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J. Phys. A: Math. Theor. 41 No 37 (19 September 2008) 372001 (12pp)

doi:10.1088/1751-8113/41/37/372001

FAST TRACK COMMUNICATION

Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential

Yan V Fyodorov¹ and Jean-Philippe Bouchaud²

¹ School of Mathematical Sciences, University of Nottingham, Nottingham NG72RD, UK
² Science & Finance, Capital Fund Management 6-8 Bd Haussmann, 75009 Paris, France

Received 28 May 2008, in final form 19 July 2008
Published 7 August 2008

Abstract. We investigate some implications of the freezing scenario proposed by Carpentier and Le Doussal (CLD) for a random energy model (REM) with logarithmically correlated random potential. We introduce a particular (circular) variant of the model, and show that the integer moments of the partition function in the high-temperature phase are given by the well-known Dyson Coulomb gas integrals. The CLD freezing scenario allows one to use those moments for extracting the distribution of the free energy in both high- and low-temperature phases. In particular, it yields the full distribution of the minimal value in the potential sequence. This provides an explicit new class of extreme-value statistics for strongly correlated variables, manifestly different from the standard Gumbel class.

PACS numbers: 05.40.–a, 75.10.Nr

1. Introduction

The random energy model (REM) introduced by Derrida in [1] is characterized by the partition function $Z_{\beta}=\sum_{i=1}^M\,{\rm e}^{-\beta V_i}$ , where β is the inverse temperature, and V_i for i = 1, ..., M are random variables with the typical variance langle V²_i rangle = V² = O(log M) for M → ∞. Such a model, as well as its generalized version (GREM) [2, 3] continues to play a paradigmatic role in statistical mechanics of disordered systems. Simple enough to allow for a detailed analytical investigation by various methods, the freezing transition exemplified by REM appears to be a rather generic phenomenon. It emerges with surprising regularity in a variety of physical situations, ranging from transparency of random media [4], directed polymers in random environment [5], p-spin glass models and the glass transition [6], random heteropolymers and models of protein folding [7] to properties of quantum particles in a random magnetic field [8, 9], and thermodynamics of a single particle in random Gaussian landscapes [10]. It is also a rich model from purely probabilistic point of view [11], and has an interesting dynamical counterpart: aging [12].

As the low-temperature behaviour in statistical mechanics is obviously controlled by the lowest available energies, it is not surprising that a detailed description of the freezing phenomenon is intimately related to the so-called extreme-value statistics of random variables [13]. For independent, identically distributed variables V_i, the cumulative probability distribution P_m(x) of V_m = min{V₁, ..., V_M} is well understood in mathematical literature and, provided the support of the distribution extends to –∞ but decays faster than any power-law, is given in the limit M 1 by the Gumbel law

where the constants a_M and b_M depend explicitly on the distribution of V_i, but the double-exponential shape of the Gumbel law is very robust (universal). This universality extends to a very broad class of correlated variables provided those correlations decay fast enough, see a detailed description in [14]. For a mathematically rigorous analysis of REM and GREM based, in particular, on the extreme-value statistics, see [15]; see also [16] for recent developments.

A few years ago Carpentier and Le Doussal (CLD) [9] studied a specific case of correlated random variables that is arguably the richest, most challenging, and relevant for applications. More precisely, thinking of the index i as referring to the sites of a certain lattice, the covariance langle V_iV_j rangle considered by CLD depended on the distance d(i, j) between those sites logarithmically. An independent support of the fact that logarithmically correlated potentials should play a special role was obtained recently through a thorough analysis of statistical mechanics of a single particle in high-dimensional random energy landscapes [10].

To understand the extreme-value statistics in the logarithmic case and to relate it to a REM-like freezing transition CLD developed a powerful, albeit non-rigorous, real space renormalization group approach to the distribution ${\cal P}(Z)$ of the partition function Z_β. In this way they discovered that logarithmic models represent, in a sense, continuous analogues of directed polymers on disordered trees, a result somewhat anticipated in [8]. The statistical mechanics of the directed polymer problem on a tree is known to be amenable to a travelling wave analysis, see the celebrated paper [5]. Similarly, for the logarithmic case the Laplace transform of the distribution ${\cal P}(Z)$ was also shown to satisfy a kind of travelling wave equation (see also works [17] for a relation between the travelling waves and the extreme-value statistics, in particular, in the context of the zero temperature directed polymer problem on a tree). Solutions of equations of that type are known to exhibit a characteristic change of the shape at some critical front velocity, and following [5] that change was interpreted by CLD as a signal of a REM-like freezing. In particular, the CLD analysis revealed that such a transition implies, among other properties, a universal non-Gumbel shape of the far-left tail for the cumulative distribution of the minimal value of logarithmically correlated variables: P_m(x → –∞) ≈ 1 – const|x| e^ax which is clearly different from the Gumbel tail P_m(x → –∞) ≈ 1 – const e^ax.

Although CLD's renormalization group is able to predict successfully the universal far-tail features of the distribution expected to be shared by all logarithmically correlated potentials, the calculation of the full distribution for a given potential is beyond the scope of that method. Indeed, the renormalization procedure needs specifying what the authors called a `fusion of environments' rule [9]. As CLD convincingly argue, the precise form of that rule is not important for recovering universal properties. But the actual shape of the travelling wave equation certainly depends on the particular fusion rule employed. As no guiding principle for the fusion rule selection had been provided, a more detailed analysis of specific models appears to be problematic.

The present paper grew out of attempts to overcome the above difficulty. Our main observation is that for a particular variant of the logarithmically-correlated REM a more detailed analysis seems to be possible. In particular, we are able to conjecture the full distribution function P_m(x) pertinent for the case considered. Namely, after appropriate rescaling the cumulative distribution P_m(x) of the minimum turns out to be given by the central result of this work

where K₁(x) stands for the modified Bessel (Macdonald) function, and the parameter β_c is given by the inverse transition temperature in the model. As such, the corresponding distribution is manifestly different from the standard Gumbel double-exponential form.

Our method draws its inspiration from CLD analysis and is essentially based on the pattern of the freezing transition revealed in [5, 9]. However, we do not resort to the renormalization group construction or travelling wave technique, and in this way circumvent the need to know the microscopic fusion rule. Instead, we demonstrate below that assuming CLD freezing scenario provides a way to extend the moments of the partition function from the high-temperature phase to the region below the transition. Roughly speaking, if those moments can be explicitly calculated above the transition—as is the case for both the standard REM as well as our variant of the logarithmic model—they can be used to conjecture the shape of the Laplace transform of the probability density above the transition point. Then the CLD approach can be used to recover the full distribution of the partition function/free energy below the transition, yielding in particular the extreme-value statistics.

The structure of the paper is the following. We start with a short general discussion of CLD freezing scenario, in particular its implications for the moments of the partition function in the low-temperature phase. As an illustration we show how CLD scenario can be used in the standard REM case to recover the exact low-temperature expressions for the partition function moments, obtained long ago by Gardner and Derrida via a rather tedious analysis [3]. After that we introduce and analyse the particular (circular) version of the one-dimensional logarithmically correlated REM.

2. CLD freezing scenario: general relations and implications

The central object of the subsequent analysis is the Laplace-transform G_β(p) of the probability distribution ${\cal P}(Z_{\beta})$ of the partition function: G_β(p) = langle exp{–pZ_β} rangle . Here and henceforth the angular brackets stand for the expectation with respect to the distribution of random variables V_i. Our approach is based on the assumption that such a Laplace transform can be efficiently found in the high-temperature phase.

Following [5, 9] we introduce the variable x via p = e^βx and consider the function $\tilde{G}(x)=\langle\exp\{-{\rm e}^{\beta x}Z_{\beta}\}\rangle$ . Extending the Derrida–Spohn scenario, CLD postulate that in the thermodynamic limit M 1 the latter function has a shape of a travelling wave, that is

where we introduced the parameter L to identify with notations used in [9], and l.o.t. stands for the lower order terms when L → ∞.^Note1 In general, both the travelling wave profile g_β(y) and the wave velocity c(β) depend on the inverse temperature β. The REM-like transition in this approach is described by a `freezing' of both the velocity and the profile function at a certain transition temperature β = β_c so that in the full low-temperature phase β ≥ β_c one has

By the very definition of the function $\tilde{G}(x)$ we then have the following relation for β ≥ β_c:

which fixes the shape of the Laplace transform $G_{\beta\ge \beta_c}(p)=\langle\exp\{-pZ_{\beta}\}\rangle$ in the whole temperature range below the transition point.

Equipped with such a scenario, we will exploit that the knowledge of the Laplace transform allows one to calculate the moments of the partition function below the transition by employing the standard identity:

where Γ(x) stands for the Euler Gamma function, and we used that $x=\frac{1}{\beta}\ln{p}$ .

According to the freezing scenario, the mean value of the free energy in the low-temperature phase is to leading order in L 1 temperature independent, and is given by $\langle F\rangle=-\frac{1}{\beta}\langle \ln{Z_{\beta}}\rangle=-c(\beta_c)L$ . Being interested in the fluctuations of the free energy, we introduce the random variable f = F – langle F rangle whose probability density we denote ${\cal P}_{\beta}(f)$ . After changing the integration variable in (6) to $y=\frac{1}{\beta}\ln{p}+c(\beta_c)L$ , introducing s = –βν and integrating once by parts we observe that (6) takes the following form:

Thus, the only function needed for investigating the low-temperature phase is the shape of the travelling wave profile g_β(y) at the critical point β = β_c. For a general nonzero temperature β_c < β < ∞ one can extract the explicit form of the free-energy distribution ${\cal P}_{\beta}(f)$ by noting that the analytical continuation s → is converts the relation (7) to a Fourier transform which as we shall shortly see can be frequently inverted explicitly. A particular simple relation is obtained in the zero-temperature limit β → ∞ where the free energy simply reduces to the minimum value of all random energies in the sample F → V_m = min_i{V_i}. It is immediately clear from (7) that

yielding a very general relation between the shape of the critical profile $g_{\beta_c}(x)$ and the probability density ${\cal P}_m(x)\equiv -\frac{\rm d}{{\rm d}x}P_m(x)$ of the fluctuations of the extreme values in the sample: ${\cal P}_m(x) =-\frac{\rm d}{{\rm d}x} g_{\beta_c}(x)$ .

Let us briefly demonstrate how this method works for the standard REM with i.i.d. Gaussian sequence of V_i. Introducing the notation $Z^{(0)}={\rm e}^{(1+\frac{\beta^2}{\beta_c^2}) \ln{M} }$ , the analysis of [9] demonstrated that everywhere in the high-temperature phase β ≤ β_c the Laplace transform is given by

Identifying L = ln M as in (3) we find from (9) and the correspondence $G(p)|_{p={\rm e}^{\beta x}}\equiv g_{\beta}(x+c({\beta})L)$ the REM travelling wave profile which is given by

In particular, when approaching the transition point β = β_c the shape and velocity of the travelling wave tends to the limiting values

According to the general discussion, the Laplace transform $\langle{\rm e}^{-pZ_{\beta}}\rangle=G_{\beta}(p)$ of the probability distribution of the partition function in the low-temperature phase β > β_c can be found from

where we again used the correspondence $x=\frac{1}{\beta}\ln{p}$ . In turn, the last expression can be immediately employed to recover the (non-integer) moments of the partition function from the first of relations in (6). Substituting there $G_{\beta\gt \beta_c}(p)$ from (12) yields after a straightforward manipulation

valid as long as $-\nu<\frac{\beta_c}{\beta}$ . The latter moments are, to leading approximation, precisely those obtained by Gardner and Derrida [3], and coincide, as expected from general arguments, with the moments of a totally asymmetric Lévy stable distribution of index β_c/β [13, 15, 18]. Finally, the above moments allow one to recover the distribution of the free-energy fluctuations in the low-temperature phase of the REM, which seems not to be written explicitly in the literature. Namely, making an analytic continuation ν → is/β, and introducing $f=-\frac{1}{\beta}\ln{Z/Z_a}$ we note that (13) takes a form of the Fourier transform of the probability density for f, see (7). Inverting that transform gives

In particular, the zero-temperature limit β → ∞ coincides with the general relation (8) between ${\cal P}_m$ and $g_{\beta_c}$ , as given by equation (11), which immediately yields the famous Gumbel distribution for the minimal energy. It is also evident that the far-left tail f → –∞ of the probability density ${\cal P}^{\rm REM}_{\beta}(f)$ is of the Gumbel form everywhere in the low-temperature phase β > β_c, again expected from general arguments [13, 15, 18].

3. The circular logarithmic REM

3.1. Definition of the model and the moments in the high-temperature phase

Consider the lattice of M points positioned equidistantly at the circumference of a unit circle. Their angular coordinates are given by $\theta_k=\frac{2\pi}{M}k, k=1,2,\ldots,M$ . With each point we associate a Gaussian random variable V_i, with position-independent variance langle V²_i rangle = V² and covariances chosen to be

For the consistency of the procedure we have to choose variance V² in a way ensuring positive definiteness of the full covariance matrix with entries V²δ_kl + (1 – δ_kl)C_kl. The condition amounts to V² > –λ_max, with λ_max being the largest eigenvalue of the matrix $\hat{C}$ with entries C_kl. The matrix $\hat{C}$ is by definition a circulant real symmetric, with zero diagonal. Hence, its eigenvalues are given by λ_q = ∑^M_l = 2 C_1lω^l–1_q, where $\omega_q=\exp \big\{\frac{2\pi{\rm i} q}{M} \big\}$ are roots of Mth degree from unity. The largest eigenvalue corresponds to q = 0, when ω_q = 1. Then we have

$\lambda_{\rm max}=-g^2\sum_{s=1}^{M-1}\ln\bigg\{4\sin^2{\frac{\pi s}{M}}\bigg\} = -g^2\ln\left\{\prod_{s=1}^{M-1}4\sin^2{\frac{\pi s}{M}}\right\}.$

Using the identity: $\frac{M}{2^{M-1}}=\prod_{s=1}^{M-1}\sin\frac{\pi s}{M}$ we see that λ_max = –2g²ln M, so we have to choose the variance to satisfy V² = 2g²ln M + W, with an arbitrary positive W > 0. In what follows we assume that W = O(1) when M 1 and therefore can be safely neglected if we are interested only in the leading terms in the thermodynamic limit.

We define the partition function for our model in the standard way through $Z_{\beta}=\sum_{i=1}^M\,{\rm e}^{-\beta V_i}$ , with the goal to evaluate positive integer moments langle [Z_β]ⁿ rangle in the limit M 1. The expected value of the partition function is obviously independent of the covariance and is given by the standard REM expression, langle Z_β rangle = exp{ln M[1 + β²g²]}, so the first nontrivial moment is

Introducing $\tilde{\theta}_k=\pi k/M$ for k = 1, 2, ..., M we see that the second term in (16) is a kind of Riemann sum, and that in the limit M → ∞ can be approximated as

The first line in (17) corresponds to the convergent integral in the limit M → ∞ which can be easily evaluated by reducing it to the standard Euler's integral of the first kind, see [22], p 898. In contrast, the second line is obtained by extracting the leading term of the divergent integral. Comparing now second `off-diagonal' term in (16) with the first `diagonal' one, we find that for 2β²g² > 1 the diagonal and off-diagonal contributions are of the same order, whereas for 2β²g² < 1 the off-diagonal contribution dominates. Finally, we arrive at

The case of a general positive integer moment can be treated along the same lines. Denoting $x_i={\rm e}^{-\beta V_i}$ we use

where in the second line we regrouped the terms according to partitions of the integer n into the sum of nonnegative integers with length k (i.e. the number of nonzero parts) taking values k = 1, ..., n. For example, partitions of the length k = 1 are sets {p₁, ..., p_M} with all but one p_j equal to zero, and with the remaining nonzero integer taking the value p = n. The total contribution of those partitions is obviously $\sum_{i_1=1}^M x_{i_1}^n$ which is the first term in (19). Similarly, the contribution of all partitions of the length k = 2 is precisely the second term in the above expression, and so on. Finally, we perform the ensemble averaging of the above sum using the identity

valid in the case of all different indices in the set i₁, ..., i_k. In the limit M → ∞ we then find by inspection the dominating terms. After manipulations generalizing those we performed earlier for n = 2 case we find the following general expression:

where

The explicit expression for the factor O(1) in the second line of (21) is rather complicated, but is actually not needed for our purposes. Finally, using the symmetry of the integrand in (22) and noting that $|{\rm e}^{2{\rm i}\theta_p}-{\rm e}^{2{\rm i}\theta_q}|^2=4\sin^2{(\theta_p-\theta_q)}$ one observes that (22) is a particular case of the so-called Morris integral (related to the famous Selberg integral) whose value was first conjectured by Dyson in his studies of the Coulomb gas problem (see a very informative historic account and further references in [19])

The integral is clearly finite provided β²g² < 1/n ≤ 1, and is divergent otherwise. The condition βg < 1 defines the high-temperature phase of the model. Note a certain similarity between our calculations and those arising in the framework of the multifractal random walk model of Bacry, Muzy and Delour [20].

The crucial point of our analysis is the ability to offer the explicit form of the probability density ${\cal P}(Z)$ of the partition function Z_β = Z > 0 which precisely reproduces the expressions for the moments (21). It is given by

where we defined Z_* = e^{2ln M} and introduced for βg < 1 the two functions:

and

To understand the structure of ${\cal P}(Z)$ note that the growth rate of the moments in the second line of equation (21) dictates that the far tail of the distribution must be of a log-normal nature. This is exemplified by the choice (26) for ${\cal P}_{\gt }(Z)$ . On the other hand, the first line in (21) and the expressions (23) yield the probability density of the form ${\cal P}_{<}(Z)$ in equation (25). The crossover value Z = Z_* is determined from the requirement for the leading exponential terms in the two pieces of the probability density to match smoothly, i.e. ${\cal P}_{<}(Z_*)\approx {\cal P}_{\gt }(Z_*)$ for M 1. The factor f(x) in (26) is assumed to be of the order of unity when its argument is of the order of unity, and is otherwise left unspecified. Finally, we verify in the appendix that the choice of P(z) in equations (24)–(26) ensures the required change in moments langle Zⁿ rangle _M1 to occur precisely at n = 1/g²β².

At the next step we use the probability density (24) for evaluating the Laplace transform function G_β(p) in the high-temperature phase. A somewhat lengthy but straightforward calculation reveals that the log-normal tail ${\cal P}_{\gt }(Z)$ gives for M 1 a negligible relative contribution to the Laplace transform, as long as we keep finite the value pZ_e < ∞. Effectively, it means that for our goals we can assume the partition function Z_β to be distributed with the probability density ${\cal P}_{<}(Z_{\beta})$ given in equation (25). After a simple transformation of variables this implies a rather simple asymptotic formula

Using such an expression, we can, for example, easily calculate the mean logarithm of the partition function, hence the mean free energy

$\langle \ln Z\rangle=\lim_{\epsilon\to 0}\left[\Gamma(\epsilon)-\int_0^{\infty}\,{\rm d}p\, p^{\epsilon-1}\langle{\rm e}^{-pZ_{\beta}}\rangle\right]=\ln{Z_e}-a\Gamma^{\prime}(1),$

so that the mean free energy is given by

The leading term yields the expected universal REM expression for the mean free energy valid in the high-temperature phase, the rest corresponds to system-specific corrections. Those corrections diverge logarithmically when approaching the critical temperature β = 1/g = β_c, signalling of the phase transition. Note that the same result for the free energy can be recovered by the standard replica trick using moments (21).

The fluctuations of the free energy around its mean value can be easily recovered as well, using the explicit form of the distribution ${\cal P}_{<}(Z_{\beta})$ . Namely, introducing

equation (25) implies the following probability density in the high-temperature phase β < β_c:

According to our previous discussion, a central role is played by $\tilde{G}(x)=G_{\beta}(p={\rm e}^{\beta x})$ . Identifying L = ln M we observe that $\tilde{G}(x)\equiv g_{\beta}(x+m_L)$ where $m_L= \frac{1}{\beta}\ln{Z_e}$ . Using equation (28) we further see that

$m_L|_{L\gg 1}\approx \frac{1+a}{\beta}L+O(1)=c(\beta) L+O(1), \qquad c(\beta)=\frac{1}{\beta}+\frac{\beta}{\beta_c^2},$

again in full agreement with CLD results in the high-temperature phase, with c(β) interpreted as the travelling wave velocity, and the wavefront profile given by

3.2. Transition to the low-temperature phase in the circular logarithmic REM

To investigate the low-temperature phase for β ≥ β_c we rely upon the CLD freezing scenario. When approaching the transition point a = β²/β²_c = 1 the profile (31) tends to a well-defined limit

where K₁(x) is the modified Bessel (Macdonald) function. According to the freezing arguments this shape via relation (8) is translated into the extreme-value probability density (2), which is our central result. This expression is non-Gumbel as the cumulative distribution behaves for x → –∞ as $P^{\rm CLM}_m(x)\approx 1+\beta_c x\, {\rm e}^{\frac{\beta_c x}{2} }$ in full agreement with the analysis of [9]. The opposite tail for x → ∞ has a generalized Gumbel-like shape $P^{\rm CLM}_m(x)\propto \exp \big\{\frac{\beta_c x}{4}-2\, {\rm e}^{\frac{\beta_c x}{2}} \big\}$ . Note a certain similarity of these two asymptotes to those of the probability density for the magnetization in the low-temperature phase of the XY model [21].

Moreover, for all temperatures below the transition β ≥ β_c the value of the leading term in m_L and the shape of the wavefront profile should be frozen to the critical point values, i.e. those for β = β_c. Thus, to the leading order m_L(β > β_c) = c(β_c)L ≡ m*_L,^Note2 whereas the profile g_β(y) is given by equation (32) for any β > β_c. In the same way as in the REM case this fact allows one to extract the moments of the partition function everywhere in the low-temperature phase when $\tilde{G}(x)=g_{\beta_c}(x+m_L^*)$ . Employing now the critical profile shape equation (32) and substituting $x=\frac{1}{\beta}\ln{p}$ we recover the Laplace transform G_β(p) of the probability density of the partition function below the transition

This gives us the possibility of calculating negative moments of the partition function as

where we have used the identity [22]

Substituting here the explicit values of γ and b, and changing ν → –ν we finally get

Although we used ν < 0 in the course of derivation, a slight modification of the above procedure, see [3], shows that the above expression is valid in a wider region, as long as ν < β_c/β < 1.

The mean value of the free energy F in the low-temperature phase is found in a similar way and the leading order term is simply langle F rangle = –β^–1 langle ln Z rangle = –m*_L. Introducing the probability density ${\cal P}_{\beta}(f)$ of f = F + m*_L we can now rewrite equation (36) as

In particular, similarly to the REM case after an analytic continuation s → is the probability density of the free energy for the circular logarithmic model (CLM) can be extracted for any β > β_c by inverting the corresponding Fourier transform. The corresponding formula takes a form of an infinite series

where ψ(x) = Γ '(x)/Γ(x). Exploiting the series expansion for the Macdonald function, see, e.g. p 909 of [22], it is easy to check that in the zero temperature limit β → ∞ the free-energy distribution indeed reduces to the extreme-value probability density of form equation (2) in full agreement with the general relation (8). Equation (39) shows that the same non-Gumbel behaviour holds for the far-left tail f → –∞ of the free-energy distribution at any temperature below the transition.

3.3. Conclusion, discussions and open problems

In the present paper, we attempted to investigate some implications of the CLD freezing scenario [9] for a particular type of REM-like model with logarithmically correlated random potential on a circle. The chosen model seems to be especially attractive due to relatively simple expressions for the integer moments of the partition function in the high-temperature phase, given by the well-known Dyson Coulomb gas integral. We argue that in such a case the Laplace transform of the probability density of the partition function can be efficiently recovered. When combined with the freezing scenario this knowledge allows us to continue the Laplace transform to the low-temperature phase. We first check that the method indeed works for the standard REM example by recovering the well-known, yet nontrivial Gardner–Derrida formulae [3] for the moments of the partition function below the freezing point. The same method is then applied to the logarithmic model in question. In particular, we are able to recover the full distribution of the lowest minimum in the potential, equation (2), and this extreme-value statistics is manifestly non-Gumbel.

Although we think our results are supported by rather convincing arguments, the calculations are very essentially based on a few plausible but not yet fully verified assumptions. As such, mathematically our conclusions have the status of well-grounded conjectures. It would be certainly very desirable to find alternative ways of investigating the model, as well as to perform accurate numerical verification of the precise form of the extreme-value statistics. Another open problem is the universality of our result, equation (2), for logarithmically correlated random variables, in particular the shape of the right tail (see [17]). We hope our results provide enough incentive for further research in this direction.

Finally, it might be useful to provide an alternative view on our choice of the logarithmically correlated potential, equation (15). By employing the known identity: $-\ln{\big(4\sin^2\frac{x_1-x_2}{2}\big)}=2\sum_{l=1}^{\infty}\frac{1}{l}\cos{l(x_1-x_2)}$ we see that the covariance function (15) represents, in fact, a 2π-periodic real-valued Gaussian random process $V(x)=\sum_{l=1}^{\infty}(v_l\, {\rm e}^{{\rm i}lx}+\bar{v}_l\, {\rm e}^{-{\rm i}lx})$ with a self-similar spectrum $\langle v_l \bar{v}_m \rangle=g^2 l^{-(2H+1)}\delta_{lm}$ characterized by the particular choice of the Hurst exponent H = 0. Such a process therefore represents a version of the so-called 1/f noise. To this end it is worth mentioning that the extreme-value statistics of the `roughness' associated with 1/f noise was investigated in [23], and found to be of Gumbel form.

Acknowledgments

YF acknowledges support of this work by the Leverhulme Research Fellowship project `A single particle in random energy landscapes'. Discussions with M Feigelman S Nechaev V Vargas and V Yudson are acknowledged with thanks.

Appendix.

The positive integer moments of the distribution ${\cal P}(Z)$ , see Equations (24)–(26), are given in the high-temperature phase βg < 1 by the sum of two terms

The first contribution corresponding to equation (25) is given by

In the limit ln M → ∞ we have from (24) and (25) $B\propto{\rm e}^{-\frac{1}{\beta^2g^2}(1-\beta^2g^2)\ln{M}}\to 0$ in view of β²g² < 1. After a simple calculation we find

As to the second contribution, a saddle-point analysis justified by ln M 1 shows that:

Comparing the two contributions m⁽ⁿ⁾_> and m⁽ⁿ⁾_< within the high-temperature phase βg < 1 we see that

(1)

m⁽ⁿ⁾_< m⁽ⁿ⁾_> as long as $1<n< \frac{1}{\beta^2 g^2}$ . Indeed

$n(1+\beta^2g^2)-\left(1+2n-\frac{1}{\beta^2 g^2}\right)=(1-\beta^2g^2)\left(\frac{1}{\beta^2 g^2}-n\right)\gt 0$

which implies

$m^{(n)}_{<}\sim Z_e^n\sim{\rm e}^{n(1+\beta^2g^2)\ln{M}}\gg{\rm e}^{(1+2n-\frac{1}{\beta^2 g^2})\ln{M}}\sim m^{(n)}_{\gt }.$

(2)

If $n\gt \frac{1}{\beta^2 g^2} \mbox{ we have } m^{(n)}_{<}\ll m^{(n)}_{\gt }$ , as in this case

$m^{(n)}_{<}\sim Z_e^{\frac{1}{\beta^2g^2}}Z_{<}^{n-\frac{1}{\beta^2g^2}}\sim{\rm e}^{(1+2n-\frac{1}{\beta^2g^2})\ln{M}}\ll m^{(n)}_{\gt }\sim{\rm e}^{(1+\beta^2 g^2n^2)\ln{M}},$

which follows from

$(1+\beta^2 g^2n^2)-\left(1+2n-\frac{1}{\beta^2g^2}\right)=\left(\beta g n-\frac{1}{\beta g} \right)^2\gt 0.$

Accounting for equation (23) and the definition of Z_e in equation (25) we indeed see that the moments langle

Zⁿ

coincide for ln M

1 with the expressions for the partition function moments (21).

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Notes

Note1: Establishing the precise form of those terms, both above and below the transition, was one of the central points of CLD analysis. It is, however, of no direct relevance for us in the present paper.
Note2: As O(1) terms in m_L above the transition diverge logarithmically when β → β_c it is natural to expect that at the transition point they should be replaced with constln L. Actually, the analysis of [9] predicts the precise value const = 1/2 at the transition point. Unfortunately, the verification of this interesting prediction goes beyond the precision of our analysis.

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