φ(R) = 0, R(u) = u, and $F_k= {2k \choose k}/(k+1)$ is the kth Catalan number.

For a critical sequence, the asymptotic behavior depends on whether the (negative) quantity u''(R(1)) = −φ''(R(1)) is finite or not. For a generic critical sequence, this quantity is finite and we have that

$$\begin{equation} F_k \sim {R(1)^2\varphi ^{\prime \prime }(R(1))\over \sqrt{\pi }} {(4 R(1))^k \over k^{5/2}} . \end{equation} \tag{ 3.6 }$$

This form of the asymptotics is observed for instance for critical random quadrangulations obtained by choosing the sequence g_k = (1/12)δ_{k, 2}, in which case φ(R) = R²/4, R(1) = 2 and φ''(R(1)) = 1/2, so that (3.6) is consistent with the exact formula $F_k= 2^{k+1} {(2k)!\over k!(k+2)!}$.

In order to escape from this generic critical behavior, we must have φ''(R(1)) infinite, which implies that R(1) must coincide with the radius of convergence R_c of φ. Such non-generic critical behaviors are analyzed in the next section. Before proceeding, let us mention the following simple lemma.

• Monotonicity property: If (g_k)_{k ⩾ 1} and $(\tilde{g}_k)_{k \ge 1}$ are two sequences, such that $g_k \le \tilde{g}_k$ for all k, the inequality being strict for at least one k, and if $(\tilde{g}_k)_{k \ge 1}$ is admissible, then (g_k)_{k ⩾ 1} is subcritical.

Indeed, denoting by $\tilde{R}(\cdot )$ and $\tilde{\varphi }(\cdot )$ the functions associated with the sequence $(\tilde{g}_k)_{k \ge 1}$, we have $R(1) < \tilde{R}(1)$ since 2R(1) − 1, being a map generating function, is a strictly increasing function of g_k for each k. Then, $\varphi ^{\prime }(R(1)) < \tilde{\varphi }^{\prime }(\tilde{R}(1)) \le 1$; hence, by definition, (g_k)_{k ⩾ 1} is subcritical.

In order to obtain a non-generic critical behavior, we must ensure simultaneously the two conditions φ'(R(1)) = 1 (criticality) and φ''(R(1)) = ∞ (non-genericity). This turns out to highly constrain the possible form of the sequence (g_k)_{k ⩾ 1}. As shown by Le Gall and Miermont [13], a natural candidate for such a sequence is

$$\begin{equation} g_k=c \left({1\over 4 R_c}\right)^{k-1} g_k^\circ \end{equation} \tag{ 3.7 }$$

where (g°_k)_{k ⩾ 1} is an arbitrary sequence of reference, such that

$$\begin{equation} g_k^\circ \sim k^{-a} \qquad \hbox{for}\quad k\rightarrow \infty \qquad \hbox{with}\quad \case{3}{2}<a<\case{5}{2} . \end{equation} \tag{ 3.8 }$$

Note that R_c is indeed the radius of convergence of φ(R) since we have φ(R) = cRf_⚬(R/(4R_c)), where

$$\begin{equation} f_\circ (x)=\sum _{k\ge 1} {2k-1 \choose k} g_k^\circ x^{k-1} \end{equation} \tag{ 3.9 }$$

is an analytic function with the radius of convergence 1/4. Since a lies in the range ]3/2, 5/2[, we have φ'(R_c) < ∞ and φ''(R_c) = ∞. The non-genericity condition, therefore, reduces to demanding that

$$\begin{equation} R(1)=R_c , \end{equation} \tag{ 3.10 }$$

or equivalently 1 = R_c − φ(R_c). Together with the criticality condition φ'(R_c) = 1, this fixes the values of c and R_c as

$$\begin{equation} c={4\over 4 f_\circ (1/4)+f_\circ ^{\prime }(1/4)}\qquad {\rm and} \qquad R_c=1+{4 f_\circ (1/4)\over f_\circ ^{\prime }(1/4)} . \end{equation} \tag{ 3.11 }$$

At these values, by standard transfer theorems [18], u(R) has a singular expansion around R = R_c given by

$$\begin{equation} u(R)=1-{2 c R_c \Gamma (1/2-a) \over \sqrt{\pi }} \left(1-{R\over R_c}\right)^{a-1/2}+O\left(1-{R \over R_c} \right)^2. \end{equation} \tag{ 3.12 }$$

Note that Γ(1/2 − a) > 0 since 3/2 < a < 5/2. Inverting this expansion, we see that the generating function R(u) behaves, when we fix the face weights g_k as above and let the vertex weight u tend to 1 from below (i.e. we use the parameter u to control the approach to the critical point), as

$$\begin{eqnarray} &&\fl R(u)=R_c(1-\kappa (1-u)^{2/(2a-1)})+O(1-u), \qquad \kappa =\left({\sqrt{\pi }\over 2 c R_c \Gamma (1/2-a)} \right)^{2/(2a-1)}.\nonumber\\ \end{eqnarray} \tag{ 3.13 }$$

Again, by transfer theorems, this implies that the coefficient of u^N in R(u), and hence the probability of having N vertices in the Boltzmann ensemble of pointed rooted maps, decays as N^{−(2a + 1)/(2a − 1)} as N → ∞.

Returning to the asymptotics of F_k, we find by Laplace's method that, for large k,

$$\begin{equation} \int _0^1 R(u)^k\, {\rm d}u\sim {2 c R_c (a-1/2)\Gamma (1/2-a)\Gamma (a-1/2) \over \sqrt{\pi }}{R_c^{k}\over k^{a-1/2}} \end{equation} \tag{ 3.14 }$$

and, using ${2k \choose k}\sim 4^k/\sqrt{\pi k}$, (a − 1/2)Γ(1/2 − a)Γ(a − 1/2) = π/sin π(a − 3/2) and R_c = R(1),

$$\begin{equation} F_k\sim {2 c R(1) \over \sin \pi (a-3/2)} {(4 R(1))^k\over k^a}. \end{equation} \tag{ 3.15 }$$

It is instructive to compare this expression with the 'input', namely the form (3.7)–(3.8) for non-generic critical face weights, which implies the asymptotics

$$\begin{equation} g_k\sim 4 c R(1) {(4 R(1))^{-k}\over k^a} . \end{equation} \tag{ 3.16 }$$

We note a striking similarity: the prefactor is simply divided by 2sin π(a − 3/2), while the exponential factor is inverted. This similarity is not merely anecdotal but will have important consequences for the O(n) loop model (see section 4).

As a concluding remark, note that the condition R(1) = R_c for non-generic criticality implies that the face degree distribution has a fat tail. As seen from the derivation [15] of (3.1), the probability that the external face in a rooted pointed map has a degree k is

$$\begin{equation} {g_k {2k-1 \choose k} R(1)^k \over R(1)-1 }, \end{equation} \tag{ 3.17 }$$

which decays exponentially as (R(1)/R_c)^k, when R(1) < R_c, but only as a power law k^{−a − 1/2} at a non-generic critical point. Similarly, from (3.3), the probability that the external face in a rooted (but unpointed) map has a degree k is

$$\begin{equation} {g_k F_k \over F_1-1 } \end{equation} \tag{ 3.18 }$$

which also decays exponentially as (R(1)/R_c)^k, when R(1) < R_c, and as a power law, now as k^−2a, at a non-generic critical point. Note finally that the condition R(1) = R_c is necessary but not sufficient for non-generic criticality as one may construct examples of subcritical and generic critical sequences such that R(1) = R_c. Such situations will however not appear in the context of the O(n) model with non-negative weights.

For the purposes of section 6, it is useful to gather facts about the so-called resolvent (this terminology being borrowed from the matrix integral formalism) defined as

$$\begin{equation} W(\xi ) = \sum _{k \ge 0} {F_k \over \xi ^{2k+1}}. \end{equation} \tag{ 3.19 }$$

Using (3.2) and performing the change of variable u → R, we obtain

$$\begin{equation} W(\xi ) = {1 \over \xi } \int _0^{R(1)} {u^{\prime }(R) \,{\rm d}R \over \sqrt{1-4R/\xi ^2}}. \end{equation} \tag{ 3.20 }$$

From this expression, it is seen that W(ξ) is analytic in the complex plane minus the segment [ − γ, γ], with $\gamma =2\sqrt{R(1)}$. Along this segment, it has a discontinuity encoded into the so-called spectral density

$$\begin{equation} \fl \rho (\xi )={W(\xi -{\rm i}0) - W(\xi +{\rm i}0) \over 2{\rm i}\pi } = {1 \over \pi } \int _{\xi ^2/4}^{R(1)} {u^{\prime }(R) \,{\rm d}R \over \sqrt{4R-\xi ^2}} , \qquad \xi \in [-\gamma ,\gamma ]. \end{equation} \tag{ 3.21 }$$

Since u'(R) ⩾ 0 for R ⩽ R(1), ρ(ξ) is nonnegative on [ − γ, γ]. Note that, by applying the Cauchy formula around the cut, we may recover W(ξ) from ρ(ξ) via

$$\begin{equation} W(\xi )=\int _{-\gamma }^{\gamma } {\rho (\xi ^{\prime }) \,{\rm d}\xi ^{\prime } \over \xi - \xi ^{\prime }} ,\qquad \xi \notin [-\gamma ,\gamma ], \end{equation} \tag{ 3.22 }$$

and in particular, since W(ξ) ∼ 1/ξ for ξ → ∞, we have the normalization ∫^γ_{− γ}ρ(ξ) dξ = 1. Asymptotics (3.5), (3.6) and (3.15) translate into the respective singularities of ρ(ξ) at $\xi =\pm \gamma =\pm 2\sqrt{R(1)}$:

$$\begin{eqnarray} &&\rho (\xi ) \sim {1 - \varphi ^{\prime }(R(1)) \over 2 \pi } (4 R(1) -\xi ^2)^{1/2} \quad \hbox{subcritical} \\&&\ms \rho (\xi ) \sim {\varphi ^{\prime \prime }(R(1)) \over 12 \pi } (4 R(1) -\xi ^2)^{3/2} \quad \hbox{generic critical}\\&&\ms \rho (\xi ) \sim {4^{1-a} R(1)^{3/2-a} c \over \Gamma (a) \sin \pi (a-3/2)}(4 R(1) - \xi ^2)^{a-1} \quad \hbox{non-generic critical} . \end{eqnarray} \tag{ 3.23 }$$

For simplicity, we first concentrate on the rigid case. In this section, (g_k)_{k ⩾ 1} denotes a sequence satisfying the fixed point condition (2.3). It immediately implies the asymptotic condition

$$\begin{equation} g_k \sim n h_1^{2k} F_k \qquad (k \rightarrow \infty ). \end{equation} \tag{ 4.1 }$$

In section 3, we have classified the possible asymptotic behaviors of F_k depending on the nature of the weight sequence (g_k)_{k ⩾ 1}: equations (3.5), (3.6) and (3.15) describe, respectively, the subcritical, generic critical and non-generic critical cases. Here, we assume that, in the non-generic critical case, the weight sequence is of the form (3.7)–(3.8). It is shown in section 6 that no other more exotic non-generic behavior exists.

We now remark that equation (3.2) implies that (F_k)^1/k → 4R(1) for large k, regardless of the nature of the weight sequence. By comparing with the general relation $\lim \sup _{k\rightarrow \infty } (g_k)^{1/k} = 1/(4 R_c)$, where R_c is the radius of convergence of the function φ(R) defined in equation (3.4), equation (4.1) implies the relation

$$\begin{equation} 16 h_1^2 R_c R(1) = 1. \end{equation} \tag{ 4.2 }$$

For later use, we introduce the parameter

$$\begin{equation} \tau = 4 h_1 R(1) . \end{equation} \tag{ 4.3 }$$

Since R(1) ⩽ R_c, we have from (4.2) that τ ⩽ 1 and the condition τ = 1 amounts to R(1) = R_c. We now prove the following.

• Non-genericity criterion: R(1) = R_c (i.e. τ = 1) iff the sequence (g_k)_{k ⩾ 1} is critical and non-generic.

The first implication ('if') was shown in section 3 without recourse to the fixed point condition (see equation (3.10)). Conversely, assume that the sequence (g_k)_{k ⩾ 1} is either subcritical or generic critical. Its asymptotic behavior is directly deduced from that of F_k via equation (4.1). In the subcritical case, due to the k^−3/2 factor in equation (3.5), φ'(R_c) is infinite, while, by definition of subcriticality, we have φ'(R(1)) < 1, and hence R(1) ≠ R_c. In the generic critical case, due to the k^−5/2 factor in equation (3.6), φ''(R_c) is infinite, while, by definition of genericity, we have φ''(R(1)) < ∞, and hence R(1) ≠ R_c. This completes the proof. (Note that when we do not impose the fixed point condition, we can easily construct examples of subcritical and generic critical sequences such that R(1) = R_c.) In view of the discussion at the end of section 3.3, we deduce that the face degree distribution of the gasket, i.e. the outer loop length distribution in the O(n) model, has a fat tail iff the model is at a non-generic critical point.

If we now assume that the sequence (g_k)_{k ⩾ 1} is critical and non-generic and takes the general form (3.7) for a suitable sequence (g°_k)_{k ⩾ 1} satisfying (3.8), then, in addition to the condition τ = 1, we see by comparing the prefactors of (3.15) and (3.16) that

$$\begin{equation} n=2 \sin \pi (a-3/2) . \end{equation} \tag{ 4.4 }$$

In particular, n must be in the range ]0, 2[ for non-generic criticality to be possible.

To summarize, we have shown that the fixed point condition (2.3) is compatible with the model being either subcritical, generic critical, with τ < 1 or, more interestingly, non-generic critical with τ = 1 and a = 2 ± b, $\pi b = \arccos (n/2)$.

So far, we did not tell which behavior corresponds to a given value of the 'physical' parameters n, g and h₁. Since the generating function F^loop_p(n; g, h₁, 0) is an increasing function of its parameters, there exists a decreasing function h_c(n; g) such that the model is well defined for h₁ < h_c(n; g) and ill defined for h₁ > h_c(n; g). In particular, h_c(n; g) > 0 if g < 1/12. By the monotonicity property of section 3.2, we see that the model is subcritical whenever h₁ < h_c(n; g). In particular, any critical point necessarily lies on the line h₁ = h_c(n; g). An exact expression for h_c(n, g) will be given in section 6 for 0 < n < 2. There we find that the model is critical all along the line h₁ = h_c(n; g). Furthermore, along this line, there exists g* such that (i) for g* < g ⩽ 1/12, the model is generic critical, (ii) for 0 ⩽ g < g*, the model is non-generic critical with a = 2 − b (dense model) and (iii) at g = g*, the model is non-generic critical with a = 2 + b (dilute model). The corresponding phase diagram is displayed in figure 12.

Let us now consider the general case of arbitrary non-negative values of h₁ and h₂ and analyze the consequences of the fixed point condition (2.2) on the asymptotics of g_k. To treat the subcritical, generic critical and non-generic critical cases simultaneously, we write the asymptotics of F_k in the form

$$\begin{equation} F_k\sim \chi {(z_c)^{-k}\over k^a},\qquad z_c={1\over 4R(1)} , \end{equation} \tag{ 4.5 }$$

with now 3/2 ⩽ a ⩽ 5/2 and where χ may be read off equations (3.5), (3.6) and (3.15), respectively. Here again, we assume that, in the non-generic critical case, the weight sequence is of the form (3.7)–(3.8).

Let us introduce the generating function

$$\begin{equation} F(z)=\sum _{k\ge 0} F_k z^k\ \end{equation} \tag{ 4.6 }$$

related to the resolvent via W(ξ) = F(1/ξ²)/ξ. By transfer theorems, the function F(z) is singular, when z → z_c, with a singular part given by

$$\begin{equation} F(z)\vert _{\rm sing}\sim \chi \Gamma (1-a) \left(1- {z\over z_c}\right)^{a-1}, \end{equation} \tag{ 4.7 }$$

where we temporarily exclude the case a = 2.

To estimate the rhs of (2.2), we use equation (2.5) to write

$$\begin{equation} \sum _{k^{\prime }\ge 0} A_{k,k^{\prime }}(h_1,h_2) F_{k^{\prime }} = \oint {{\rm d}z\over 2 {\rm i} \pi z} F(z) \left(\lambda _+(z)^k+\lambda _-(z)^k\right), \end{equation} \tag{ 4.8 }$$

where me may choose as an integration contour the circle |z| = z₀ for any h₂ < z₀ < z_c. (Note that, since $A_{k,k^{\prime }}(h_1,h_2)\ge A_{k,k^{\prime }}(0,h_2)= 2 {k+k^{\prime }-1 \choose k-1} h_2^{k+k^{\prime }}$, z_c has to be larger than h₂ in order for the lhs of (4.8) to be finite, and hence the model to be well defined.) At large k, this quantity may be evaluated via the method of Hankel contours. Indeed, the contour may be deformed as shown in figure 8 so that the dominant contribution to the integral comes from the vicinity of z_c. Using the scaling

$$\begin{equation} z=z_c\left(1+{\delta \over k}\right) , \end{equation} \tag{ 4.9 }$$

the variable δ is now to be integrated back and forth from ∞ to 0 (below and above the cut of F(z)). Using the expansion

$$\begin{equation} \lambda _+(z)=\lambda _+(z_c) \left(1-\mu _+ {\delta \over k}+O\left({1\over k^2}\right)\right) \end{equation} \tag{ 4.10 }$$

with

$$\begin{equation} \mu _+= - \left.\left(z {{\rm d} \ \over {\rm d}z} \log \lambda _+(z) \right)\right|_{z=z_c} , \end{equation} \tag{ 4.11 }$$

we obtain at large k

$$\begin{eqnarray} &&\sum _{k^{\prime }\ge 0}A_{k,k^{\prime }}(h_1,h_2) F_{k^{\prime }} \sim \chi \Gamma (1-a) {\lambda _+(z_c)^k \over k^a} {\sin \pi (1-a)\over \pi } \int _0^{\infty } {\rm d}\delta \,\delta ^{a-1}\, {\rm e}^{-\mu _{+}\delta } \nonumber\\ &&\hphantom{\sum _{k^{\prime }\ge 0}A_{k,k^{\prime }}(h_1,h_2) F_{k^{\prime }}}= \chi {\lambda _+(z_c)^k \over (\mu _{+})^a k^a} . \end{eqnarray} \tag{ 4.12 }$$

Here, we assumed that λ₊(z_c) > λ₋(z_c), which is valid as soon as h₁ > 0. The case h₁ = 0 will be discussed later. From the fixed point condition, this behavior is transferred into the asymptotics of g_k:

$$\begin{equation} g_k\sim n \chi {\lambda _+(z_c)^k \over (\mu _{+})^a k^a} . \end{equation} \tag{ 4.13 }$$

By a slight modification of our reasoning, one may easily show that this asymptotics also holds in the case a = 2. Using again the property that (g_k)^1/k → 1/(4R_c) for k → ∞, this yields a first consistency relation

$$\begin{equation} {1\over 4 R_c}=\lambda _+\left({1\over 4 R(1)}\right) . \end{equation} \tag{ 4.14 }$$

This relation generalizes (4.2), which is recovered by noting that λ₊(z) = h²₁/z in the rigid case.

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Repeating the proof of the previous section, we have again R(1) ⩽ R_c and R(1) = R_c iff the sequence (g_k)_{k ⩾ 1} is non-generic critical. Focusing on such a non-generic critical solution, relation (4.14) allows one to identify 1/(4R(1)) as the fixed point z* of the mapping z↦λ₊(z), namely

$$\begin{equation} {1\over 4 R(1)}=z^*= h_1+2h_2 . \end{equation} \tag{ 4.15 }$$

A second consistency relation is now obtained by comparing the prefactors in (4.13) and (3.16) and using the precise value χ = 2cR(1)/sin π(a − 3/2). We eventually deduce the relation

$$\begin{equation} \fl n= 2 (\mu ^*)^a \sin \pi (a-3/2), \qquad \mu ^*=-\left.\left(z {{\rm d} \ \over {\rm d}z} \log \lambda _+(z) \right)\right|_{z=z^*}=-\lambda _+^{\prime }(z^*). \end{equation} \tag{ 4.16 }$$

Now we recall that z↦λ₊(z) is an involution in the vicinity of its fixed point z*, which implies that λ'₊(z*) = −1 (since λ₊(z) obviously decreases with z), a result which may trivially be checked by a direct calculation. This fixes μ* = 1 and we recover the same relation (4.4) as in the rigid case.

To summarize, the fixed point condition (2.2) is compatible with having the model subcritical, generic critical or non-generic critical with $a=2\pm b,\ \pi b =\arccos (n/2)$ as long as h₁ > 0. The case h₁ = 0 leads to a different relation since in this case λ₊(z) = λ₋(z) = h₂z/(z − h₂), and hence the contribution of λ₋(z) in (4.8) cannot be neglected. Then, equation (4.12) must be replaced by

$$\begin{equation} \sum _{k^{\prime }\ge 0}A_{k,k^{\prime }}(0,h_2) F_{k^{\prime }}\sim 2 \chi {\lambda _+(z_c)^k \over (\mu _{+})^a k^a} . \end{equation} \tag{ 4.17 }$$

At a non-generic point, we obtain the same relation z_c = 1/(4R(1)) = z* as before (with now z* = 2h₂) and the same value μ* = 1, but comparing the prefactors as was done above now yields the new relation

$$\begin{equation} n=\sin \pi (a-3/2) , \qquad h_1=0 \end{equation} \tag{ 4.18 }$$

without the factor 2 in front. We thus expect that our O(n) loop model for h₁ = 0 and 0 < n < 1 should have the same non-generic critical behavior as the O(2n) loop model for h₁ > 0.

We start from the general expression (3.4) for the function u(R), defining implicitly the generating function R(u). Differentiating (3.4) with respect to R, we readily obtain

$$\begin{equation} u^{\prime }(R)=1 - \sum _{k \ge 1} g_k {2k-1 \choose k} k R^{k-1}. \end{equation} \tag{ 5.1 }$$

Independently, we observe that, upon performing the change of variable $u\rightarrow {\tilde{R}}=R(u)$ in (3.2), we have

$$\begin{equation} F_k = {2k \choose k} \int _0^{R(1)} {\tilde{R}}^k u^{\prime }({\tilde{R}})\, {\rm d}{\tilde{R}}. \end{equation} \tag{ 5.2 }$$

So far, we have only rewritten some equations of section 3.2, related to bipartite maps. Let us now combine them with the fixed point condition (2.3) transcribing the gasket decomposition for the O(n) rigid loop model: assuming that n, g and h₁ are such that the model is well defined, we subtitute (5.2) into (2.3), and then the result into (5.1), and obtain a linear integral equation for u'(R):

$$\begin{equation} u^{\prime }(R)=1-6 g R-{n\over 2} \sum _{k\ge 1}k {2k \choose k}^2 h_1^{2k} R^{k-1} \int _0^{R(1)} {\tilde{R}}^k u^{\prime }({\tilde{R}})\, {\rm d}{\tilde{R}}. \end{equation} \tag{ 5.3 }$$

(Note that ${2k-1 \choose k}={2k \choose k}/2.$) By the change of variables

$$\begin{equation} \fl x={R\over R(1)}, \quad y={{\tilde{R}}\over R(1)}, \quad f(x)=u^{\prime }(R), \quad \rho = 6 g R(1), \quad \tau = 4 h_1 R(1) , \end{equation} \tag{ 5.4 }$$

the linear integral equation is rewritten in the more compact form

$$\begin{eqnarray} &&f(x)+{n\over 2 \pi }\int _0^1 K_\tau (x,y) f(y)\, {\rm d}y= 1-\rho x \qquad (0 \le x \le 1) \\&&\ms K_\tau (x,y)= \tau ^2 y \psi (\tau ^2 x y), \qquad \psi (t)= \sum _{k\ge 1} {\pi k\over 16^k} {2k \choose k}^2 t^{k-1}. \end{eqnarray} \tag{ 5.5 }$$

We recognize a Fredholm integral equation of the second kind [19] whose unknown is the function f defined on the range [0, 1], and which depends on the non-negative parameters n, ρ and τ. Note the consistency relation

$$\begin{equation} \int _0^1 f(x)={1\over R(1)}, \end{equation} \tag{ 5.6 }$$

which follows from $1=u(R(1))= \int _0^{R(1)} u^{\prime }({\tilde{R}})\,{\rm d}{\tilde{R}} = R(1)\int _0^1 f(x)\,{\rm d}x$.

Let us now discuss the conditions on the function f and the parameters ρ and τ that arise from our derivation. First, the existence of the inverse function R(u) for u between 0 and 1 implies that u'(R(1)) ⩾ 0 and u'(R) > 0 for 0 ⩽ R < R(1). This immediately translates into the positivity conditions

$$\begin{equation} f(1) \ge 0, \qquad f(x)>0\qquad {\rm for}\quad 0\le x < 1 \end{equation} \tag{ 5.7 }$$

with f(1) = 0 iff the model is critical in the sense of section 3.2. Furthermore, we also obtain that ρ ⩽ 1, since u'(R) ⩽ 1 − 6 g R by (5.3). Second, by the discussion of section 4.1, we have τ ⩽ 1 and the case τ = 1 corresponds to a non-generic critical point. Note that this is precisely the range on which equation (5.5) is well defined. Indeed, since $(\pi k /16^k) {2k \choose k}^2 \rightarrow 1$ for k → ∞, the radius of convergence of ψ(t) is 1, with ψ(t) ∼ 1/(1 − t) for t → 1; hence, K_τ(x, y) is a smooth function of (x, y) ∈ [0, 1]² except for τ = 1, where it has a polar singularity at x = y = 1. The integral equation still holds for x = 1, which implies that f(y) → 0 for y → 1 sufficiently fast.

Conversely, given ρ and τ both between 0 and 1 and such that equation (5.5) admits a solution f satisfying the positivity conditions (5.7), we may return to the original variables as follows. We first compute R(1) via (5.6), then deduce u'(R), g and h₁ via (5.4), and finally obtain u(R) as the primitive of u'(R) satisfying u(0) = 0. Following the steps of the above derivation backward, we find that u(R) satisfies (3.4) with g_k given by (2.3) so that we are indeed 'solving' the O(n) rigid loop model. The positivity conditions ensure that the inverse function R(u) and, thus, the model are well defined. In particular, the generating function F^loop_p(n; g, h₁, 0) = F_p is directly expressed from f as

$$\begin{equation} F_p = {2p \choose p} {\int _0^1 x^p f(x)\, {\rm d}x \over \big( \int _0^1 f(x)\, {\rm d}x \big)^{p+1}} . \end{equation} \tag{ 5.8 }$$

In conclusion, we have shown that generating functions for the O(n) rigid loop model may be obtained by solving the linear integral equation (5.5). In practice, the main difficulty is that the change of parameters (g, h₁) → (ρ, τ) is rather intricate. Nevertheless, if we were able to compute the function f for arbitrary parameters n, ρ and τ, then we could deduce F^loop_p(n; g, h₁, 0) in a parametric form. Before further analyzing the solutions of (5.5), let us extend our formalism to the non-rigid case.

We may repeat the above analysis in the case of arbitrary values of h₁ and h₂, replacing the fixed point condition (2.3) by the more general one (2.2). The main complication is that an extra sum over a variable k' is involved. Equation (4.8) allows one to rewrite this sum as a contour integral. Furthermore, from (5.2), we have

$$\begin{equation} F(z)= \int _0^{R(1)} {u^{\prime }({\tilde{R}})\, {\rm d}{\tilde{R}} \over \sqrt{1 - 4 z {\tilde{R}}}}. \end{equation} \tag{ 5.9 }$$

Substituting these expressions into (2.2), and then into (5.1), we may explicitly evaluate the sum over the variable k and derive a linear integral equation for u'(R). Again a more compact form is obtained after a suitable change of variables: we let x, y, f(x) and ρ to be as given in (5.4), while we now define

$$\begin{equation} \tau =4R(1)(h_1+2h_2). \end{equation} \tag{ 5.10 }$$

After some work, we arrive at the same form for the linear integral equation

$$\begin{equation} f(x)+{n\over 2 \pi }\int _0^1 K_\tau (x,y) f(y)\, {\rm d}y= 1-\rho x \end{equation} \tag{ 5.11 }$$

but with a different kernel K_τ(x, y), now given by

$$\begin{equation} \fl K_\tau (x,y)= \oint {{\rm d}\zeta \over 2 {\rm i}\zeta } {1\over (1-\tau y \zeta )^{1/2}} {1\over 2}\left( { \tau \Lambda _+(\zeta )\over (1-\tau x \Lambda _+(\zeta ))^{3/2}} +{ \tau \Lambda _-(\zeta )\over (1-\tau x \Lambda _-(\zeta ))^{3/2}} \right), \end{equation} \tag{ 5.12 }$$

where we may choose as an integration contour the circle |ζ| = ζ₀ for any Λ⁻¹₊(1/(τx)) < ζ₀ < 1/(τy), and where we introduced the rescaled functions

$$\begin{equation} \Lambda _{\pm }(\zeta )={\lambda _\pm \left(\left(h_1+2h_2\right)\zeta \right) \over h_1+2h_2} . \end{equation} \tag{ 5.13 }$$

Note that Λ₊ and Λ₋ are functions of the ratio h₁/h₂, and therefore, K_τ(x, y) depends implicitly on this ratio. Despite the apparent complication in the expression for the kernel K_τ(x, y), much of the discussion of section 5.1 can be generalized to the non-rigid case. Using exactly the same arguments, we find that:

• the consistency relation (5.6),
• the positivity conditions (5.7), where again f(1) = 0 iff the model is critical,
• the inequality ρ ⩽ 1,
• expression (5.8) for the O(n) loop model generating functions F^loop_p(n; g, h₁, h₂) = F_p,

all still hold in the non-rigid case. The discussion of τ, in view of that of section 4.2 and of the general expression (5.10), becomes slightly more involved. We nevertheless find that τ is still between 0 and 1, with τ = 1 corresponding to a non-generic critical point. Furthermore, K_τ(x, y) is a smooth function of (x, y) ∈ [0, 1]² except in the case τ = 1, where, remarkably, we obtain the same singular behavior as in the rigid case, provided that h₁ > 0. More precisely, when x and y tend to 1 (keeping (1 − x)/(1 − y) finite), we have

$$\begin{equation} K_1(x,y) \sim {1\over 1-xy} . \end{equation} \tag{ 5.14 }$$

For the record, let us briefly explain how this property results from (5.12) by a saddle-point approximation. The contour integral in the latter equation is dominated, for x, y → 1 by the vicinity of ζ = 1, and we set

$$\begin{equation} \zeta =1+{\rm i} \epsilon Z ,\qquad x=1-\epsilon X , \qquad y=1-\epsilon Y \end{equation} \tag{ 5.15 }$$

with → 0. At leading order in , we have

$$\begin{equation} \fl \Lambda _+(\zeta )=1-\epsilon \mu ^* Z\ +\cdots , \qquad {\rm with}\quad \mu ^*=-\Lambda ^{\prime }_{+}(1)=-\lambda ^{\prime }_{+}(z^*)=1, \end{equation} \tag{ 5.16 }$$

using again the 'miraculous' involutivity of λ₊(z) around z*. We finally obtain the estimate

$$\begin{equation} K_1(x,y)\sim {1\over \epsilon } \int _{-\infty }^\infty {{\rm d}Z\over 4} {1\over (Y-{\rm i}Z)^{1/2}(X+{\rm i}Z)^{3/2}}= {1\over \epsilon }{1\over X+Y}\sim {1\over 1-xy} . \end{equation} \tag{ 5.17 }$$

In the case h₁ = 0, this estimate must be doubled since Λ₋ then has an equal, instead of negligible, contribution. We thus have K₁(x, y) ∼ 2/(1 − xy) in this case.

In summary, the generating functions F^loop_p(n; g, h₁, h₂) for the O(n) loop model on tetravalent maps may be expressed, via a change of parameters (g, h₁, h₂) → (ρ, τ, h₁/h₂), in terms of the solution of the linear integral equation (5.11).

As mentioned previously, equation (5.11) is a Fredholm integral equation of the second kind for the function f(x). This equation depends on n, ρ and τ and the ratio h₁/h₂ as follows: −n/(2π) is the so-called parameter of integral equation, ρ appears only on the right-hand side, and τ and h₁/h₂ determine the kernel (5.12). (In particular, for h₁/h₂ = ∞, we recover the rigid case of (5.5).) Here, our terminology is borrowed from [19]. In this section, we shall assume that n and h₁/h₂ are fixed quantities and will look at the dependence of f(x) on τ and ρ, both varying a priori between 0 and 1. We shall successively consider the case τ < 1 and the case τ = 1, since the latter is special and corresponds, as shown before, to non-generic criticality. Under certain reasonable assumptions, Fredholm theory implies the qualitative phase diagram of figure 9 in the (ρ, τ) plane. This picture will be corroborated by the exact results of section 6.

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The case τ < 1:

Here, K_τ(x, y) is a smooth function of (x, y) varying in the domain [0, 1]², and it is thus square integrable. We may then apply Fredholm theory: assuming that −(2π)/n is not a characteristic value of the integral equation (i.e. there exists no non-zero function f(x) such that f(x) + (n/2π)∫¹₀K_τ(x, y)f(y) dy = 0; we expect this assumption to be valid when n is between 0 and 2), equation (5.11) has a unique solution f(x) for each value of ρ. More precisely, by linearity, we have

$$\begin{equation} f(x)=f_1(x)-\rho f_{\rm id}(x) \end{equation} \tag{ 5.18 }$$

where f₁(x) (resp. f_id(x)) is the solution of the linear integral equation obtained by changing its right-hand side into 1 (resp. x). Note that f₁(x) and f_id(x) do not depend on ρ but implicitly depend on τ as well as n and h₁/h₂. These functions might be expressed, for instance, via Neumann series at least for n small enough.

However, the solution f(x) does not necessarily satisfy the positivity conditions (5.7) ensuring that the model is well defined. In view of (5.18), we conjecture that these conditions amount to ρ ⩽ ρ_c, where ρ_c = f₁(1)/f_id(1). In particular, when ρ = ρ_c, f(1) = 0 and the model is critical. We identify this critical point as generic in the sense of section 3.2, since we expect f'(1), and hence u''(R(1)), to be finite for any τ < 1. As ρ_c depends on τ, we obtain a generic critical line in the (ρ, τ) plane; see figure 9. Note that, for any values of n and h₁/h₂, the generic critical line starts from (ρ = 1, τ = 0), corresponding to the critical point for pure quadrangulations without loops. Indeed, since K₀ = 0, we have f(x) = 1 − ρx for τ = 0; therefore, the critical value of ρ is 1. (We also recover the known critical values R(1) = 2 and g = 1/12). Moreover, for 0 < n < 2, we expect the line of generic critical points to connect continuously with the non-generic critical line τ = 1 (to be discussed later): ρ_c should be positive for all τ < 1 and have a positive limit ρ* as τ → 1.

The case τ = 1:

Now, K₁(x, y) is no longer a smooth function of (x, y) in the domain [0, 1]², but diverges as (5.14) for x, y → 1. We do not know whether a general theory applies to such kernels. Nevertheless, from [20], we expect that equation (5.11) has, for all ρ, a unique continuous solution satisfying f(1) = 0. Thanks to this cancellation, the integral ∫¹₀K₁(1, y)f(y)dy may still be well defined.

Furthermore, we see that the solution f(x) cannot be regular for x → 1 (i.e. vanish as an integer power of x − 1) as otherwise, the integral in (5.11) would contain singular terms (with logarithms), which are not present on the rhs, regular at x = 1. This suggests to assume that the solution behaves as

$$\begin{equation} f(x)\sim C (1-x)^\alpha \end{equation} \tag{ 5.19 }$$

for some constant C and some positive non-integral exponent α (the condition α > 0 ensures both that f(1) = 0 and that the integral in (5.11) converges at x = 1). Remarkably, α cannot take arbitrary values, but is related to n via

$$\begin{equation} n=2 \sin \pi \alpha \end{equation} \tag{ 5.20 }$$

as seen from the following argument. Subtracting from (5.11) its value at x = 1, we obtain

$$\begin{equation} f(x)+{n\over 2 \pi }\int _0^1 (K_1(x,y)-K_1(1,y)) f(y)\, {\rm d}y= \rho (1- x) \end{equation} \tag{ 5.21 }$$

with K₁(x, y) − K₁(1, y) ∼ −(1 − x)y/((1 − y)(1 − xy)), when x and y tend to 1. For x → 1, the dominant singular term in the integral arises from y such that 1 − y = O(1 − x). More precisely, writing y = 1 − s(1 − x), we obtain at leading order

$$\begin{equation} \left. \int _0^1 (K_1(x,y)-K_1(1,y)) f(y)\, {\rm d}y\right|_{\rm sing.} \sim - C (1-x)^\alpha \int _0^\infty {s^{\alpha -1} \over 1+s}\, {\rm d}s \end{equation} \tag{ 5.22 }$$

with the rhs integral evaluated as π/sin πα. As such, this equation holds only for α < 1 so that the integral on the rhs is convergent. It remains valid for α > 1 provided the integral is analytically continued, with again the value π/sin πα (note that the integral on the lhs also contains regular terms that, for α > 1, dominate the leading singular term). Combining (5.19) and (5.21), whose rhs contains no singular term, we deduce (5.20).

Note that, for a given value of n between 0 and 2, the possible α > 0 satisfying (5.20) is of the form α = 1/2 ± b + m, with $\pi b = \arccos (n/2)$ and m being a non-negative integer. So far, we have only discussed the dominant exponent but, by linearity, all exponents appearing in the expansion of f(x) at x = 1 should be of this form. We, therefore, expect f(x) to be of the general form

$$\begin{equation} f(x)=(1-x)^{1/2-b}\Phi _{-}(x)+(1-x)^{1/2+b}\Phi _{+}(x) \end{equation} \tag{ 5.23 }$$

where Φ₋(x) and Φ₊(x) are functions with a regular expansion at x = 1 (i.e. contain only integral powers of x − 1). By considering the expansion of (5.11) at x = 1 and splitting it into singular and regular parts (i.e. separating terms with non-integral and integral exponents), we find that Φ₋(x) and Φ₊(x) should satisfy

$$\begin{equation} \fl (1-x)^{1/2\pm b}\Phi _{\pm }(x)+{\cos (\pi b)\over \pi }\left. \left\lbrace \int _0^1 K_1(x,y) (1-y)^{1/2\pm b} \Phi _{\pm }(y)\, {\rm d}y\right\rbrace \right|_{\rm sing.}= 0 \end{equation} \tag{ 5.24 }$$

and

$$\begin{equation} \fl {\cos (\pi b)\over \pi }\left. \left\lbrace \int _0^1 K_1(x,y) ((1-y)^{1/2-b} \Phi _{-}(y) +(1-y)^{1/2+b} \Phi _{+}(y) )\, {\rm d}y\right\rbrace \right|_{\rm reg.}= 1-\rho x . \end{equation} \tag{ 5.25 }$$

Now we expect that Φ₋(1) is not zero generically so that the leading singularity at x = 1 is of the form (1 − x)^{1/2 − b}. Furthermore, Φ₋(1) must be non-negative in order to satisfy the positivity conditions (5.7). This situation should hold for any ρ < ρ* and corresponds to the dense phase of the O(n) loop model, see figure 9. At ρ = ρ*, we expect Φ₋(1) = 0 and Φ₊(1) > 0 so that, at this special point, the leading singularity becomes of the form (1 − x)^{1/2 + b}, corresponding now to the dilute O(n) loop model. For ρ > ρ*, we expect Φ₋(1) < 0 so that the model is ill defined. The fact that the critical condition f(1) = 0 for τ < 1 coincides when τ → 1 with the condition Φ₋(1) = 0 is rather natural and makes us believe that the transition from the line of generic critical points to that of non-generic ones should be continuous. Note that having an effective value of α larger than 1 for the leading singularity simply corresponds to a situation, where both Φ₋(1) and Φ₊(1) would vanish. This may occur only for a particular class of right-hand side in the integral equation and it is not expected in the present case.

Finally, recalling section 2.2, we see that behavior (5.19) for f(x) = u'(R) corresponds precisely to expansion (3.12) for u(R). In particular, we identify

$$\begin{equation} a=\alpha +\case{3}{2} \end{equation} \tag{ 5.26 }$$

and relation (5.20) is nothing but (4.4).

The starting point is a linear integral equation, not for the function u'(R) as in section 5.1, but for the resolvent W(ξ) defined as in (3.19). Those two quantities are related via the general formula (3.20). Moreover, substituting the fixed point condition (2.3) for the rigid loop model into (3.4), we have for R < R(1)

$$\begin{eqnarray} &&u(R)=R-3 g R^2-{n\over 2} \sum _{k \ge 1} {2k \choose k} h_1^{2k} R^{k} F_k \nonumber\\ &&\hphantom{u(R)}= R-3 g R^2-{n\over 2} \oint {{\rm d}\xi W(\xi ) \over 2{\rm i}\pi } \left({1\over \sqrt{1-4 h_1^2 R \xi ^2}}-1\right) \end{eqnarray} \tag{ 6.1 }$$

where the contour of integration is, say, the circle of radius $\gamma =2 \sqrt{R(1)}$ (note that 4h²₁Rγ² is always smaller than 1 from equation (4.2)). Upon differentiating with respect to R, and substituting into (3.20), we obtain

$$\begin{eqnarray} &&\fl W(\xi ) = S(\xi ) - {n \over \xi } \oint {{\rm d}\xi ^{\prime } W(\xi ^{\prime }) \over 2 {\rm i} \pi } \int _0^{R(1)} {\rm d}R \left( 1 - {4 R \over \xi ^2} \right)^{-1/2} h_1^2 \xi ^{\prime 2} \left( 1 - 4 h_1^2 \xi ^{\prime 2} R \right)^{-3/2} \nonumber\\ &&\hphantom{\fl W(\xi ) }= S(\xi ) - {n \over 2 \xi } \oint {{\rm d}\xi ^{\prime } W(\xi ^{\prime }) \over 2 {\rm i} \pi } {h_1^2 \xi ^{\prime 2} \over h_1^2 \xi ^{\prime 2} - 1/\xi ^2} \left( \sqrt{ 1 - 4 R(1)/\xi ^2 \over 1 - 4 R(1) h_1^2 \xi ^{\prime 2}} -1 \right), \end{eqnarray} \tag{ 6.2 }$$

where

$$\begin{equation} S(\xi ) = \case{1}{2} ( \xi - g \xi ^3 - \xi ( 1 - 2 g R(1) - g \xi ^2) \sqrt{1 - 4R(1)/\xi ^2} ) \end{equation} \tag{ 6.3 }$$

corresponds to the first two terms on the rhs of (6.1).

Equation (6.2) is a linear integral equation for the resolvent, which rephrases that of section 5.1 for f(x) = u'(R). It implies a simpler functional equation for W(ξ) as follows: for ξ ∈ [ − γ, γ], we have

$$\begin{equation} S(\xi + {\rm i} 0) + S(\xi - {\rm i} 0) = \xi - g \xi ^3 \end{equation} \tag{ 6.4 }$$

so that, from (6.2), we may write

$$\begin{equation} W(\xi + {\rm i} 0) + W(\xi - {\rm i} 0) = \xi - g \xi ^3 + {n \over \xi } \oint {{\rm d}\xi ^{\prime } W(\xi ^{\prime }) \over 2 {\rm i} \pi } {h_1^2 \xi ^{\prime 2} \over h_1^2 \xi ^{\prime 2} - 1/\xi ^2}. \end{equation} \tag{ 6.5 }$$

The latter integral may be evaluated by the residue theorem: the integrand has poles at ξ' = ±(h₁ξ)⁻¹, each with residue (h₁ξ)⁻¹W((h₁ξ)⁻¹)/2, and a pole at ξ' = ∞ with residue −F₀ = −1. Hence, the resolvent satisfies the functional equation

$$\begin{equation} \fl W(\xi + {\rm i} 0) + W(\xi - {\rm i} 0) = \xi - g \xi ^3 + {n \over \xi } - {n \over h_1 \xi ^2} W \left( {1 \over h_1 \xi } \right),\qquad \xi \in [-\gamma ,\gamma ]. \end{equation} \tag{ 6.6 }$$

Note the similarity with equation (1.2) for the different model of section 1.2. Equation (6.6) can also be obtained as a consequence of loop equations in the matrix model formulation of our O(n) model [21, equation (V-22)].

Solving (6.2) boils down to finding a solution of (6.6), which is bounded, odd in ξ and such that W(ξ) ∼ 1/ξ as ξ → ∞. As we shall see, these requirements fix W(ξ) completely. By linearity, we may write

$$\begin{equation} W(\xi ) = W_{\rm part}(\xi ) + W_{\rm hom}(\xi ) \end{equation} \tag{ 6.7 }$$

Here, W_part(ξ) is the easy particular solution of (6.6):

$$\begin{equation} W_{\rm part}(\xi ) = { 2 ( \xi - g \xi ^3 ) - n \big( {1 \over h_1^2 \xi ^3 } - {g \over h_1^4 \xi ^5}\big) \over 4 - n^2} + {n \over (2 + n) \xi } \end{equation} \tag{ 6.8 }$$

and W_hom(ξ) is now an odd solution of the homogeneous equation:

$$\begin{equation} W_{\rm hom}(\xi + {\rm i} 0) + W_{\rm hom}(\xi - {\rm i} 0) + {n \over h_1 \xi ^2} W_{\rm hom} \left( {1 \over h_1 \xi } \right)=0 . \end{equation} \tag{ 6.9 }$$

The condition that W(ξ) ∼ 1/ξ for ξ → ∞ and W is bounded amounts to demanding that

$$\begin{eqnarray} &&\fl W_{\rm hom}(\xi ) = {2 g \over 4 - n^2} \xi ^3 - {2 \over 4 - n^2} \xi + {2 \over 2 + n} \xi ^{-1} + O(\xi ^{-2}),\qquad \xi \rightarrow \infty , \\&&\ms \fl W_{\rm hom}(\xi ) = {-n \over 2}\left({2 g \over 4 - n^2} \xi ^{-5} - {2 \over 4 - n^2} \xi ^{-3} + {2 \over 2 + n} \xi ^{-1} + O(1)\right),\qquad \xi \rightarrow 0 . \end{eqnarray} \tag{ 6.10 }$$

We shall present in section 6.3 the general expression for W_hom(ξ), when 4h₁R(1) < 1, which corresponds to a subcritical or generic critical situation. Something special happens on the non-generic critical line 4h₁R(1) = 1 because then the cut [ − γ, γ] collides with its image under ξ↦1/(h₁ξ) (see figure 10). As we shall now see, W_hom(ξ) has a simple expression along this line.

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As seen in section 4.1, the non-generic critical line is characterized by τ = 4h₁R(1) = 1 (or equivalently R(1) = R_c). This implies that the extremity of the cut $\gamma =2\sqrt{R(1)}=1/\sqrt{h_1}$ is a fixed point of ξ↦1/(h₁ξ). It is then possible to guess the general solution of the homogeneous equation (6.9), which is odd in ξ, namely

$$\begin{eqnarray} &&\fl W_{\rm hom}(\xi )= \left(B(\xi ) - {\gamma ^2 \over \xi ^2} B\left({\gamma ^2\over \xi }\right)\right)\left({\xi -\gamma \over \xi +\gamma } \right)^b -\left(B(-\xi ) - {\gamma ^2 \over \xi ^2} B\left(-{\gamma ^2\over \xi } \right)\right)\left({\xi +\gamma \over \xi -\gamma } \right)^b\nonumber\\ \end{eqnarray} \tag{ 6.11 }$$

where B(ξ) is some arbitrary analytic function and, again, $\pi b=\arccos (n/2)$. The fact that this form satisfies (6.9) can be checked directly and one can prove that it describes all the solutions. This in turn leads to a spectral density supported on [ − γ, γ]:

$$\begin{eqnarray} &&\fl \rho (\xi ) = -{\sin (\pi b)\over \pi }\left( \left(B(\xi ) - {\gamma ^2 \over \xi ^2} B\left({\gamma ^2\over \xi }\right) \right)\left({\gamma - \xi \over \gamma + \xi }\right)^b\right.\nonumber\\ &&\left. + \left(B(-\xi ) - {\gamma ^2 \over \xi ^2} B\left(-{\gamma ^2\over \xi }\right)\right) \left({\gamma + \xi \over \gamma - \xi }\right)^b\right). \end{eqnarray} \tag{ 6.12 }$$

The requirement that W_hom is holomorphic in C∖[ − γ, γ] imposes that B is an entire function. To satisfy (6.10), B(ξ) must be a polynomial of degree 3, whose four coefficients are determined from

$$\begin{equation} \fl B(\xi )\left({\xi -\gamma \over \xi +\gamma } \right)^b = {g \over 4 - n^2} \xi ^3 - {1\over 4- n^2} \xi + {1 \over 2 + n} \xi ^{-1} + O(\xi ^{-3}) \qquad {\rm as}\quad \xi \rightarrow \infty . \end{equation} \tag{ 6.13 }$$

Note that we could a priori imagine terms of orders ξ² and ξ⁰ in this expansion since they would be canceled by parity in the expansion of W_hom(ξ) at large ξ. However, such terms would create poles for ρ(ξ) at ξ = 0, which are not allowed because ρ should be integrable. Condition (6.13) amounts to five equations: four of them fix the coefficients of B(ξ) and the last one yields some additional relation between g and h₁ that is nothing but the equation for the non-generic critical line. We find explicitly

$$\begin{equation} \fl B(\xi )={g\over 4-n^2} \left(\xi ^3+2 b \gamma \xi ^2 +2 b^2 \gamma ^2 \xi + {2\over 3}(b+2 b^3)\gamma ^3 \right) - {1\over 4-n^2} (\xi +2 b \gamma ), \end{equation} \tag{ 6.14 }$$

while the equation for the non-generic critical line reads

$$\begin{equation} g={3\over 2+b^2}\left(h_1- {2-n\over 2b^2} h_1^2\right). \end{equation} \tag{ 6.15 }$$

In order for expression (6.11) to be consistent with the positivity of u'(R), the spectral density (6.12) must be positive on ] − γ, γ[. In particular, expanding ρ(ξ) for ξ → ±γ, we have

$$\begin{eqnarray} &&\fl \rho (\xi )={\sin (\pi b)\over \pi } \left( 4^b (B(-\gamma )-\gamma B^{\prime }(-\gamma )) \left(1-{\xi ^2\over \gamma ^2}\right)^{1-b}\right.\nonumber\\ &&\left.+\, 4^{-b} (B(\gamma )+\gamma B^{\prime }(\gamma )) \left(1-{\xi ^2\over \gamma ^2}\right)^{1+b}\right) +O\left(1-{\xi ^2\over \gamma ^2}\right)^{2-b}. \end{eqnarray} \tag{ 6.16 }$$

Demanding the positivity of ρ(ξ) requires B( − γ) − γB'( − γ) ⩾ 0, which yields the condition

$$\begin{equation} g\le {3 h_1\over 2(b^2-2 b+3)} . \end{equation} \tag{ 6.17 }$$

The non-generic critical line (6.15) cannot extend outside of the region defined by (6.17) and, therefore, ends when it hits the boundary of this region, i.e. at a point (g*, h*₁) with

$$\begin{equation} g^*= {3 b^2 (2-b)^2 \over 2(2-n)(b^2-2 b+3)^2} , \qquad h_1^*= {b^2 (2-b)^2 \over (2-n)(b^2-2 b+3)} . \end{equation} \tag{ 6.18 }$$

As shown in section 6.4, the critical line becomes generic beyond that point. When (g, h₁) = (g*, h*₁), we may check that the coefficient of the first subleading term is positive, i.e. B(γ) + γB'(γ) > 0. Comparing (6.16) with (3.23), we therefore read the value of the exponent a, namely a = 2 − b for g < g* and a = 2 + b at g = g*.

Equation (6.6) can be recast more elegantly in terms of the differential form ω(ξ) = W_hom(ξ) dξ, namely

$$\begin{equation} \forall \,\xi \in [-\gamma ,\gamma ],\qquad \omega (\xi + {\rm i}0) + \omega (\xi - {\rm i}0) - n \omega (s(\xi )) = 0 , \end{equation} \tag{ 6.19 }$$

where s is the involution ξ↦1/(h₁ξ). Note that (s(ξ))² = 1/λ₊(1/ξ²), where λ₊ is the involution of section 2.3, specialized to the rigid case. We underline the similarity of equation (6.19) with that relevant in the O(n) model discussed in section 1.2, where loops visit only vertices of degree 3; in this case, we had a different involution $\xi \mapsto {\tilde{h}}^{-1} - \xi$. So, the techniques already developed for the O(n) model, where loops visit only vertices of degree 3 [11, 21], can be applied to equation (6.19) with few modifications.

The general strategy is to introduce a parametrization, which opens the cut [ − γ, γ] as well as its image ] − ∞, −(h₁γ)⁻¹]∪[(h₁γ)⁻¹, +∞[ under the involution (see figure 10), e.g.,

$$\begin{equation} v(\xi ) = V \int _{-(h_1\gamma )^{-1}}^{\xi } {{\rm d}\eta \over \sqrt{-(\eta ^2 - \gamma ^2)(\eta ^2 - (h_1\gamma )^{-2})}} . \end{equation} \tag{ 6.20 }$$

The new variable v can be expressed in terms of Jacobi elliptic functions and is a multivalued function of ξ, depending on the path followed in the complex plane from the origin (h₁γ)⁻¹ to ξ. Conversely, one may view ξ as a function of v ∈ C. We choose the constant V by demanding that v((h₁γ)⁻¹) = −1/2, when following a path with small positive imaginary part from −(h₁γ)⁻¹ to ∞ and then back to (h₁γ)⁻¹. This leads to

$$\begin{equation} V = {{\rm i} \over 4h_1\gamma K(h_1\gamma ^2)} , \end{equation} \tag{ 6.21 }$$

where K is the complete elliptic integral. We then denote T = v( − γ) (by the most straightforward path), and the fact that the square-root discontinuity is included in the real axis implies that T = i |T|. The function ξ(v) is easily seen to have the following properties:

$$\begin{eqnarray} &&\xi (v + 1) = \xi (v) ,\qquad \xi (-v) = \xi (v) ,\\&&\ms \xi (v + 1/2) = -\xi (v) , \qquad \xi (T-v)= (h_1\xi (v))^{-1} , \end{eqnarray} \tag{ 6.22 }$$

from which one can also deduce the properties of its derivative ξ'(v). For bookkeeping, we mention the expansion when v → v_∞ = −1/4:

$$\begin{equation} \xi (v) = {\Xi _{-1} \over v - v_{\infty }} + \Xi _{1}(v - v_{\infty }) + O((v - v_{\infty })^3) \end{equation} \tag{ 6.23 }$$

with

$$\begin{eqnarray} &&\Xi _{-1} = {\rm i}V = {-1\over 4h_1\gamma K(h_1\gamma ^2)} = {{\rm i} T \over h_1\gamma K^{\prime }(h_1\gamma ^2)} \\&&\ms \Xi _{1}\Xi _{-1}={1 \over 6}(\gamma ^2 + (h_1\gamma )^{-2}), \end{eqnarray} \tag{ 6.24 }$$

where we denote $K^{\prime }(\tau ) = K(\sqrt{1 - \tau ^2})$.

The image of the points of C∖([ − γ, γ]∪] − ∞, −(h₁γ)⁻¹]∪[(h₁γ)⁻¹, +∞[) reached by a path that does not cross the segment [γ, (h₁γ)⁻¹] (for instance, a straight path) is the domain ${\mathcal V} = \lbrace v: {\rm Re}[v] \in ]-1/2,1/2[ ,{\rm Im}[v] \in ]0,|T|[\rbrace$, as shown in figures 10 and 11. Let us define

$$\begin{equation} \varpi (v) = W_{\rm hom}(\xi (v))\xi ^{\prime }(v), \end{equation} \tag{ 6.25 }$$

which is an analytic function on ${\mathcal V}$. Since W_hom(ξ) has no discontinuity when ξ ∈ ] − ∞, −(h₁γ)⁻¹]∪[(h₁γ)⁻¹, ∞[, ϖ(v) takes opposite values when v↔ − v along the segment [ − 1/2, 1/2]. Thus, ϖ can be extended to an analytic function defined on ${\mathcal V}_2 = {\mathcal V}\cup (-{\mathcal V})$ by setting

$$\begin{equation} \forall v \in -{\mathcal V},\qquad \varpi (v) = -\varpi (-v) . \end{equation} \tag{ 6.26 }$$

Likewise, the absence of discontinuity along ξ ∈ [γ, (h₁γ)⁻¹] allows one to extend ϖ as an analytic function defined on the strip ${\mathcal V}_{\rm strip} = \bigcup _{m \in {\bf Z}} \ ({\mathcal V}_2 + m)$ by setting

$$\begin{equation} \forall v \in ({\mathcal V}_2 + m),\qquad \varpi (v) = \varpi (v - m) . \end{equation} \tag{ 6.27 }$$

The top boundary of this strip, ${\mathcal V}_{\rm cut}=\lbrace v:\ {\rm Im}[v] = |T|\rbrace$, maps to points ξ ∈ [ − γ, γ]. Eventually, looking at equation (6.22) and using equation (6.26), equation (6.19) turns into

$$\begin{equation} \forall v \in {\mathcal V}_{{\rm cut}},\qquad \varpi (v) + \varpi (v - 2T) - n\varpi (v - T) = 0 . \end{equation} \tag{ 6.28 }$$

This allows one to extend recursively and without ambiguity ϖ as an analytic function on ${\bf C} = \bigcup _{m \in {\bf Z}} ({\mathcal V}_{\rm strip} + 2mT)$. For instance, for all $v\in {\mathcal V}_{\rm strip} + 2T$, ϖ(v) can be defined as nϖ(v − T) − ϖ(v − 2T), noting that v − T and v − 2T belong to ${\mathcal V}_{\rm strip}$. Since (6.26)–(6.28) are linear analytic relations between analytic functions, they are now valid for any value v ∈ C.

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To summarize, we reduced the problem to that of finding an analytic function v↦ϖ(v), which is odd, 1-periodic, and satisfies

$$\begin{equation} \varpi (v - 2T) - n\varpi (v - T) + \varpi (v) = 0 . \end{equation} \tag{ 6.29 }$$

Equation (6.10) demands that ϖ(v) behaves as

$$\begin{eqnarray} &&\fl \varpi (v)=-{2 g \Xi _{-1}^4 \over 4-n^2} {1 \over (v-v_{\infty })^5} + {2 \Xi _{-1}^2 \over 4 - n^2}\left[1 - {g \over 3}(\gamma ^2 + (h_1\gamma )^{-2})\right]{1 \over (v - v_{\infty })^3}\nonumber\\ &&-\, {2 \over 2 + n} {1 \over v - v_{\infty }} + O(1) \end{eqnarray} \tag{ 6.30 }$$

when v → v_∞ = −1/4, and

$$\begin{eqnarray} &&\fl \varpi (v) = {n g\Xi _{-1}^4 \over 4 - n^2 }{1 \over (v - v_{0})^5} - {n \Xi _{-1}^2 \over 4 - n^2}\left[1 - {g \over 3}\big (\gamma ^2 + (h_1\gamma )^{-2}\big )\right]{1 \over (v - v_{0})^3}\nonumber\\ &&+\, {n \over 2 + n} {1 \over v - v_{0}} + O(1) \end{eqnarray} \tag{ 6.31 }$$

when v → v₀ = T − 1/4. We now assume n∉{0, 2}, such that e^2iπb ≠ 1. Defining

$$\begin{equation} \varpi _\pm (v)={\varpi (v-T)-{\rm e}^{\mp {\rm i}\pi b}\varpi (v) \over {\rm e}^{\pm {\rm i}\pi b}-{\rm e}^{\mp {\rm i}\pi b}} , \end{equation} \tag{ 6.32 }$$

we have ϖ(v) = ϖ₊(v) + ϖ₋(v), while, from the 1-periodicity of ϖ and from (6.29), ϖ_± must satisfy

$$\begin{equation} \varpi _{\pm }(v + 1) = \varpi _{\pm }(v),\qquad \varpi _{\pm } (v + T) = {\rm e}^{\pm {\rm i}\pi b}\varpi _{\pm }(v). \end{equation} \tag{ 6.33 }$$

Such functions are generalizations of elliptic functions, and they can be constructed by taking appropriate ratios of the Jacobi theta function of nome q = e^iπT. Let us just state the existence of a unique analytic function ζ_b that satisfies

$$\begin{equation} \zeta _b(v + 1) = \zeta _b(v),\qquad \zeta _b(v+T)={\rm e}^{{\rm i}\pi b} \zeta _b(v) \end{equation} \tag{ 6.34 }$$

and has a unique pole when v = 0 mod Z⊕TZ , which is simple and is such that ζ_b(v) ∼ 1/v when v → 0. Its construction and main properties are listed in appendix B. One may generate functions satisfying (6.33) with poles of higher degree at v = 0 by considering the derivatives of ζ_b or of ζ_−b(v) = −ζ_b( − v), and put this pole at any given point w by shifting the argument v to v − w. Since holomorphic functions satisfying (6.33) must vanish identically, one may determine ϖ by matching the divergent behavior at its poles with a linear combination of the previous functions. This leads eventually to

$$\begin{equation} \fl \varpi (v) = -{1 \over 2 + n}{\mathcal D}\left\lbrace \zeta _{b}(v - 1/4) + \zeta _b(v + 1/4) - \zeta _b(- v + 1/4) - \zeta _b(-v - 1/4)\right\rbrace \end{equation} \tag{ 6.35 }$$

where ${\mathcal D}$ is the differential operator:

$$\begin{equation} {\mathcal D} = {g \Xi _{-1}^4 \over 24(2 - n)} \partial _v^4 + {\Xi _{-1}^2 \over 2(2 - n)}\left[-1 + {g \over 3} (\gamma ^2 + (h_1\gamma )^{-2} ) \right]\partial _v^2 + 1 . \end{equation} \tag{ 6.36 }$$

Then, the spectral density is given by

$$\begin{eqnarray} &&\fl \rho (\xi (v))\,{\rm d}\xi (v) = {\xi ^{\prime }(v) \over 2{\rm i}\pi } \left(W_{{\rm hom}}(\xi (2T - v)) - W_{{\rm hom}}(\xi (v))\right) dv \nonumber\\ &&\hphantom{\fl \rho (\xi (v))\,{\rm d}\xi (v) }= -{dw \over 2\pi } \sqrt{{2 - n \over 2 + n}} {\mathcal D} \left\lbrace \zeta _b\left(w - {1\over 4}\right) + \zeta _b \left(w+ {1\over 4}\right)\right.\nonumber\\ &&\hphantom{\fl \rho (\xi (v))\,{\rm d}\xi (v) =}\left.+\,\zeta _b\left(-w+ {1\over 4}\right)+ \zeta _b\left(-w-{1\over 4}\right)\right\rbrace , \end{eqnarray} \tag{ 6.37 }$$

where we have set w = T − v so that w ∈ [0, 1/2] corresponds to ξ ∈ [ − γ, γ]. The value of $\gamma = 2\sqrt{R(1)}$ is determined a posteriori as a function of g and h₁ by requiring from (3.21) that $\rho (\xi ) \propto \sqrt{\xi \pm \gamma }$ when ξ → ∓γ, which is equivalent to demanding that ρ(ξ(v))ξ'(v) = O(w²) when w → 0.

In the solution above, the generic critical line is the relation between g and h₁ obtained by demanding that ρ(ξ)∝(ξ ± γ)^3/2 when ξ → ∓γ (see equation (3.23)). In other words, in the Taylor expansion of ρ(ξ(v))ξ'(v), as given by (6.37), when w → 0, the generic critical line is characterized by the vanishing of the terms of order 1 and w². We may write these two conditions in a parametric way, with parameter τ = h₁γ² = 4h₁R(1), as

$$\begin{eqnarray} &&Z_0 + {\Xi _{-1}^2 \over 2(2 - n)}\left(-1 + {g \over 3h_1} (\tau + \tau ^{-1})\right) Z_2 + {\Xi _{-1}^4 g \over 24(2 - n)} Z_4 = 0 , \\&&\ms Z_2 + {\Xi _{-1}^2 \over 2(2 - n)}\left(-1 + {g \over 3h_1} (\tau + \tau ^{-1})\right) Z_4 + {\Xi _{-1}^4 g \over 24(2 - n)} Z_6 = 0 , \end{eqnarray} \tag{ 6.38 }$$

where

$$\begin{equation} Z_{2j} = \partial ^{(2j)}_{w = 0}\left(\zeta _{b}(w - 1/4) + \zeta _{b}(w + 1/4)\right) . \end{equation} \tag{ 6.39 }$$

Note that Z_2j depends implicitly on τ via ζ_b, which depends on T, itself related to τ via

$$\begin{equation} \tau = \left({\vartheta _2(0|4T) \over \vartheta _3(0|4T)}\right)^2 = \left({\vartheta _4\big(0|{-1 \over 4T}\big) \over \vartheta _3\big(0|{-1 \over 4T}\big)}\right)^2. \end{equation} \tag{ 6.40 }$$

The solution is

$$\begin{eqnarray} &&h_1 = {1 \over 2(2 - n)\tau [K^{\prime }(\tau )]^2} {\Delta _8 \over \Delta _6 + 4(\tau ^2 + 1)[K^{\prime }(\tau )]^2 \Delta _4} , \\&&\ms g = {6 \over 2 - n} {\Delta _8 \Delta _4 \over (\Delta _6 + 4(\tau ^2 + 1)[K^{\prime }(\tau )]^2 \Delta _4)^2}, \end{eqnarray} \tag{ 6.41 }$$

where $K^{\prime }(\tau )=K(\sqrt{1-\tau ^2})$ and Δ₄, Δ₆ and Δ₈ are functions of τ defined via

$$\begin{equation} \fl \Delta _4 = T^{4}\big(Z_0Z_4 - Z_2^2\big),\qquad \Delta _6 = -T^{6}(Z_0Z_6 - Z_2Z_4),\qquad \Delta _8 = T^{8}\big(Z_2Z_6 - Z_4^2\big) . \end{equation} \tag{ 6.42 }$$

Those equations become simpler in the neighborhood of the special points τ → 0 (i.e. h₁ → 0), or τ → 1 (i.e. h₁ → h*₁, the tip of the non-generic critical line). When h₁ → 0, it is convenient to use the variable q = e^iπT → 0 for asymptotics, and we find

$$\begin{eqnarray} &&g = {1 \over 12} - {n \over 18} q^{4} + {n \over 36}(7 + n) q^{8} + O(q^{12}) \\&&\ms h_1 = {q^2 \over 2} - \left(2 + {n \over 6}\right)q^{6} + \left(7 + 2n + {n^2 \over 18}\right) q^{10} + O(q^{14}). \end{eqnarray} \tag{ 6.43 }$$

As expected, the generic critical line meets the critical point of quadrangulations at (g = 1/12, h₁ = 0), and near this point it behaves as

$$\begin{equation} h_1 = {3 \over 2} \sqrt{2 \over n} \sqrt{{1 \over 12} - g},\qquad g \rightarrow 1/12. \end{equation} \tag{ 6.44 }$$

When τ → 1, it is convenient to use the variable q' = e^−iπ/T → 0 for asymptotics. The computation confirms that the generic critical line ends at the tip (g*, h*₁) of the non-generic critical line found in equation (6.18), and we find near this point that

$$\begin{eqnarray} &&\fl \left({g - g^* \over g^*}\right) = - {2(1 - 2b) \over 2 + b^2} \left({h_1 - h_1^*\over h_1^*}\right) -{(2 - b)^2 \over 2 + b^2} \left({h_1 - h_1^* \over h_1^*}\right)^2 \nonumber\\ &&+\, {64 (1 + b)\over (1 - b)(2 + b^2)} \left({(2-b)^2(1-b)^2(3-2b+b^2)\over 4b(1+b)^2(2+b^2)}\right)^{1/b} \left({h_1 - h_1^* \over h_1^*} \right)^{1/b} \nonumber\\ &&+\,o\left({h_1 - h_1^* \over h_1^*} \right)^{1/b}. \end{eqnarray} \tag{ 6.45 }$$

The first two terms coincide exactly with equation (6.15): when passing from the generic to the non-generic critical line, both the slope and the curvature remain continuous. For n = 1 (i.e. b = 1/3), we find a leading discontinuity in the third derivative of g with respect to h₁, as expected for the Ising model [22].

The results above are best summarized in the phase diagram of figure 12. We have found a line of non-generic critical points given by the arc of parabola (6.15), which links the point (g = 0, h₁ = 2b²/(2 − n)) to the point (g*, h*₁) of equation (6.18). Along this line, the exponent a takes the value 2 − b. At the terminating point (g*, h*₁), a takes instead the value 2 + b. We then found a line of generic critical points with a more complicated parametrization (6.41). When τ decreases from 1 to 0, this line links the point (g*, h*₁) to the point (g = 1/12, h₁ = 0) describing pure quadrangulations. As just mentioned, the non-generic and generic critical lines connect with a continuous slope. Their concatenation forms the line h₁ = h_c(n; g) of section 4.1 as the model cannot be well defined above this line.

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Let us finally note that, in the limit n → 0, we have g* → 1/12 and h*₁ → 1/8. The non-generic critical line tends to the arc of parabola

$$\begin{equation} g=\case{4}{3} \big(h_1-4 h_1^2\big), \qquad \case{1}{8} \le h_1 \le \case{1}{4}, \end{equation} \tag{ 6.46 }$$

while the generic critical line becomes the vertical segment parametrized by g = 1/12, 0 ⩽ h₁ ⩽ 1/8 (see figure 13). This may be understood as follows: the small n expansion of F^loop_p describes quadrangulations equipped with a fixed finite number of rigid loops. Using exact enumeration results for quadrangulations with multiple boundaries, it can be seen that the contribution to F^loop_p from loops of large length 2k behaves as (4h₁R_Q)^2k, where R_Q is the generating function R(1) for pure quadrangulations, solution of

$$\begin{equation} R_Q=1+3 g R_Q^2 . \end{equation} \tag{ 6.47 }$$

Having a finite contribution from large loops requires that 4h₁R_Q ⩽ 1. Note that R_Q ranges from 1 to 2 when g ranges from 0 to 1/12. Criticality may be obtained in two ways: either we set 4h₁R_Q = 1 so that the contribution from large loops decays sub-exponentially (note then the equivalence between equations (6.46) and (6.47)) or we may set g = 1/12 so that the pure quadrangulations are themselves critical. This last situation requires h₁ ⩽ 1/8, since R_Q = 2 in this case.

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At this stage, it should be clear to the reader that relation (4.4) between the loop weight n and the exponent a only depends on a few properties of the ring transfer matrix M(z). Denoting by λ₊(z) the largest eigenvalue of M²(z), we used the estimate $\sum _{k^{\prime }\ge 0}A_{k,k^{\prime }}z^{-k^{\prime }}\sim \lambda _+(z)^k$ to eventually obtain the relation

$$\begin{equation} n= 2 (\mu ^*)^a \sin \pi (a-3/2), \qquad \mu ^*=-\lambda _+^{\prime }(z^*), \end{equation} \tag{ 7.1 }$$

where z* is the fixed point of the mapping z↦λ₊(z). In the symmetric case discussed so far, this mapping is an involution in the vicinity of z* so that λ'₊(z*) = −1. Let us now consider a slightly modified, non-symmetric version of our model defined as follows: the squares of type (c) in figure 1 can be classified into two classes—those whose two edges not crossed by the loop belong to the inner contour, and those in which these two edges belong to the outer contour. We can view these two species as corresponding to outward, resp. inward turns of the loop at hand. For instance, the loop in figure 3 makes eight inward and six outward turns. Assigning different weights, say h_{2, out} and h_{2, in}, respectively, to these squares, the transfer matrix M(z) is replaced by

$$\begin{equation} M(z)=\left(\begin{array}{@{}cc@{}} \displaystyle {h_1 z^{-1/2} \over 1-h_{2,{\rm out}} z^{-1}}& \displaystyle {{1 \over 1-h_{2,{\rm out}} z^{-1}}} \\\ms h_{2,{\rm in}} & 0 \end{array}\right) . \end{equation} \tag{ 7.2 }$$

Relation (2.7) becomes

$$\begin{equation} ( h_{2,{\rm out}}\lambda + h_{2,{\rm in}} z - \lambda z )^2 - h_1^2 \lambda z =0 \end{equation} \tag{ 7.3 }$$

whose largest solution λ₊(z) now has a fixed point at z* = h₁ + h_{2, out} + h_{2, in}. Since λ and z do not play symmetric roles in (7.3), the mapping z↦λ₊(z) is no longer an involution and we now have a non-trivial value μ* = (h₁ + 2h_{2, out})/(h₁ + 2h_{2, in}), leading to the new relation

$$\begin{equation} n=2 \sin \pi (a-3/2)\left( { h_1+2h_{2,{\rm out}}\over h_1+2 h_{2,{\rm in}} }\right)^a. \end{equation} \tag{ 7.4 }$$

Note in particular that the range of n allowing for non-generic criticality is modified. It is instructive to compare this result to that obtained on a regular square lattice. In this context, the parametrization n = 2sin π(a − 3/2) = 2cos πb naturally appears in the Coulomb gas approach to the model [23]. Many critical exponents of the model have simple (typically polynomial) expressions in terms of a (or b). On such a regular lattice, we also have a well-defined notion of exterior and interior of a loop and we may give different weights to outward and inward turns. On a regular lattice, there are, however, four more inward than outward turns so that the symmetry of the model may be restored at the price of a rescaling n → n(h_{2, in}/h_{2, out})² of the weight per loop. This in turn changes relation (4.4) into n = 2sin π(a − 3/2)(h_{2, out}/h_{2, in})².

Returning to the O(n) model on quadrangulations, the case h₁ = 0 is, as before, special since the largest eigenvalue is degenerate in this case; hence, (7.4) is replaced by

$$\begin{equation} n= \sin \pi (a-3/2)\left( { h_{2,{\rm out}}\over h_{2,{\rm in}} }\right)^a. \end{equation} \tag{ 7.5 }$$

Returning to the symmetric case, we may impose some restriction on the lengths of the loop by demanding, for instance, that they should be multiples of a fixed integer N. Such a restriction may occur, for instance, when the loop model is inherited from some underlying edge-coloring problem. Now the length of a loop whose outer and inner contours have the lengths 2k and 2k' is simply (k + k'). The consistency relation (2.2) has to be modified to account for the new constraint, and we are naturally led to now consider the quantity

$$\begin{eqnarray} &&\fl \sum _{{\ssty k^{\prime }\ge 0}\atop {\ssty k+k^{\prime }=0 \ {\rm mod}\ N}} A_{k,k^{\prime }}(h_1,h_2) z^{-k^{\prime }}={1\over N} \sum _{j=0}^{N-1}((\omega ^j \lambda _+(\omega ^{-j} z))^k+(\omega ^j \lambda _-(\omega ^{-j} z))^k), \qquad\! \omega ={\rm e}^{2{\rm i}\pi /N},\nonumber\\ \end{eqnarray} \tag{ 7.6 }$$

to be estimated as before in the vicinity of z* and for large k. The above sum behaves as (1/N) (λ₊(z*))^k with now a 1/N prefactor provided |λ₊(z*ω^−j)| < λ₊(z*) when j = 1, ..., N − 1, which holds for h₂ > 0 (again we also suppose that h₁ > 0 to avoid that λ₋(z*) = λ₊(z*)). The correcting factor (1/N) trivially results in a change of relation (4.4) into

$$\begin{equation} n= 2 N \sin \pi (a-3/2) \end{equation} \tag{ 7.7 }$$

(for h₁, h₂ > 0). In particular, imposing an even size for the loops takes the O(n) loop model in the universality class of the O(n/2) loop model without the parity constraint. This fact was already recognized in [24] in the slightly different context of loops living on triangles.

In the rigid case h₂ = 0, loops are automatically of even length by construction. We have ω^jλ₊(ω^−jz*) = ω^2jλ₊(z*) and the above sum behaves as λ₊(z*)^k, as before, provided k is a multiple of N for N odd (respectively, a multiple of N/2 for N even), while it vanishes otherwise. For h₂ = 0, relation (4.4) is therefore unchanged. Finally, for h₁ = 0, we obtain n = Nsin π(a − 3/2) instead.

Our results are easily extended to the case of maps whose faces have arbitrary even degrees, provided these degrees remain bounded, say, by 2M + 2 (M ⩾ 1). Faces of degree 2m not visited by a loop receive a non-negative weight g^(m) (1 ⩽ m ⩽ M + 1) and those visited by a loop receive a weight $h^{(m_1,m_2)}$ (m₁, m₂ ⩾ 0) if the face has m₁ (resp. m₂) incident edges belonging to the outer (resp. inner) contour of the loop at hand. Since the total degree of such a face is m₁ + m₂ + 2, we will implicitly assume in the following that $h^{(m_1,m_2)}$ is nonzero only if m₁ + m₂ ⩽ 2M, and m₁ and m₂ have the same parity. With these new weights, the O(n) loop model is now described by the fixed point condition

$$\begin{equation} g_k=\sum _{m=1}^{M+1}g^{(m)}\delta _{k,m} +n \sum _{k^{\prime }\ge 0} A_{k,k^{\prime }} F_{k^{\prime }} \end{equation} \tag{ 7.8 }$$

where $A_{k,k^{\prime }}$ is the generating function for (rooted) rings (made up of faces with arbitrary even degrees) with sides of lengths 2k and 2k'. As before, this generating function is best encoded in the quantity

$$\begin{equation} \sum _{k^{\prime }\ge 0} A_{k,k^{\prime }}z^{-k^{\prime }} = \mathop {\mathrm{tr}}\nolimits Mz^2(k) \end{equation} \tag{ 7.9 }$$

involving a new transfer matrix M(z) of size 2M × 2M given by

$$\begin{equation} M(z)=\left(\begin{array}{@{}cccccc@{}} \displaystyle {\sum h^{(1,m_2)}z^{-{m_2\over 2}}\over 1-\sum h^{(0,m_2)}z^{-{m_2\over 2}}} & \displaystyle{1\over 1-\sum h^{(0,m_2)}z^{-{m_2\over 2}}} & 0 & 0 & \cdots & 0 \\\ms \displaystyle\sum h^{(2,m_2)}z^{-{m_2\over 2}} & 0 & 1 & 0 & \cdots & 0 \\\ms \displaystyle \sum h^{(3,m_2)}z^{-{m_2\over 2}} & 0 & 0 & 1 & \cdots & 0 \\\ms \displaystyle \sum h^{(4,m_2)}z^{-{m_2\over 2}} & 0 & 0 & 0 & \ddots & 0 \\\ms \displaystyle \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\\ms \displaystyle\sum h^{(2M,m_2)}z^{-{m_2\over 2}} & 0 & 0 & 0 & 0 & 0 \end{array}\right). \end{equation} \tag{ 7.10 }$$

Here, the sum in $\sum h^{(i,m_2)}z^{-{m_2\over 2}}$ runs over m₂ ⩾ 0 (and in practice over values of m₂ ranging from 0 to 2M − i and having the same parity as i). As shown in appendix A, the eigenvalues λ of M²(z) are the solutions of the characteristic equation

$$\begin{equation} \fl \Bigg((\lambda z)^M -\displaystyle \sum _{{\ssty m_1,m_2\ge 0}\atop{\ssty m_1,m_2\ {\rm even}}} h^{(m_1,m_2)} \lambda ^{{2M-m_1\over 2}} z^{{2M-m_2\over 2}}\bigg)^2= \bigg(\sum _{{\ssty m_1,m_2\ge 0}\atop{\ssty m_1,m_2\ {\rm odd}}} h^{(m_1,m_2)} \lambda ^{{2M-m_1\over 2}} z^{{2M-m_2\over 2}} \bigg)^2 . \end{equation} \tag{ 7.11 }$$

Repeating the analysis of section 4, we again find at a non-generic critical point the consistency relation between n and exponent a characterizing the large k asymptotics of F_k:

$$\begin{equation} n= 2 (\mu ^*)^a \sin \pi (a-3/2), \qquad \mu ^*=-\lambda _+^{\prime }(z^*) \end{equation} \tag{ 7.12 }$$

in terms of the largest eigenvalue λ₊(z) of M²(z) and its fixed point z*. Note that this relation holds when the largest eigenvalue is not degenerate. In a symmetric model, we must set $h^{(m_1,m_2)}=h^{(m_2,m_1)}$ so that λ and z play symmetric roles in (7.11). This implies as before that the mapping z → λ₊(z) is an involution in the vicinity of z* and that μ* = 1 so that the simple relation (4.4) is recovered.

This generic relation (7.12) is modified whenever $h^{(m_1,m_2)}=0$ for all odd values of m₁ and m₂. In this case, the rhs of equation (7.11) vanishes and the largest eigenvalue is degenerate, resulting in the suppression of the factor 2 in (7.12), namely n = (μ*)^asin π(a − 3/2) with again μ* = 1 in the symmetric case.