Two-point functions of random-length random walk on high-dimensional boxes

We study the two-point functions of a general class of random-length random walks (RLRWs) on finite boxes in Zd with d⩾3 , and provide precise asymptotics for their behaviour. We show that in a finite box of side length L, the two-point function is asymptotic to the infinite-lattice two-point function when the typical walk length is o(L2) , but develops a plateau when the typical walk length is Ω(L2) . We also numerically study walk length moments and limiting distributions of the self-avoiding walk and Ising model on five-dimensional tori, and find that they agree asymptotically with the known results for the self-avoiding walk on the complete graph, both at the critical point and also for a broad class of scaling windows/pseudocritical points. Furthermore, we show that the two-point function of the finite-box RLRW, with walk length chosen via the complete graph self-avoiding walk, agrees numerically with the two-point functions of the self-avoiding walk and Ising model on five-dimensional tori. We conjecture that these observations in five dimensions should also hold in all higher dimensions.


Introduction
The effects of boundary conditions on the finite-size scaling of statistical-mechanical lattice models in high dimensions has a rather long history [1,2,3,4,5], but has remained a very active area; see e.g.[6,7,8,9,10,11,12,13,14].One particular topic of interest has been the scaling of the Ising susceptibility at the infinite volume critical point, where it has been observed numerically that on boxes of side-length L, periodic boundary conditions produce a scaling L d/2 , in contrast to the L 2 behaviour observed with free boundary conditions.
Significant progress explaining this phenomenon mathematically was recently presented in [15], where a rigorous renormalisation group analysis was performed of the weakly coupled hierarchical |ϕ| 4 model in d ≥ 4 dimensions, on finite boxes of volume L d with both periodic and free boundary conditions.In particular, it was shown that while the effective critical point for the model with periodic boundary conditions coincides with the infinite volume critical point, the effective critical point for free boundary conditions is shifted away from the infinite-volume value by an amount of order L −2 .Moreover, it was shown that for both boundary conditions, increasing the temperature above the effective critical point by an amount of order L −λ leads the susceptibility to scale as L λ , for any 3/2 ≤ λ < d/2, but as L d/2 for any λ ≥ d/2.By universality, one would expect the same behaviour to hold for the self-avoiding walk (SAW) and Ising model on boxes in Z d , and indeed numerical evidence supporting this belief has been presented in [16].
Another striking, and related, feature of high-dimensional models with periodic boundary conditions is the so-called plateau which emerges in their two-point function, sufficiently close to the critical point, so that the initial simple random walk decay |x| 2−d becomes subdominant to a term which is independent of x but decaying in L. For the Ising model on tori T d L with d > 4, it was proved in [17] that at the (infinite volume) critical point the two-point function is bounded below by c 1 |x| 2−d + c 2 L −d/2 , and it was conjectured that an upper bound of the same order should exist.This conjecture was extended in [16], where it was predicted that for the SAW and Ising model on T d L with d > 4, at temperatures shifted above the infinite volume critical point by L −λ , the twopoint function behaves as c 1 |x| 2−d + c 2 L λ−d when 2 ≤ λ ≤ d/2, and as c 1 |x| 2−d + c 2 L −d/2 when λ ≥ d/2.The latter behaviour has recently been established rigorously [18] for the Domb-Joyce model with d > 4, for sufficiently weak interaction strength.Moreover, analogous behaviour is also now known for bond percolation [19] when d ≥ 11 for the nearest-neighbour model, and d > 6 for spread-out models.We refer to the regime with 0 < λ < d/2 as the high-temperature scaling window, and to the regime λ > d/2 as the critical window.It was observed numerically in [16] that for the SAW and Ising model with λ < 2, the two-point function decays faster than a power-law on large scales, but no conjecture for the precise nature of this behaviour was made.
The general plateau behaviour conjectured in [16] was supported by numerical simulations of the SAW and Ising model on five-dimensional tori, but was motivated by considering a model of simple random walk in which the walk length is chosen to be finite and random, and distributed as a SAW on the complete graph.The behaviour of SAW on the complete graph has been recently studied [20,21,22].Yet the behaviour of the two-point function of random-length random walk, with arbitrary walk length distributions, appears not to have been studied in significant detail.This question was recently addressed for random-length random walk on Z d in [23].One contribution of the current work is to present sharp asymptotic results for the two-point function of random-length random walks on finite boxes in Z d , for three distinct choices of boundary conditions, and with only modest assumptions on the walk length distribution.In summary, we find that if the walk length is concentrated on a scale o(L 2 ), then the finitebox and infinite lattice two-point functions are asymptotic.This is to be expected, since simple random walks of length N typically explore distances of order √ N. By contrast, for walks whose expected length is Ω(L 2 ), we establish a plateau given by the ratio of the mean walk length to the system volume.Specialising to the case that the walk length distribution is that of SAW on the complete graph, we find a universal model of high-dimensional torus two-point function behaviour, that agrees numerically with the SAW and Ising model on five-dimensional tori, in both the critical window and the high-temperature scaling window for any 0 < λ < d/2.Moreover, we find that the high-temperature scaling window consists of two separate regimes; universal exponential decay in terms of the continuum Green function for λ < 2, and λ-dependent plateau for 2 < λ < d/2.
We note that in the special case in which the walk length is geometrically distributed, the two-point function of random-length random walk on Z d corresponds to the lattice Green function, which is very well studied; see [24] and references therein.In that case, random-length random walk is generally referred to as killed random walk [25].We also note that plateau behaviour of the geometrically-killed simple random walk two-point function on tori was recently established in [18,Theorem 1.4], as a corollary of their weakly self-avoiding walk result.The results we present here for random-length random walk are both sharper and more general than given in [18,Theorem 1.4].
The key assumption underlying the use of the complete graph SAW length in the random-length random walk to describe the SAW and Ising two-point functions on T d L , is that for d > 4, the large L behaviour of the length of the SAW or Ising walk on T d L should behave in the same way as SAW on the complete graph; see Section 2 for a definition of the Ising walk.We therefore now summarise the known behaviour [20,21,22] of the SAW on the complete graph, K n .At fugacities 1/n(1 + an −p ), the mean walk length scales as n −p for p ∈ (0, 1/2) and a > 0, but scales as √ n for all p ≥ 1/2 and a ∈ R. Analogous behaviour has also been established [26] for SAW on the hypercube, Z N 2 , and for weakly self-avoiding walk on the torus T d L with d > d c when the interaction strength is sufficiently small [27].Moreover, the variance and limiting distributions of the appropriately scaled/standardised length of SAW on K n are also known in detail [21,22].For p ∈ (0, 1/2) with a > 0 the variance scales as n 2p , and the walk length divided by its mean converges to a mean-1 exponential distribution, while for p > 1/2 the variance scales as n and the standardised walk length converges to a half-normal distribution.In Section 4.1 we provide strong numerical evidence that the same behaviours hold for the SAW and Ising walk length on T d L with d = 5, and we conjecture that they in fact hold for all d ≥ 5.

Outline
Let us outline the remainder of this article.Section 1.2 lists some notational conventions.Section 2 defines the specific quantities of interest for the SAW and Ising model.Section 3 describes the random-length random walk models considered, and presents our main results for their two-point functions.Section 4 then presents numerical results for the SAW and Ising model on five-dimensional tori, both in the critical window and high-temperature scaling window.Specifically, Section 4.1 presents numerical results for the SAW and Ising walk length, while Section 4.2 considers their two-point functions.Finally, Sections 5 and 6 present proofs of Propositions 3.2 and 3.3, respectively, and Section 7 presents a proof of Lemma 3.1.

Notation
For integer d ≥ 1 and L > 2, we let we denote its Euclidean norm by |x| := √ x • x.We let T d L denote the d-dimensional discrete torus, of linear size L. We view T d L both as a graph, whose vertex set is taken to be B d L , and also, when convenient, as a module over the commutative ring T L := [−L/2, L/2) ∩ Z in which addition and multiplication are defined modulo L.
The standard asymptotic symbols such at O, o etc will refer to large L asymptotics.The definition of random-length random walk requires a sequence of random walk lengths, (N L ) L .In the asymptotic results we present for random-length random walk, the implied constants may depend on d and the choice of the sequence of distributions corresponding to (N L ) L .Statements such as f = O(g) in that context then mean that for any particular choice of d and the sequence of walk length distributions, there exists a constant c > 0 such that f (L) ≤ c g(L) for all sufficiently large L. In the case that constants depend on additional parameters, we will highlight this via subscripts; e.g. if for fixed λ we have f (L, λ) ≤ c(d, λ)g(L, λ) for all L ≥ N(d, λ), then we will write f = O λ (g).If f = O(g) and g = O(f ) we write f ≍ g.We find it convenient to also use the Vinogradov symbols, so that f ≪ g is equivalent to f = O(g), and f ≫ g is equivalent to g = O(f ).We also find it convenient to write f = Ω(g) to denote g = O(f ) and f = ω(g) to denote g = o(f ).

The SAW and Ising model
Let G = (V, E) be a rooted graph, with root 0. For n ∈ N, let Ω n G denote the set of all n-step walks on G which start at 0; i.e. all sequences ω 0 , . . ., ω n such that ω i ∈ V , ω 0 = 0 and ω i ω i+1 ∈ E. We set Ω G := n∈N Ω n G .For ω ∈ Ω n G , the notation ω : 0 → v implies ω n = v, and we denote the end of ω by e(ω) = ω n .In what follows, we let |ω| denote the number of steps, or length, of the walk ω ∈ Ω G , so that A walk ω ∈ Ω G is self-avoiding if ω i = ω j for all i = j.We consider the variablelength ensemble of self-avoiding walks on G, and let S denote a random SAW with this distribution, so that for all ω ∈ Ω G with The quantity z > 0 is the fugacity.We will be interested in the distribution of the walk length |S|, and the two-point function defined by [28] g(x) := Our simulations of S, discussed below, were performed using a lifted version [29] of the Berretti-Sokal algorithm [30].
We also consider analogous quantities for the Ising model.The zero-field ferromagnetic Ising model on finite graph G = (V, E) at inverse temperature β ≥ 0 is defined by the measure The corresponding two-point function is defined by The Ising two-point function can be conveniently re-expressed via the high-temperature expansion, as follows.For v ∈ V \ 0, let C v denote the set of all A ⊆ E such that the set of all vertices of odd degree in (V, A) is precisely {0, v}, and let C 0 denote the set of all A ⊆ E such that (V, A) has no vertices of odd degree.For a family of edge sets By analogy with the SAW case, we refer to z = tanh(β) as the Ising fugacity.The high-temperature expansion for the Ising model (see e.g.[31, (3.5)] or [32, Lemma 2.1]) implies that we can re-express (7) so that for all x ∈ V g This high-temperature representation of the Ising model can also be used to provide a natural definition of the Ising walk, first discussed in [33,31], and studied numerically in [23].Fix an (arbitrary) ordering, ≺, of V .We define T : such that from v i we choose v i+1 to be the smallest neighbour of v i such that v i v i+1 ∈ A and v i v i+1 has not previously been traversed by the walk.It is clear that T (A) defines an edge self-avoiding trail from 0 to v. Now let A denote a random element of ∪ v∈V C v with distribution The distribution of A is precisely the stationary distribution of the Prokofiev-Svistunov worm algorithm [34], in which the worm tail is fixed at the root.Our simulations of A, discussed below, were performed using such a worm algorithm.We will be interested in the induced distribution of T := T (A), and particularly in the distribution of its length, |T |, which we refer to as the Ising walk length.We note that by partitioning Ω G in terms of T , we can re-express the Ising two-point function (7) in precisely the form (4) but with Moreover, it can also be re-expressed in the form (5) with S replaced by T .The simulations of the SAW and Ising model to be presented in Section 4 were performed on five-dimensional tori.As we will demonstrate, the asymptotic behaviour of both |S| and |T | appear to coincide with the known [21,22] asymptotic behaviour of |S| on the complete graph, which we now summarise.Let G be the complete graph K n , rooted at a fixed vertex, and suppose the fugacity z satisfies 1/z = n(1 + an −p ).Let K denote S in this setting.It is known [20] that the critical fugacity is z = 1/n.Moreover, if p < 1/2 and a > 0 then we have for large n that Furthermore, let X be a standard normal random variable, and let Y be an exponential random variable with mean 1.Then as n → ∞ we have for p < 1/2 and a > 0 that For later reference, we shall denote by F the law of the standardised version of |X|, i.e. for x ∈ R

Numerical details
Our simulations of the SAW and Ising model were performed on 5-dimensional tori, at pseudocritical points z L = z c (1 − L −λ ) for various λ > 0, where z c denotes the estimated location of the infinite-volume critical point.In the Ising case we used the estimate z c = 0.113 424 8(5) [14], while in the SAW case we used z c = 0.113 140 84(1) [29].A detailed analysis of integrated autocorrelation time is presented in [35] for the worm algorithm and in [29] for the lifted Berretti-Sokal algorithm.Our fitting methodology and corresponding error estimation follow standard procedures, see for instance [36,37].

Definitions and boundary conditions
Let (C n ) n∈N be an i.i.d.sequence of uniformly random elements of {±e 1 , ..., ±e d }, where e i = (0, . . ., 1, . . ., 0) ∈ Z d is the standard unit vector along the ith coordinate axis, and let N be an N-valued random variable independent of (C n ) n∈N .The corresponding random-length random walk (RLRW) on Z d is the process Z := (Z t ) N t=0 defined so that Z 0 = 0 and Z t = Z t−1 + C t for each 1 ≤ t ≤ N .We also consider RLRWs (X P t ) N t=0 , (X R t ) N t=0 , and (X H t ) N t=0 on B d L , with periodic, reflecting, and holding boundary conditions, respectively, defined so that X * 0 = 0, and for all 1 ≤ t ≤ N we have when where * denotes either P, R or H, as appropriate.We define the two-point function of Z to be As noted in the Introduction, in the special case in which N is geometrically distributed, the two-point function of RLRW on Z d corresponds to the lattice Green function.Analogous definitions hold for X P , X R , X H . Specifically, for such a process on These two-point functions are closely-related to one another, as the next lemma illustrates.Recall that we consider T d L as a module over the commutative ring T L , with addition and scalar multiplication defined modulo L in each entry.For each x ∈ T d 2L , we can then define The partition of T d 2L into the sets [x] L defines an equivalence relation on T d 2L , in which the sets [x] L are the equivalence classes.The case of d = 1, corresponding to projecting a cycle onto a path, is illustrated in Fig. 1.
The proof of Lemma 3.1, which is based on Markov chain projection arguments, is discussed in Section 7. with L = 7.Note that in both cases the set of equivalence classes are in bijection with B 1 L .

Main results for RLRW two-point functions
We now state our main results for the asymptotic behaviour of RLRW two-point functions on B d L , for periodic, reflecting and holding boundary conditions.We defer proof of these results to Sections 5 and 6.We provide numerical evidence of the connection of these results to the Ising and SAW models in Section 4.2.
The RLRW model on B d L is most easily understood by relating it to the RLRW on the infinite lattice.The asymptotic behaviour of the latter was studied in detail in [23].See also [24].
Suppose N L is chosen so that its typical scale a L grows with L. One would expect the behaviour of g * ,L,N L (x) to differ qualitatively depending on whether or not a L grows fast enough that the RLRW can explore distances from the origin of order L, so that the presence of the boundary can be felt.We therefore present two separate results relating g * ,L,N L to g N L , depending on the asymptotics of N L .
Let ∆ denote the standard degenerate distribution function, i.e. ∆ is the indicator function for [0, ∞).Proposition 3.2.Consider a sequence of N-valued random variables N L , for which there exists a sequence a L > 0 satisfying: (iii) There exists r, C > 0 such that E(e rN L /a L ) ≤ C for all L.
(iv) There exists a distribution function Then, with * denoting P, R or H, as L → ∞ we have: The assumptions on N L given in Proposition 3.2 imply that typical N L will have length of order a L = O(L λ ) with λ < 2. Since a SRW walk of length N typically explores distances from the origin of order √ N, it then follows that a typical such RLRW will explore distances of order o(L) from the origin, and will therefore be too short to feel the boundary.It is therefore unsurprising that the finite-box and infinite lattice twopoint functions are asymptotic on such a scale, for any of the three choices of boundary conditions studied.The spatial scales probed by Proposition We note that, in particular, if G corresponds to the distribution function of a mean-1 exponential random variable, then we have lim with and where K ν (•) denotes the modified Bessel function of the second-kind [38].As discussed in [24], E is intimately related to the Green function of the continuum Laplacian on R d .As elaborated on in Section 4, the exponential choice for G here is motivated by the behaviour of the self-avoiding walk on the complete graph.We now turn our attention to the case that E(N L ) = Ω(L 2 ).Theorem 1.4] for the case of a geometric walk length distribution, giving upper and lower bounds for the difference between the torus and infinite lattice two-point functions in terms of the susceptibility, but without control of the constants, and with a lower bound that is weaker by a logarithmic factor when d = 4.
Suppose that N L is such that N L /E(N L ) converges in distribution, and that the limiting distribution function is continuous at the origin.E.g., this occurs if N L = |K L d | in either the critical window or the high-temperature scaling window.Now suppose that E(N L ) ≍ L λ with λ > 2. It then follows ‡ from [23, Proposition 3.1] that for any sequence Therefore, the exponential decay displayed by (24) for λ < 2 cannot be observed when λ > 2. Consequently, Proposition 3.3 implies that g * ,L,N L (x L ) decays as a powerlaw for ), but is then dominated by a term of order ).We note that the scale 2) is o(L), and therefore realisable inside To probe the crossover from power-law to plateau behaviour we choose  We note that the same critical window behaviour for the Ising and SAW mean, variance and limit distribution were observed at the estimated infinite-volume critical fugacity, z c , on 5-dimensional tori in [23].One can formally view this case as λ = +∞.

Universal walk length distributions
To support the claim that the boundary of the critical window lies at d/2, Table 1 provides estimates of the scaling exponent µ obtained by fitting E(|S|) and E(|T |) to an ansatz a + bL µ , for various values of λ.As expected from the complete graph SAW results, we indeed observe that µ = λ for each λ ≤ d/2, but µ = d/2 for all λ ≥ d/2.
and there exist constants A S,d,λ,a , A T ,d,λ,a , B S,d,λ,a , B T ,d,λ,a > 0 such that and While if λ > d/2, then for any a ∈ R and there exist constants and var(|S|

Universal two-point functions
We now provide numerical evidence that in both the high-temperature scaling window and the critical window, the two-point functions of the SAW and Ising model display the same asymptotic behaviour as does a RLRW whose walk length distribution is chosen to be that of a corresponding complete-graph SAW.
We first consider the high-temperature scaling window with λ < 2. Assuming the validity of the conjectures on S and T outlined in ( 28) and ( 29), it follows from standard convergence of types arguments (see e.g.[39, pp. 193]) that for all y ∈ R lim Assuming the validity of (34), and that the assumptions of Proposition 3.2 hold for this choice of N L , it follows from( 24) that as Universality then makes it natural to conjecture that the asymptotics of |x L | d−2 g(x L ) for the SAW and Ising model on the torus should be given by for suitable values of the model-dependent constants, α, γ, depending d, λ, and a. Figures 4a and 4b provide strong evidence in favour of these conjectures.In Figure 4a, the constants for SAW are set to α = 0.75, γ = 0.83/ A S,d,λ,a , while in Figure 4b the constants for the Ising model are set to α = 0.75, γ = 0.87/ A T ,d,λ,a , where A S,d,λ,a and A T ,d,λ,a were estimated by fitting the mean walk length; cf.(29).We now consider λ > 2. Assuming the validity of the conjecture (29), the discussion in Section 3.2 suggests that we should observe a plateau in this case.Moreover, we expect the order of the plateau to be L Universality then makes it natural to conjecture that for both the SAW and Ising model on the torus for suitable model-dependent parameters α, γ, depending on d, λ and a.

Proof of Proposition 3.2
Let (S n ) ∞ n=0 be a simple random walk on Z d , starting from the origin, and let We say that n ∈ N and z ∈ Z d have the same parity, and write n ↔ z, iff n+ |z| 1 is even, where The proof of Proposition 3.2 will utilise the following three lemmas, whose proofs are deferred until the end of this section.
In all that follows, any reference to reflecting or holding boundary conditions on B d L assumes L is odd.
First note that Lemma 3.1 implies with * = P, R, H.It therefore suffices to show that the left-hand side of ( 42) is o(1) as L → ∞.
We first consider the case of periodic boundary conditions.Part (i) of Lemma 3.1 implies and since p 0 (z) = 0 for z = 0, Eq. ( 40) gives We begin by showing that the second term on the right-hand side of (44) is exponentially small in L. First consider the large n terms.Combining assumption (iii) with the Chernoff bound implies that P(N L ≥ n) ≤ Ce −r n/a L .Letting ζ L := ⌊L 1+λ/2 ⌋, and recalling assumption (ii), it then follows that for any S ⊆ Z d we have for some κ > 0 which depends on d and on the specific sequence of distributions corresponding to (N L ).Now consider the small n terms.Fix c > 0 and let y ∈ Z d satisfy |y| ≥ c L. Lemma 5.1 implies But Lemma 5.2 implies |x L + zL| ≥ |z|L/2 for all z ∈ Z d \ 0 and x L ∈ B d L , and so (46) implies Therefore combining ( 45) and ( 47) with (44) shows that We now prove an analogous result for reflecting/holding boundaries.From Parts (ii) and (iii) of Lemma 3.1 we have for where L = L if * = H, and L = L − 1 if * = R.The second and third terms on the right-hand side in (49) can be shown to be exponentially small by arguing analogously to the periodic case.Indeed, applying (45) immediately shows that 2L of fixed cardinality, 2 d , and so arguing analogously to (47) implies Similarly, since |y| ≥ L/4 for all y ∈ [x L ] L \ x L , it follows immediately from (46) that Summarising then, we have for * = P, R, H that , and so it follows from (53) that for Proof of Lemma 5.1.Let a ≥ 0, and for j ∈ [d] let S j n denote the jth coordinate of S n .We begin with the elementary observation that if (S j n It then follows from the union bound that for any λ n > 0, where in the last step we utilised the Chernoff bound.Consequently, for any x ∈ Z d we have A simple calculation shows that for all t ∈ R E(e Therefore, taking λ n = d/n we have as n → ∞ that log E(e The stated result then follows from (58) and the specialisation of (56) to λ n = d/n.

It follows that for any
which establishes the lower bound in (41).Now, since This establishes the upper bound in (41).
Proof of Lemma 5.3.Since N L , a L ≥ 0 and G = ∆, there must exist s > 0 such that G(s) < 1.It then follows that there exists ε < 1 and α ∈ (0, s) such that G is continuous at α and G(α) < ε, which implies Therefore there exists α > 0 and ǫ < 1 such that for all sufficiently large L Now let m L := ⌊αa L ⌋ − 1(x L ⌊αa L ⌋).Then m L ≥ 0 for all sufficiently large L, and By construction, x L ↔ m L and m L → ∞ as L → ∞, and so it follows from [40, Theorem 1.2.1, (1.10)] that there exists C ∈ (0, ∞) such that for all sufficiently large L ] and so Combining (65) with (63) then yields the stated result.
6. Proof of Proposition 3.3 The local central limit theorem for simple random walk approximates p n via pn .We will make use of the following lemma, whose proof is deferred to the end of this section.Let Γ(s, z) := denote the (upper) incomplete gamma function, and let denote the complementary error function.
Lemma 6.1.Let d ≥ 1 be an integer, let a > 0 and let b ∈ R d .Then, and there exist c d , C d > 0 such that Proof of Proposition 3.3.Rearranging (44) yields where Similarly, rearranging (49) yields, for * = R, H and with L = where It is convenient in what follows to define the sequence l L via and to then let y L denote an arbitrary sequence in Z d satisfying y L ∈ B d l L .With this notation, we introduce the abbreviations The first task is to show that E(L), Ẽ(L), F (L) and F (L) are all o(E(N L )/L d ).We begin by considering E(L) and Ẽ(L).Since 1(w via a change of variables n → n − 1 in the first sum.Similarly, changing variables in the second sum yields It then follows that Now define δ := (1+d/2) −1 ; the motivation for this will become clear following (91).We first consider the small n terms in (83) and 84.Let  86) with z = 1 that we can bound the small n terms in (84) via But since l δ L |z| δ − l δ L − |z| δ + 1 = (l δ L − 1)(|z| δ − 1) ≥ 0 for any z ∈ Z d \ 0, we have that Therefore, combining the fact that Lemma 5.2 implies |y L + l L z| ≥ |z|l L /2, with (86) and (88), it then follows that We therefore see that the sums of the small n terms in E(L) and Ẽ(L) are exponentially small in L.
We now consider the large n terms in ( 83) and ( 84).An elementary argument shows that there exists c d > 0 such that for any n ∈ Z + and y ∈ Z (90) Using Markov's inequality, it then follows that for any y, z Since |z| −d−1 is summable over Z d \ 0, combining (89), ( 87) and (91) shows that The bounds for F (L) and F (L) are obtained similarly, with the aid of the local central limit theorem.If n ≤ ⌈|zl L | 2−δ ⌉ and w, z ∈ Z d \ 0 satisfy |w| ≥ |z| l L /4, then it follows from [25, Proposition 2.1.2]that for some ϕ = ϕ(d) > 0. It then follows from (93), (85) and the triangle inequality that Just as (87) and (89) were obtained from (86) and (88), analogous arguments using (94) yield and Arguing as in (91), and combining with (95) and (96) then yields Next we consider B(L).Let a L be a positive sequence.For any y ∈ Z d we have If now |y| ≥ L/2, then since e −t < t −γ for all t, γ > 0, taking γ = d/2 we have and therefore Consequently, if E(N L ) = Ω(L 2 ) then choosing a L → 1 in (99) and (101) implies It now remains to study the asymptotic behaviour of A(L).We first consider Part (i), and therefore assume E(N L ) = Ω(L 2 ).It follows from Lemma 5.2 and (70) of Lemma 6.1 that Since E(N L ) = Ω(L 2 ), there exists γ > 0 such that E(N L ) ≥ γ L 2 for all sufficiently large L. It follows again Lemma 5.2 and (70) of Lemma 6.1 that It follows in particular that Together with (102), ( 92), ( 98), ( 71) and (75) this establishes Part (i).
We now focus on Part (ii), and therefore assume E(N L ) = ω(L 2 ).Let a L > 0 be a sequence satisfying a L → ∞ and a L = o(E(N L )/L 2 ).From Lemma 5.2 and (70) of Lemma 6.1 we have But (69) of Lemma 6.1 implies and, since erfc is decreasing and continuous and erfc(0 But, by assumption on a L , and so 1 2 Combining ( 110) and (114) we then have In particular, for periodic boundary conditions we have l L = L and so (115) implies while for reflecting or holding boundary conditions we have l L = 2 L and so (115) implies Together with (103), ( 92), ( 98), ( 71) and (75) this establishes Part (ii).
Proof of Lemma 6.1.Let ǫ ∈ (−1, 1).For any z ≥ 2 we have e −a(z+ǫ) 2 ≤ e −a(t−1/2+ǫ) 2 for all t ∈ [z − 1/2, z + 1/2], and so A similar argument also produces the lower bound Now let α ∈ R, and define {α} := α − ⌊α⌋ ∈ [0, 1).Since z → z − ⌊α⌋ is a bijection of Z, we have It follows from ( 118) and (119) that Eq. ( 69) then follows by observing that We now consider (70).Let n ∈ Z + and Therefore there exist c d , C d > 0 such that for all n ∈ Z + Now observe that if |z| ∞ denotes the sup norm on R d , then A lower bound is obtained similarly: (126) Lemma 7.3.Let L ≥ 3, and let P P,L , P H,L , and P R,L denote the transition matrices of simple random walk on B d L with periodic, holding and reflective boundary conditions, respectively.For any odd L we have for all x, y ∈ B L .
Proof of Lemma 3.1.Let P denote the transition matrix of simple random walk on Z d , and let P * ,L denote the transition matrix of simple random walk on B d L with * boundary conditions, where * can denote P, R or H, for periodic, reflecting or holding, respectively.Fix a random variable N , and let g denote the two-point function defined in (127) corresponding to P and N .Likewise, let g * ,L denote the two-point function corresponding to P * ,L and N .The corresponding two-point functions defined in Section 3 are the specialisations of g and g * ,L to the case x 0 = 0. We therefore freely omit the first argument of (127) when convenient, with the understanding that in such instances it takes the value 0.
We begin by proving Part (i).Consider the equivalence relation on Z d defined so that to each x ∈ B d L there corresponds the equivalence class It is straightforward to verify that P respects this equivalence relation, and a simple calculation shows that P # ([x], [y]) = P P,L (x, y) for all x, y ∈ B d L .We therefore have for any where the penultimate step follows from Lemma 7.1.This establishes Part (i).where the second step follows from Lemmas 7.2 and 7.3, and the last step follows from Lemma 7.1.This establishes Part (ii).Part (iii) is proved similarly.

Proof of Lemmas 7.2 and 7.3
Recall the definition of [x] L given in (22).If x, y ∈ T d 2L and i ∈ [d], we will write y i ∼ x i iff y i ∈ {x i , −L − x i }, where we emphasise that addition and multiplication are modulo 2L.If y i ∼ x i for all i ∈ [d], then we will also write y ∼ x, so that y ∼ x iff y ∈ [x] L .To avoid confusion, in this section we will denote adjacency between two vertices x, y ∈ T d L by x ↔ y, where the value of L will be clear from the context.
Clearly, z ∈ N(x).Moreover, for all i = k we have so that in either case z k ∼ y k .It then follows that z ∼ y and so z ∈ N(x) ∩ [y] L .In particular, we see that N(x ′ ) ∩ [y] L is nonempty iff N(x) ∩ [y] L is nonempty.
Suppose now that N(x)∩[y] L is nonempty, and define f : The above argument implies that for any z ′ = x ′ + δe k ∈ N(x ′ ) ∩ [y] L , if we define z ∈ N(x) ∩ [y] L as in (136) then and so f is surjective.Now suppose z, z ′ ∈ N(x) ∩ [y] L satisfy f (z) = f (z ′ ).Without loss of generality, suppose z = x + δe k , so that

Figure 1 :
Figure 1: Illustration of the equivalence classes defined by (22) with d = 1.(a) Equivalence classes, [x] L−1 , on T 1 2(L−1) with L = 7.(b) Equivalence classes, [x] L , on T 1 2L 3.2 correspond to distances of the order of √ a L , where a L is the typical scale of N L .Under the assumptions of Proposition 3.2, it follows from Proposition 3.2 and [23, Proposition 3.1] that lim

Figure
Figure 2a shows the simulated results for E(|S|) and E(|T |) at fugacity z = z c (1 − L −λ ) with λ = 1, 9/4, 3.If the boundary of the critical windows for the SAW and Ising model on T d L occur at the square root of the volume, as occurs for the complete graph SAW, then for d = 5 the value λ = 3 > d/2 should lie inside the critical window while λ = 1, 9/4 should lie outside the critical window.Figure 2a is clearly consistent with the conjecture that E(|S|) and E(|T |) scale like L λ when λ < d/2, but like L d/2 for λ > d/2.Likewise, the results in Figure 2b are consistent with the conjecture that var(|S|) and var(|T |) scale like L 2λ when λ < d/2, but like L d for λ > d/2.This behaviour is precisely analogous to complete graph SAW behaviour shown in (12) and (13).Similarly, Figure 3 illustrates the appropriately scaled/standardised distribution functions of |S| and |T | for λ = 1 and λ = 3. Figure 3a strongly suggests for λ = 1 that |S|/E(|S|) and |T |/E(|T |) converge in distribution to a mean-1 exponential random variable, precisely as stated in (14) for the complete graph SAW in the hightemperature scaling window.Likewise, Figure 3b strongly suggests for λ = 3 that (|S| − E(|S|))/ var(|S|) and (|T | − E(|T |))/ var(|T |) converge in distribution to a standardised half-normal distribution, precisely as stated in (15) for the complete graph SAW in the critical window.‡ Although the statement of [23, Proposition 3.1] specifies ξ > 0, its proof also holds when ξ = 0.

Figure 4 :
Figure 4: (a) Two-point functions on five-dimensional tori of SAW at fugacity z L = z c (1 − L −λ ) and λ = 1.The dashed curve corresponds to the ansatz (36) with (25), with constants α, γ set to the values described in the text, with A S,d estimated via simulation.(b) Analogous plot to (a), for Ising case.(c) Two-point functions on five-dimensional tori of SAW at fugacity z L = z c (1 − L −λ ) and λ = 9/4.The dashed curve corresponds to the plateau ansatz (38), with constants α, γ set to the values described in the text, with B S,d and B T ,d estimated via simulation.(d) Analogous plot to (c), with λ = 3.

Table 1 :
Estimated µ values for E(|S|) and E(|T |) on T d L with d = 5 at fugacity z L = z c (1 − L λ ) at various values of λ.Based on the above observations, we conjecture that the following holds for any d ≥ 5 and z