Fluctuations of the partition function in the generalized random energy model with external field

Fluctuations of the partition function in the generalized random

energy model with external field

Citation for published version (APA):

Bovier, A., & Klymovskiy, A. (2008). Fluctuations of the partition function in the generalized random energy model with external field. Journal of Mathematical Physics, 49(12), 125202-1/27.

https://doi.org/10.1063/1.2962982

DOI:

10.1063/1.2962982 Document status and date: Published: 01/01/2008 Document Version:

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Fluctuations of the partition function in the generalized

random energy model with external field

Anton Bovier1,2,a兲 and Anton Klimovsky2,b兲 1

Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany

2

Institut für Mathematik, Technische Universität Berlin, Straße des 17 Juni 136, 10623 Berlin, Germany

共Received 12 May 2008; accepted 26 June 2008; published online 3 December 2008兲

We study Derrida’s generalized random energy model共GREM兲 in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of param-eters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the ther-modynamic limit tend to have a certain optimal magnetization, as prescribed by the strength of the external field and by parameters of the GREM. © 2008 American Institute of Physics. 关DOI:10.1063/1.2962982兴

I. INTRODUCTION

Despite the recent substantial progress due to Guerra,1Aizenman et al.,2,3 and Talagrand4in establishing rigorously the Parisi formula for the free energy of the celebrated Sherrington– Kirkpatrick共SK兲 model, understanding of the corresponding limiting Gibbs measure is still very limited.

Due to the above mentioned works, it is now rigorously known that the generalized random energy model共GREM兲 introduced by Derrida5is closely related to the SK model at the level of free energy, see, e.g., Bovier6 Sec. 11.3. Recently Bovier and Kurkova7–9 have performed a detailed study of the geometry of the Gibbs measure for the GREM. This confirmed the predicted in the theoretical physics literature hierarchical decomposition of the Gibbs measure in rigorous terms.

As pointed out by Bovier and Kurkova7共see also Ben Arous et al.10兲, the GREM-like models may represent an independent interest in various applied contexts, where correlated heavy-tailed inputs play an important role, e.g., in risk modeling.

One of the key steps in the results of Bovier and Kurkova7 is the identification of the fluc-tuations of the GREM partition function in the thermodynamic limit with Ruelle’s probability cascades. In this paper we also perform this step and study the effect of external field on the fluctuations共i.e., the weak limit laws兲 of the partition function of the GREM in the thermodynamic limit. We find that the main difference introduced by the presence of external field, comparing to the system without external field, is that the coarse graining mechanism should be altered. This change reflects the fact that the coarse-grained parts of the system tend to have a certain optimal magnetization as prescribed by the strength of the external field and by parameters of the GREM. We use the general line of reasoning suggested by Bovier and Kurkova,7i.e., we consider the point

a兲_{Electronic mail: bovier@wias-berlin.de.} b兲_{Electronic mail: klimovsk@math.tu-berlin.de.}

49, 125202-1

processes generated by the scaling limits of the GREM Hamiltonian. We streamline the proof of the weak convergence of these point processes to the corresponding Poisson point process by using the Laplace transform.

A. Organization of the paper

In the following subsections of the Introduction we define the model of interest and formulate our main results on the fluctuations of the partition function of the random energy model共REM兲 and GREM with external field and also on their limiting free energy共Theorems 1.1–1.4兲. Their proofs are provided in the subsequent sections. Section II is devoted to the large deviation results providing an expression for the free energy for a class of models with Gaussian Hamiltonians and external field 共Theorem 2.1兲. In Sec. III we resort to more refined analysis and perform the calculations of the fluctuations of the ground state and of the partition function in the REM with external field in the thermodynamic limit. Section VI contains the proofs of the results on the fluctuations of the ground state and of the partition function for the GREM with external field. B. Definition of the model

Derrida’s GREM was proposed as a mean-field spin-glass model with a Gaussian Hamiltonian and hierarchical correlation structure. In this paper, we consider the GREM with uniform external 共magnetic兲 field. In contrast to the work of Derrida and Gardner,11

we consider here the model with the external field which depends linearly on the total magnetization共i.e., the uniform mag-netic field兲. Derrida and Gardner11

considered the “lexicographic” external field which is particu-larly well adapted to the natural lexicographic distance generated by the GREM Hamiltonian.

Given N苸N, consider the standard discrete hypercube ⌺N⬅兵−1;1其N. It will play the role of

the index set. Define the 共normalized兲 lexicographic overlap between the configurations

␴共1兲_,␴共2兲_苸⌺ Nas qL共␴共1兲,␴共2兲兲 ⬅

冦

0, ␴₁共1兲⫽␴₁共2兲 1 Nmax兵k 苸 关1;N兴 艚 N:关␴ 共1兲_兴 k=关␴共2兲兴k其, otherwise.

冧

共1.1兲 We equip the index set with the lexicographic distance defined as

dL共␴共1兲,␴共2兲兲 ⬅ 1 − qL共␴共1兲,␴共2兲兲.

This distance is obviously an ultrametric, that is, for all␴共1兲,␴共2兲,␴共3兲苸⌺N, we have

dL共␴共1兲,␴共3兲兲 ⱕ max兵dL共␴共1兲,␴共2兲兲,dL共␴共2兲,␴共3兲兲其.

Throughout the paper, we assume that we are given a large enough probability space共⍀,F,P兲 such that all random variables under consideration are defined on it. Without further notice, we shall assume that all Gaussian random variables共vectors and processes兲 are centered.

Let GREMN⬅兵GREMN共␴兲其␴苸⌺Nbe the Gaussian random process on the discrete hypercube

⌺Nwith the covariance of the following form:

E关GREMN共␴共1兲兲GREMN共␴共2兲兲兴 =␳共qL共␴共1兲,␴共2兲兲兲, 共1.2兲

where ␳:关0;1兴→关0;1兴 is the nondecreasing right-continuous function such that ␳共0兲=0 and

␳共1兲=1. Given h苸R+, consider the Gaussian process X⬅XN⬅兵XN共h,␴兲其␴苸⌺Ndefined as

XN共h,␴兲 ⬅ GREMN共␴兲 +

h

冑

N

兺

i=1

N

␴i, ␴苸 ⌺N. 共1.3兲

The second summand in共1.3兲is called the external field. The parameter h represents the strength of external field. Denote the total magnetization by

mN共␴兲 ⬅

1 N

兺

_i=1

N

␴i, ␴苸 ⌺N. 共1.4兲

The random process共1.3兲induces the Gibbs measureGN共␤, h兲苸M1共⌺N兲 in the usual way,

GN共␤,h兲共兵␴其兲 ⬅

1 ZN共␤兲

exp关␤

冑

NXN共␤−1h,␴兲兴,

where the normalizing constant ZN共␤兲 is called the partition function ZN共␤, h兲 and is given by the

following sum of 2N_{correlated exponentials:}

ZN共␤,h兲 ⬅

兺

␴苸⌺Nexp关␤

冑

NXN共h,␴兲兴. 共1.5兲

The real parameter␤⬎0 is called the inverse temperature. The important quantities are the free energy defined as

pN共␤,h兲 ⬅

1

Nlog ZN共␤,h兲, 共1.6兲

and the ground state energy

MN共h兲 ⬅ N−1/2max

␴苸⌺NXN共h,␴兲. 共1.7兲

In what follows, we shall think of␤and h as fixed parameters. We shall occasionally lighten our notation by not indicating the dependence on these parameters explicitly.

In this paper we shall mainly be interested in the weak limit theorems共i.e., fluctuations兲 of the partition function 共1.5兲and of the ground state as N↑ +⬁. To be precise, the general results on Gaussian concentration of measure imply that 共1.7兲and共1.6兲are self-averaging. By the fluctua-tions of the ground state, we mean the weak limiting behavior of the rescaled point process generated by the Gaussian process共1.3兲. This behavior is studied in Theorems 1.1 and 1.2 below. These theorems readily imply the formulas for the limiting free energy共1.6兲and the ground state

共1.7兲. A recent account of the mathematical results on the GREM without external field and, in particular, on the behavior of the limiting Gibbs measure can be found in the paper of Bovier and Kurkova.12The GREM with external field was previously considered by Jana and Rao13共see also Jana14兲, where its free energy was expressed in terms of a variational problem induced by an application of Varadhan’s lemma. In this work, we apply very different methods to obtain precise control of the fluctuations of the partition function for the GREM with external field. As a simple consequence of these results, we also get a rather explicit1formula for the limiting free energy in the GREM with external field共see Theorem 1.4兲.

C. Main results

In this paper, we shall consider the case of the piecewise constant function ␳ with a finite number of jumps. Consider the space of discrete order parameters,

Qn

⬘

⬅ 兵q:关0;1兴 → 关0;1兴兩q共0兲 = 0,q共1兲 = 1,

q is nondecreasing, piecewise constant with n jumps其. 共1.8兲 Recall the function␳ from共1.2兲. Assume that␳苸Qn

⬘

. In what follows, we shall refer to␳ as the

discrete order parameter. In this case, it is possible to construct the process GREMN as a finite

sum of independent Gaussian processes. Assume that

␳共x兲 =

兺

k=1 n qk1_关xk;xk+1兲共x兲, 共1.9兲 where 0⬅ x₀⬍ x₁⬍ ¯ ⬍ xn= 1, 共1.10兲 0⬅ q0⬍ q1⬍ ¯ ⬍ qn= 1. 共1.11兲 Let 兵ak其k=1 n _{傺R be such that a} k 2_{= q}

k− qk−1. We assume that, for all k苸关1;n兴艚N, we have xkN

苸N2

and also ak⫽0. Denote ⌬xl⬅xl− xl−1.

Consider the family of independently and identically distributed standard Gaussian random variables,

兵X共␴共1兲_,␴共2兲_{, . . . ,␴}共k兲_{兲兩k 苸 关1;n兴 艚 N,␴}共1兲_{苸 ⌺}

x₁N, . . . ,␴共k兲苸 ⌺x_kN其.

Using these ingredients, for␴=␴共1兲 储␴共2兲 储. . .储␴共n兲苸⌺N, we have

GREMN共␴兲 ⬃

兺

k=1 n

akX共␴共1兲,␴共2兲, . . . ,␴共k兲兲. 共1.12兲

Equivalence 共1.12兲 is easily verified by computing the covariance of the right hand side. The computation gives, for␴,␶苸⌺N,

Cov关GREMN共␴兲GREMN共␶兲兴 = qNqL共␴,␶兲.

D. Limiting objects

We now collect the objects which appear in weak limit theorems for the GREM partition function and for the ground states. We denote by I :关−1;1兴→R+Cramér’s entropy function, i.e., I共t兲 ⬅1₂关共1 − t兲log共1 − t兲 + 共1 + t兲log共1 + t兲兴. 共1.13兲 Define

␮共t兲 ⬅

冑

2共log 2 − I共t兲兲, M共h兲 ⬅ max

t苸关−1;1兴共␮共t兲 + ht兲. 共1.14兲

Suppose that the maximum in共1.14兲is attained at t = t_ⴱ= t_ⴱ共h兲. 关The maximum exists and is unique, since␮共t兲+ht is strictly concave.兴 Consider the following two real sequences:

AN共h兲 ⬅ 共␮共tⴱ兲

冑

N兲−1, 共1.15兲 BN共h兲 ⬅ M共h兲

冑

N + AN共h兲 2 log

冉

AN共h兲2共I

⬙

共tⴱ兲 + h兲 2␲共1 − t_ⴱ2兲

冊

. 共1.16兲 Define the REM scaling function uN,h共x兲:R→R as

2_{This condition is for notational simplicity. It means that we actually consider instead of N the increasing sequence} 兵N␣其␣苸N傺N such that N␣↑ +⬁ as␣↑ +⬁, satisfying N␣xk苸N, for all␣苸N and all k苸关1;n兴艚N.

uN,h共x兲 ⬅ AN共h兲x + BN共h兲. 共1.17兲

Given f : D傺R→R+, we denote by PPP共f共x兲dx,x苸D兲 the Poisson point process with intensity f. We start from a basic limiting object. Assume that the point processP共1兲onR satisfies

P共1兲_{⬃ PPP共exp共− x兲dx,x 苸 R兲 共1.18兲}

and is independent of all random variables around. The point process共1.18兲is the limiting object which appears in the REM.

Theorem 1.1: If n = 1 (the REM case), then, using the above notations, we have

兺

␴苸⌺N␦uN,h −1_{共XN共h,␴兲兲}→ N→⬁ w P共1兲_{, 共1.19兲}

where the convergence is the weak one of the random probability measures equipped with the vague topology.

Remark 1.1: It is easy to check that, for h = 0, Theorem 1.1 reduces to the results of Bovier and Kurkova.15 Indeed, in this case t_ⴱ共0兲=0, and, hence, ␮共0兲=M共0兲=

冑

2 log 2, which in turn implies that the scaling constants共1.15兲and共1.16兲coincide with that of the REM without external field.

To formulate the weak limit theorems for the GREM 共i.e., for the case n⬎1兲, we need a limiting object which is a point process closely related to the Ruelle probability cascade 共Ruelle16_{兲. Define, for j,k苸关1;n+1兴艚N, j⬍k, the “slopes” corresponding to the function ␳}

in 共1.2兲as ␪j,k⬅ qk− qj−1 xk− xj−1 . Define also the following h-dependent “modified slopes”:

␪

˜

j,k共h兲 ⬅␪j,k␮共tⴱ共␪j,k

−1/2_h兲兲−2_{. 共1.20兲}

Define the increasing sequence of indices兵Jl共h兲其_l=0 m_共h兲

傺关0;n+1兴艚N by the following algorithm. Start from J0共h兲⬅0 and define iteratively

Jl共h兲 ⬅ min兵J 苸 关Jl−1;n + 1兴 艚 N:˜␪J_l−1,J共h兲 ⬎˜␪J+1,k共h兲, for all k ⬎ J其. 共1.21兲

Note that m共h兲ⱕn. The subsequence of indices共1.21兲induces the following coarse graining of the initial GREM: q ¯l共h兲 ⬅ qJ_l共h兲− qJ_l−1共h兲, 共1.22兲 x ¯l共h兲 ⬅ xJ_l共h兲− xJ_l−1共h兲, 共1.23兲 ␪ ¯ l共h兲 ⬅␪J_l−1,J_l. 共1.24兲

The parameters共1.22兲induce the new order parameter␳共J共h兲兲苸Qm

⬘

in the usual way,

␳共J共h兲兲_{共q兲 ⬅}

兺

l=1 m共h兲

qJ_l共h兲1关x_J

l共h兲;xJl+1共h兲兲共x兲.

uN,␳,h共x兲 ⬅

兺

l=1 m_共h兲

关q¯l共h兲1/2Bx¯_l共h兲N共¯␪l共h兲−1/2h兲兴 + N−1/2x.

Define the rescaled GREM process as

GREMN共h,␴兲 ⬅ uN,␳,h

−1 _共GREM

N共h,␴兲兲.

Define the point process of the rescaled GREM energiesEN as

EN共h兲 ⬅

兺

␴苸⌺N␦GREMN共h,␴兲

. 共1.25兲

Consider the following collection of independent point processes共which are also independent of all random objects introduced above兲:

兵P_␣ 1,. . .,␣l−1 共k兲 _兩␣ 1, . . . ,␣l−1苸 N;l 苸 关1;m兴 艚 N其, such that P_␣ 1,. . .,␣k−1 共k兲 _{⬃ P}共1兲_.

Define the limiting GREM cascade point processPmonRmas follows:

Pm⬅

兺

␣苸Nm ␦共P共1兲_共␣ 1兲,P␣1 共2兲_共␣ 2兲,. . .,P␣1,␣2,. . .,␣m−1 共m兲 _共␣ m兲兲. 共1.26兲

Consider the following constants:

␥

¯l共h兲 ⬅ 共␪˜J_l−1,Jl兲 1/2_, and define the function Eh,f:Rm→R as

E_h,共m兲_␳共e₁, . . . ,em兲 ⬅␥¯1共h兲e1+ ¯ +¯␥m共h兲em.

Note that due to 共1.21兲, the constants 兵␥¯l共h兲其l=1 m

form a decreasing sequence, i.e., for all l 苸关1;m兴艚N, we have

␥

¯l共h兲 ⬎␥¯l+1共h兲. 共1.27兲

The cascade point process 共1.26兲 is the limiting object which describes the fluctuations of the ground state in the GREM.

Theorem 1.2: We have EN共h兲 → N↑+⬁ w

冕

Rm ␦E h,
共m兲_共e

1,. . .,em兲Pm共de1, . . . ,dem兲 共1.28兲 and

MN共h兲 → N↑+⬁

兺

_l=1

m共h兲

关共q¯l共h兲x¯l共h兲兲1/2M共¯␪l共h兲−1/2h兲兴, 共1.29兲

almost surely and in L1_.

Theorem 1.2 allows for complete characterization of the limiting distribution of the GREM partition function. To formulate the result, we need the ␤-dependent threshold l共␤, h兲 苸关0;m兴艚N such that above it all coarse-grained levels l⬎l共␤, h兲 of the limiting GREM are in the “high temperature regime.” Below this threshold the levels lⱕl共␤, h兲 are in the “frozen state.” Given␤苸R₊, define

l共␤,h兲 ⬅ max兵l 苸 关1;n兴 艚 N:␤␥¯l共h兲 ⬎ 1其.

We set l共␤, h兲⬅0, if␤␥¯₁共h兲ⱕ1. The following gives full information about the limiting fluctua-tions of the partition function at all temperatures.

Theorem 1.3: We have

exp关−␤

冑

N

兺

l=1 l共␤,h兲

共q¯l共h兲1/2B¯x_l共h兲N共␪¯l−1/2h兲兲兴exp

冋

− N

冉

log 2 + log ch共␤h共1 − xJ_l共␤,h兲兲兲

+1 2␤ 2_{共1 − q} J_l共␤,h兲兲

冊

册

ch2/3共␤h共1 − xJ_l共␤,h兲兲兲ZN共␤,h兲 → N↑+⬁ w K共␤,h,␳兲

冕

Rl共␤,h兲

exp关␤E_h,共l共␤,h兲兲_␳ 共e1, . . . ,el共␤,h兲兲兴Pl共␤,h兲共de1, . . . ,del共␤,h兲兲, 共1.30兲

where the constant K共␤, h ,␳兲 depends on␤, h, and␳ only. Moreover, K共␤, h ,␳兲=1, if␤␥l共␤,h兲+1

⬍1 and K共␤, h ,␳兲苸共0;1兲, if␤␥l共␤,h兲+1= 1.

The above theorem suggests that the increasing sequence of the constants 兵␤l

⬅␥¯−1_其

l=1 m共h兲_傺R

+can be thought of as the sequence of the inverse temperatures at which the phase transitions occur: at␤lthe corresponding coarse-grained level l of the GREM with external field

“freezes.”

As a simple consequence of the fluctuation results of Theorem 1.3, we obtain the following formula for the limiting free energy of the GREM.

Theorem 1.4: We have lim N↑+⬁ pN共␤,h兲 =␤

兺

l=1 l共␤,h兲 关共x¯l¯ql兲1/2␮共tⴱ共¯␪l −1/2_{h兲兲 + hx¯} ltⴱ共¯␪l −1/2_{h兲兴 + log 2 + log ch共␤h共1 − x} J_l共␤,h兲兲兲 +1 2␤ 2_{共1 − q} J_l共␤,h兲兲, 共1.31兲

almost surely and in L1.

Remark 1.2: For h = 0, since␮共t_ⴱ共0兲兲=␮共0兲=

冑

2 log 2 (see Remark 1.1), we have

␪

˜

j,k共h兲 =

␪j,k

2 log 2,

which together with共1.21兲recovers the coarse-graining algorithm of Bovier and Kurkova7共1.15兲 for the GREM without external field.

II. PARTIAL PARTITION FUNCTIONS, EXTERNAL FIELDS, AND OVERLAPS

In this section, we propose a way to compute the free energy of disordered spin systems with external field using the restricted free energies of systems without external field. The computation involves a large deviations principle. For gauge invariant systems, we also show that the partition function of the system with external field induced by the total magnetization has the same distri-bution as the one induced by the overlap with fixed but arbitrary configuration. This section is based on the ideas of Derrida and Gardner.11

Fix p苸N. Given some finite interaction p-hypergraph 共VN, EN

共p兲_{兲, where V}

N=关1;n兴艚N and

XN共␴兲 ⬅

兺

i_苸EN共p兲

Ji共N,p兲␴i₁␴i₂¯␴i_p, ␴苸 ⌺N, 共2.1兲

where J共N,p兲⬅兵J_i共N,p兲其i苸E

N

共p兲is the collection of random variables having the symmetric joint distri-bution. That is, we assume that, for any␧共1兲,␧共2兲苸兵−1; +1其E_N共p兲_{and any t苸R}E_N共p兲_,

E

冋

exp

冉

i

兺

r_苸EN共p兲

tr␧r共1兲Ji共N,p兲

冊

册

=E

冋

exp

冉

i

兺

r_苸EN共p兲

tr␧r共2兲Ji共N,p兲

冊

册

, 共2.2兲

where i苸C denotes the imaginary unit.

A particular important example of共2.1兲is Derrida’s p-spin Hamiltonian given by

SKN共p兲共␴兲 ⬅ N−p/2

兺

i₁,. . .,ip=1

N

gi₁,. . .,i_p␴i₁␴i₂¯␴i_p,

where兵gi₁,. . .,i_p其i₁,. . .,i_p=1

N

is a collection of independently and identically distributed standard Gauss-ian random variables. Note that the condition共2.2兲is obviously satisfied.

Given␮苸⌺N, define the corresponding gauge transformation T␮:⌺N→⌺Nas

T_␮共␴兲i=␮i␴i, ␴苸 ⌺N. 共2.3兲

Note that the gauge transformation 共2.3兲 is obviously an involution. We say that a d-variate random function f :⌺N

d_{→R is gauge invariant, if, for any␮苸⌺}

Nand any共␴共1兲, . . . ,␴共d兲兲苸⌺N d

, f共T_␮共␴共1兲兲, ... ,T_␮共␴共d兲兲兲 ⬃ f共␴共1兲, . . . ,␴共d兲兲,

where ⬃ denotes equality in distribution. Define the overlap between the configurations ␴,␴

⬘

苸⌺Nas RN共␴,␴

⬘

兲 ⬅ 1 N

兺

_i=1 N ␴i␴i

⬘

. 共2.4兲

Note that the overlap共2.4兲and the lexicographic overlap共1.1兲are gauge invariant. Given a bounded function FN:⌺N→R, define the partial partition function as

ZN共p兲共␤,q,␧,XN,FN兲 ⬅

兺

␴:兩FN共␴兲−q兩ⱕ␧ exp共␤

冑

NXN共␴兲兲. 共2.5兲 Denote UN⬅ FN共⌺N兲, U ⬅

冉

艛 N=1 ⬁ UN

冊

. 共2.6兲

关The bar in 共2.6兲denotes closure in the Euclidean topology.兴 Note that for the case FN= RN we

obviously have

UN=

再

1 −

2k

N:k苸 关0;N兴 艚 Z

冎

, U =关− 1;1兴.

Proposition 2.1 (Ref.11): Assume that XNis given either by共2.1兲or XN⬃GREMN. Fix some

gauge invariant bivariate function FN:⌺N2→R, and q苸R.

Then, for all␴

⬘

,␶

⬘

苸⌺N, we have

ZN共p兲共␤,q,␧,XN,FN共·,␴

⬘

兲兲 ⬃ ZN共p兲共␤,q,␧,XN,FN共·,␶

⬘

兲兲. 共2.7兲

the partial partition function which corresponds to fixing the total magnetization共1.4兲, i.e., ZN共p兲共␤,q,␧,XN,RN共·,␴

⬘

兲兲 ⬃ ZN共p兲共␤,m,␧,兲 ⬅

兺

␴:兩m共␴兲−q兩⬍␧

exp共␤

冑

NXN共␴兲兲.

Remark 2.1: The proposition obviously remains valid for the Hamiltonians XN given by the

linear combinations of the p-spin Hamiltonians共2.1兲with varying p苸N. Proof:

共1兲 If XN is defined by 共2.1兲, then 共2.7兲follows due to the gauge invariance of 共2.1兲 and FN.

Indeed, there exists␮苸⌺Nsuch that␴

⬘

= T␮共␶

⬘

兲. Define

Ji共N,p,␮兲⬅ Ji共N,p兲␮i₁¯␮i_p.

Due to the symmetry of the joint distribution of J共N,p兲, we have 兵XN共␴兲其␴苸⌺_N⬃ 兵XN共␴兲兩J共N,p兲=J共N,p,␮兲其␴苸⌺_N, which implies共2.7兲.

共2兲 If XN= GREMN, then, since XNis a Gaussian process, to prove the equality in distribution, it

is enough to check that the covariance of XNis gauge symmetric. Equivalence共2.7兲follows

due to共1.2兲and the fact that the lexicographic overlap共1.1兲is gauge invariant.

䊐 The partial partition function共2.5兲induces the restricted free energy in the usual way,

pN共p兲共␤,q,␧,XN,FN兲 ⬅ 1 Nlog ZN 共p兲_{共␤,q,␧,X} N,FN兲. 共2.8兲 Given␴共1兲,␴共2兲苸⌺N, let CN共␴共1兲,␴共2兲兲 ⬅ E关XN共␴共1兲兲XN共␴共2兲兲兴, C˜N共␴共1兲兲 ⬅ CN共␴共1兲,␴共1兲兲. Define VN⬅ 兵CN共␴,␴兲:␴苸 ⌺N其, V ⬅

冉

艛 N=1 ⬁ VN

冊

.

The following result establishes a large deviation-type relation between the partial free energy and the full one.

Theorem 2.1: Assume XN=兵XN共␴兲其␴苸⌺_N is a centered Gaussian process and FN:⌺N→R are

such that, for all N , M苸N, all␴共1兲,␴共2兲苸⌺N, and all␶共1兲,␶共2兲苸⌺M,

C_N+M共␴共1兲 储␶共1兲,␴共2兲 储␶共2兲兲 ⱕ N N + MCN共␴ 共1兲_,␴共2兲_{兲 +} M N + MCM共␶ 共1兲_,␶共2兲_{兲, 共2.9兲} FN+M共␴共1兲 储␶共1兲兲 ⱕ N N + MFN共␴ 共1兲_{兲 +} M N + MFM共␶ 共1兲_{兲. 共2.10兲}

Assume that CNand FNare bounded uniformly in N.

共1兲 The following holds:

pN共␤,XN,FN兲 ⬅

1

Nlog_␴苸⌺N

兺

exp共␤

冑

NXN共␴兲 + NFN共␴兲兲 →

N↑+⬁

共2兲 The limiting free energy p共␤, X , F兲 is almost surely deterministic. 共3兲 We have lim ␧↓+0Nlim_↑+⬁ pN共p兲共␤,q,␧,XN,FN兲 ⬅ p共p兲共␤,q,X,F兲 = sup v苸V␭苸R,␥苸Rinf 共− ␭q −␥v + p共␤,X,␭F +␥C˜ 兲兲, almost surely and in L1_{. 共2.12兲} 共4兲 Finally,

p共␤,X,F兲 = sup

q苸U共p

共p兲_{共␤,q,X,F兲 + q兲. 共2.13兲}

Remark 2.2:

共1兲 If there exists 兵constN苸R+其N=1⬁ such that, for all␴苸⌺N,

C ˜

N共␴兲 = constN, 共2.14兲

then共2.12兲simplifies to p共p兲共␤,q,X,F兲 = inf

␭苸R共− ␭q + p共␤,X,␭F兲兲, almost surely and in L

1_{. 共2.15兲} 共2兲 Inequality 共2.10兲 can alternatively be substituted by the assumption (see Guerra and Toninelli17Theorem 1) that FN共␴兲= f共SN共␴兲兲, where f :R→R, f 苸C1共R兲, and SN:⌺N→R is

the bounded function such that, for all␴苸⌺N,␶苸⌺M,

SN+M共␴储␶兲 =

N

N + MSN共␴兲 + M

N + MSM共␶兲.

共3兲 It is easy to check that the assumptions of Proposition 2.1 are fulfilled, e.g., for XN⬅ c1GREMN+ c2SKN共p兲,

and

FN共·兲 ⬅ f1共RN共·,␴共N兲兲兲 + f2共qL共·,␴共N兲兲兲,

where␴共N兲苸⌺N, c1, c2苸R, and f1, f2:R→R, such that f1苸C1共R兲, f2is convex. Note that in this case, due to Proposition 2.1, the free energies共2.11兲 and共2.12兲do not depend on the choice of the sequence兵␴共N兲其_N=1⬁ 傺⌺N.

Proof: Similarly to Contucci et al.18 Theorem 1 and Guerra and Toninelli17 Theorem 1, we obtain共2.11兲. Then共2.11兲implies that

p共␤,XN,␭FN+␥C˜N兲 →

N_↑+⬁p共␤,X,␭F +␥C˜ 兲, almost surely and in L

1_.

Hence, we can apply the quenched large deviation results Bovier and Klimovsky19Theorems 3.1 and 3.2 which readily yield共2.13兲and共2.12兲关or 共2.15兲兴, in the case of共2.14兲. 䊐 Remark 2.3: Derrida and Gardner11 sketched a calculation of the free energy defined in 共2.11兲in the following case:

FN= qL and XN= GREMN. 共2.16兲

This case is easier than the case共1.6兲we are considering here, since both qLand GREMNhave

III. THE REM WITH EXTERNAL FIELD REVISITED

In this section, we recall some known results on the limiting free energy of the REM with external field. However, we give some new proofs of these results which illustrate the approach of Sec. II. Moreover, we prove the weak limit theorem for the ground state and for the partition function of the REM with external field.

Recall that the REM corresponds to the case n = 1 in共1.12兲. This implies that the process X is simply a family of 2N_{independently and identically distributed standard Gaussian random}

vari-ables. To emphasize this situation we shall write REM共␴兲 instead of GREM共␴兲. A. Free energy and ground state

Let us start by recalling the following well-known result on the REM.

Theorem 3.1 (Derrida,20 Eisele,21 and Olivieri and Picco22): Assume that n = 1 and let p共␤, h兲 be given by共1.6兲. The following assertions hold.

共1兲 We have lim N→⬁ pN共␤,0兲 =

冦

␤2 2 + log 2, ␤ⱕ

冑

2 log 2, ␤

冑

2 log 2, ␤ⱖ

冑

2 log 2,

冧

almost surely and in L1. 共3.1兲 共2兲 For all␤ⱖ

冑

2 log 2 and N苸N, we have

0ⱕ E关pN共␤,0兲兴 ⱕ␤

冑

2 log 2. 共3.2兲

See, e.g., Bovier6Theorem 9.1.2 for a short proof.

Given k苸关0;N兴艚N, define the set of configurations having a given magnetization

⌺N,k⬅

再

␴苸 ⌺N:

兺

i=1 N

␴i= N − 2k

冎

. 共3.3兲

Lemma 3.1: Set tk,N⬅N−2k/N. Given any ␧⬎0, uniformly in k苸关0;N兴艚N such that

t_k,N苸 关− 1 + ␧;1 − ␧兴, we have the following asymptotics:

冉

N k

冊

N=_↑+⬁

冑

2 ␲ 2Ne−NI共tk,N兲

冑

N共1 − t_k,N2 兲

冉

1 + 1 N

冉

1 12+ 1 3共1 − tk,N 2 _兲

冊

+O

冉

1 N2

冊

冊

. 共3.4兲

Proof: A standard exercise on Stirling’s formula. 䊐

Theorem 3.2 (Dorlas and Wedagedera23): Assume that n = 1 (the REM case) and let p共␤, h兲 be given by共1.6兲. We have p共␤,h兲 ⬅ lim N→⬁ pN共␤,h兲 =

冦

log 2 + log ch␤h + ␤2 2 , ␤ⱕ

冑

2共log 2 − I共tⴱ兲兲 ⬅␤0

␤共

冑

2共log 2 − I共tⴱ兲兲 + htⴱ兲, ␤ⱖ

冑

2共log 2 − I共tⴱ兲兲

冧

, almost surely and in L1,

共3.5兲 and t_ⴱ苸共−1;1兲 is a unique maximizer of the following concave function:

Proof: For the sake of completeness, we give a short proof based on共the ideas of兲 Theorem 2.1. Put Mk,N⬅

冦

log2

冉

N k

冊



, k苸 关1;N − 1兴 艚 N, 1, k苸 兵0,N其,

冧

where
x denotes the largest integer smaller than x. Consider the free energy 关cf. 共2.8兲兴 of the REM of volume M_k,N, p_k,N共␤兲 ⬅ 1 M_k,Nlog

兺

␴苸⌺N k exp共␤M_k,N1/2REM共␴兲兲,

where REM⬅兵REM共␴兲其␴苸⌺N is the family of standard independently and identically distributed Gaussian random variables. Let

p ˜_k,N共␤兲 ⬅Mk,N N pk,N

冉冉

N M_k,N

冊

1/2 ␤

冊

. 共3.6兲

Note that共3.6兲is the restricted free energy关cf.共2.8兲兴 of the REM, where the restriction is imposed by the total magnetization共1.4兲given by tk,N.

We claim that the family of functions P⬅兵E关pN共·兲兴其N苸N is uniformly Lipschitzian. Indeed,

uniformly in␤ⱖ0, we have

⳵␤E关pN共␤兲兴 = N−1/2E关GN共␤,0兲关XN共␴兲兴兴 ⱕ N−1/2E

关

max

␴苸⌺NX共␴兲

兴

N→_↑+⬁

冑

2 log 2.

Hence, the familyP has uniformly bounded first derivatives.

Given t苸共−1;1兲 and tk_N,N苸UN 关cf.共2.6兲兴 such that limN↑+⬁tk_N,N= t, using共3.4兲, we have lim

N↑+⬁

Mk_N,N

N = 1 − I共t兲log2e. 共3.7兲

Using共3.7兲and the uniform Lipschitzianity of the familyP, we get lim N_↑+⬁ p ˜k_N,N共␤兲 = 共1 − I共t兲log2e兲p

冉

␤

冑

1 − I共t兲log2e

冊

. 共3.8兲

Combining共3.8兲with共2.13兲and共2.15兲, we get p共␤,h兲 = max

t苸关−1;1兴

再

t␤h +共1 − I共t兲log2e兲p

冉

␤

冑

1 − I共t兲log₂e

冊

冎

. 共3.9兲 To find the maximum in共3.9兲, we consider two cases.

共1兲 If␤ⱕ

冑

2共log 2−I共t_ⴱ兲兲, then according to共3.1兲, we have

p

冉

_冑

␤ 1 − I共t兲log₂e

冊

= log 2 + ␤2 2共1 − I共t兲log2e兲 . Hence,共3.9兲implies p共␤,h兲 = max t_{苸关−1;1兴}

再

t␤h + ␤2

2 + log 2 − I共t兲

冎

= log 2 + log ch␤h +

␤2

2 , 共3.10兲 where the last equality is due to the fact that the expression in the curly brackets is concave and, hence, the maximum is attained at a stationary point. The stationarity condition reads

t = t0共␤,h兲 ⬅ tanh␤h. 共3.11兲 It is easy to check that the following identity holds:

I共t兲 = t tanh−1t − log ch tanh−1t. 共3.12兲 Combining共3.12兲and共3.11兲, we get 共3.10兲.

共2兲 If␤ⱖ

冑

2共log 2−I共t_ⴱ兲兲, then again by 共3.1兲, we have

p

冉

_冑

␤ 1 − I共t兲log2e

冊

=

_冑

␤

冑

2 log 2 1 − I共t兲log2e . Hence,共3.9兲transforms to p共␤,h兲 = max t苸关−1;1兴兵t␤

h +␤

冑

2共log 2 − I共t兲兲其 =␤共

冑

2共log 2 − I共t_ⴱ兲兲 + ht_ⴱ兲, 共3.13兲 where the last equality is due to the concavity of the expression in the curly brackets.

Combining共3.10兲and共3.13兲, we get 共3.5兲. 䊐

Remark 3.1: We note that due to the continuity of the free energy as a function of␤, we have at the freezing temperature␤0,

t0共␤0,h兲 = tⴱ共h兲. 共3.14兲

Theorem 3.2 suggests that the following holds.

Theorem 3.3: Under the assumptions of Theorem 3.2, we have lim

N_↑+⬁

1

冑

N␴苸⌺NmaxXN共h,␴兲 =

冑

2共log 2 − I共tⴱ兲兲 + htⴱ, almost surely and in L

1_{. 共3.15兲} Proof: We have 1 ␤pN共␤兲 ⱕ 1 Nlog

冉

N␤

冑

N max␴苸⌺NXN共h,␴兲

冊

= log N ␤N + 1

冑

N␴苸⌺NmaxXN共h,␴兲. 共3.16兲 In view of共3.5兲, relation共3.16兲readily implies that

冑

2共log 2 − I共t_ⴱ兲兲 + ht_ⴱⱕ lim

N_↑+⬁ N−1/2 max ␴苸⌺NXN共h,␴兲. 共3.17兲 We also have 1 ␤pN共␤兲 ⱖ 1

冑

N␴苸⌺Nmax XN共h,␴兲, which combined again with共3.5兲implies that

冑

2共log 2 − I共t_ⴱ兲兲 + ht_ⴱⱖ lim

N↑+⬁

N−1/2 max

␴苸⌺NXN共h,␴兲. 共3.18兲

Due to the standard concentration of Gaussian measure关e.g., Ledoux24共2.35兲兴 and the fact that the free energy共1.6兲is Lipschitzian with the constant␤

冑

N as a function of XN共h,·兲 with respect to the

Euclidean topology, the bounds共3.17兲and共3.18兲combined with the Borell–Cantelli lemma give

the convergence共3.15兲. 䊐

B. Fluctuations of the ground state

In this subsection, we shall study the limiting distribution of the point process generated by the properly rescaled process of the energy levels, i.e.,共1.25兲.

Proof of Theorem 1.1: Let us denote EN共h兲 ⬅

兺

␴苸⌺N␦uN,h

−1_{共XN共h,␴兲兲}. 共3.19兲

We treatEN共h兲 as a random pure point measure on R. Given some test function␸苸C0+共R兲 共i.e., a non-negative function with compact support兲, consider the Laplace transform of 共3.19兲 corre-sponding to␸, L_EN共h兲共␸兲 ⬅ E

冋

exp

再

−

兺

␴苸⌺N␸共uN,h −1_共X N共h,␴兲兲兲

冎

册

=

兿

k=0 N

再

1 2␲

冕

_Rexp

冉

−␸

冉

uN,h −1

冉

x +

_冑

h N共N − 2k兲

冊冊

− x2 2

冊

dx

冎

共

N k

兲

. 共3.20兲

Introduce the new integration variables y = u_N,h−1共x+h/

冑

N共N−2k兲兲. We have 共3.20兲 =

兿

k=0 N

再

AN共h兲 2␲

冕

_Rexp

冉

−␸共y兲 − 1 2

冉

uN,h共y兲 − h

冑

N共N − 2k兲

冊

2

冊

dy

冎

共

N k

兲

= exp

再

兺

k=0 N

冉

N k

冊

log

冉

1 − AN共h兲

冑

2␲

冕

R 共1 − e−␸共y兲_兲exp

冋

₋1 2

冉

uN,h共y兲 − h

冑

N共N − 2k兲

冊

2

册冊

冎

. 共3.21兲 Note that the integration in 共3.21兲 is actually performed over y苸supp␸, since the integrand is zero on the complement of the support. It is easy to check that uniformly in y苸supp␸ the integrand in共3.21兲and, hence, the integral itself is exponentially small共as N↑ +⬁兲. Consequently, we have 共3.21兲 = N_↑+⬁ exp

再

−

冕

supp␸ 共1 − e−␸共y兲_兲

冉

兺

k=0 N

冉

N k

冊

AN共h兲

冑

2␲ exp

冋

− 1 2

冉

uN,h共y兲 − h

冑

N共N − 2k兲

冊

2

册

冊

⫻共1 + o共1兲兲

冎

. 共3.22兲

Denote t_k,N⬅N−2k/N. Using Lemma 3.1, we get

共3.22兲 = N_↑+⬁ exp

再

−共1 + o共1兲兲

冕

supp␸ 共1 − e−␸共y兲_兲

冉

兺

k=0 N AN共h兲 ␲共N共1 − tk,N

2 _兲兲1/2exp

冋

N共log 2 − I共tk,N兲兲

−1

2共uN,h共y兲 − htk,N

冑

N兲

2

册

冊

冎

_{. 共3.23兲}

Note that despite the fact that Lemma 3.1 is valid only for t_k,N苸关−1+␧;1−␧兴, we can still write

共3.23兲, since the both following sums are negligible:

0ⱕ

兺

k:tk,N苸共关−1;−1+␧兴艛关1−␧,1兴兲

AN共h兲

␲共N共1 − tk,N

2 _兲兲1/2exp

冋

N共log 2 − I共tk,N兲兲 −

1

2共uN,h共y兲 − htk,N

冑

N兲 2

册

ⱕ KN exp共− LN兲

0ⱕ

兺

k:tk,N苸共关−1;−1+␧兴艛关1−␧,1兴兲

冉

N k

冊

AN共h兲

冑

2␲ exp

冋

− 1 2共uN,h共y兲 − htk,N

冑

N兲 2

册

_{ⱕ KN exp共− LN兲.} Consider the sum appearing in共3.23兲,

SN共h,y兲 ⬅

兺

k=0 N

AN共h兲

␲共N共1 − tk,N

2 _兲兲1/2exp

冋

N共log 2 − I共tk,N兲兲 −

1

2共uN,h共y兲 − htk,N

冑

N兲

2

册

_{. 共3.24兲} Introduce the functions fN, gN:关−1;1兴→R as

fN共t兲 ⬅ I共t兲 + 1 2

冉

uN,h共y兲

冑

N − ht

冊

2 − log 2, gN共t兲 ⬅ AN共h兲 ␲共N共1 − t2_兲兲1/2. Note that definition共1.14兲implies

I

⬘

共t_ⴱ兲 = h␮共t_ⴱ兲. 共3.25兲

A straightforward computation using共1.15兲,共1.16兲, and 共3.25兲gives

fN

⬙

共t兲 = I

⬙

共t兲 + h ⬎ 0, 共3.26兲 fN

⬘

共tⴱ兲 = − h 共2␮共tⴱ兲N兲

冋

2y + log

冉

I

⬙

共tⴱ兲 + h 4␲共1 − t_ⴱ2兲共log 2 − I共t_ⴱ兲兲N

冊

册

= O

冉

log N N

冊

, 共3.27兲 fN共tⴱ兲 = − 1 N

冋

y + 1 2log

冉

I

⬙

共t_ⴱ兲 + h 4␲共1 − t_ⴱ2兲共log 2 − I共t_ⴱ兲兲N

冊

册

+ o

冉

1 N

冊

. 共3.28兲

Hence, since 共3.27兲 vanishes even after being multiplied by

冑

N, 共3.27兲 is negligible for the purposes of the asymptotic Laplace principle. This readily implies that uniformly in y苸supp␸,

SN共h,y兲 ⬃ N_↑+⬁ NgN共tⴱ兲 2

冉

2␲fN

⬙

共tⴱ兲 N

冊

1/2 exp关NfN共tⴱ兲兴. 共3.29兲

Using共3.26兲–共3.28兲in the right hand side of共3.29兲, we obtain that uniformly in y苸supp␸, SN共h,y兲 ⬃

N_↑+⬁

exp共− y兲. 共3.30兲

Finally, combining共3.30兲and共3.23兲, we obtain

lim

N_↑+⬁

L_EN共h兲共␸兲 = exp

再

−

冕

R

共1 − e−␸共y兲_兲e−y_dy

冎

_{. 共3.31兲} The right hand side of共3.31兲is the Laplace transform of PPP共e−x_{dx , x苸R兲. Then a standard result}

implies the claim共1.19兲. 䊐

C. Fluctuations of the partition function

In this subsection, we compute the weak limiting distribution of the partition function under the natural scaling induced by共1.17兲. Define

CN共␤,h兲 ⬅ exp

冋

␤M共h兲N +

␤

2␮共t_ⴱ兲log

冉

I

⬙

共t_ⴱ兲 + h

4␲共1 − t_ⴱ2兲共log 2 − I共t_ⴱ兲兲N

冊

册

, 共3.32兲 DN共␤,h兲 ⬅ ch−2/3共␤h兲exp

冋

N

冉

log 2 + log ch␤h +

␤2 2

冊

册

, 共3.33兲 ␣共␤,h兲 ⬅ ␤ ␮共tⴱ兲. 共3.34兲 Theorem 3.4: If␤⬎␮共t_ⴱ兲, then ZN共␤,h兲 CN共␤,h兲 → N↑+⬁ w

冕

R e␣共␤,h兲xdP共1兲共x兲. 共3.35兲 If␤⬍␮共t_ⴱ兲, then ZN共␤,h兲 DN共␤,h兲 → N↑+⬁ w 1. 共3.36兲

Proof: This is a specialization of Theorem 1.3 which is proven in Sec. IV B. 䊐 IV. THE GREM WITH EXTERNAL FIELD

In this section, we obtain the main results of the paper concerning the GREM with external field. We prove the limit theorems for the distribution of the partition function and that of the ground state. As a simple consequence of these fluctuation results, we obtain an explicit formula for the free energy of the GREM with external field.

A. Fluctuations of the ground state

As in the REM, we start from the ground state fluctuations共cf. Theorem 1.1兲. The following is the main technical result of this section that shows exactly in which situations the GREM with external field has the same scaling limit behavior as the REM with external field.

Proposition 4.1: Either of the following two cases holds. 共1兲 If, for all l苸关2;n兴艚N,

log 2 − I共tⴱ共␪l,n −1/2_h兲兲 log 2 − I共t_ⴱ共h兲兲 ⬍␪l,n, 共4.1兲 then we have

兺

␴苸⌺N␦uN,h −1_{共XN共h,␴兲兲}→ N↑+⬁ w PPP共e−x_{,x苸 R}d_{兲. 共4.2兲}

共2兲 If, for all l苸关2, ... ,n兴艚N,

log 2 − I共t_ⴱ共␪l,n−1/2h兲兲

log 2 − I共t_ⴱ共h兲兲 ⱕ␪l,n, 共4.3兲

and there exists (at least one) l0苸关2;n兴艚N, log 2 − I共tⴱ共␪l−1/2₀,nh兲兲

log 2 − I共tⴱ共h兲兲 =␪l0,n, 共4.4兲

兺

␴苸⌺N␦uN,h −1_{共XN共h,␴兲兲}→ N_↑+⬁ w PPP共Ke−x,x苸 R兲. 共4.5兲

Remark 4.1: If condition共4.3兲is violated, i.e., there exists l₀苸关2;n兴艚N such that log 2 − I共t_ⴱ共␪_l,n−1/2h兲兲

log 2 − I共tⴱ共h兲兲 ⬎␪l,n, 共4.6兲

then the REM scaling [cf.共1.19兲 and共1.17兲] is too strong to reveal the structure of the ground state fluctuations of the GREM. Theorem 1.2 shows how the scaling and the limiting object should be modified to capture the fluctuations of the GREM in this regime.

Proof:

共1兲 Denote Nl⬅⌬xlN, for l苸关1;n兴. We fix arbitrary test function ␸苸CK+共R兲, i.e., a non-negative function with compact support. Consider the Laplace transform L_EN共h兲共␸兲 of the random measureEN共h兲 evaluated on the test function␸:

L_EN共h兲共␸兲 ⬅ E

冋

exp

冉

−

兺

␴苸⌺N共␸ⴰ uN,h −1_兲共X N共h,␴兲兲

冊

册

=E

冋

兿

␴苸⌺N exp共− 共␸ⴰ u_N,h−1兲共XN共h,␴兲兲兲

册

. 共4.7兲 Consider also the family of independently and identically distributed standard Gaussian random variables,

兵X共␴共l兲_,␴共2兲_{, . . . ,␴}共n兲_{兲兩l 苸 关1;n兴 艚 N,␴}共l兲_{苸 ⌺}

N_l, . . . ,␴共n兲苸 ⌺N_n其.

Given l苸关1;n兴艚N and y苸R, define LN共l,v兲 ⬅ E

冋

兿

␴共l兲储. . .储␴共n兲苸⌺_{共1−xl−1兲N} exp共−␸ⴰ uN,h−1共v + alX共␴共l兲兲 + ¯ + anX共␴共l兲, . . . ,␴共n兲兲 + h共1 − xl−1兲

冑

Nm共␴共l兲, . . . ,␴共n兲兲兲兲

册

. 共4.8兲 We readily have L_EN共h兲共␸兲 = LN共1,0兲. 共4.9兲

Due to the treelike structure of the GREM, for l苸关1;n−1兴艚N, we have the following recursion: LN共l,v兲 =

兿

␴共l兲苸⌺N_l

E关LN共l + 1,v + alX + h⌬xl

冑

Nm共␴共l兲兲兲兴, 共4.10兲

where X is a standard Gaussian random variable. Introduce the following quantities: YN共h,y,v,t,l兲 ⬅ uN,h共y兲 − h共1 − xl−1兲

冑

Nt −v.

We claim that, for any l苸关1;n兴艚N, uniformly in v苸R satisfying vⱕ

冑

N共M共h兲 −␦−共1 − ql−1兲␮共h兲 − h共1 − xl−1兲tⴱ共␪l,n

−1/2_h兲兲, we have

log LN共l,v兲 ⬃ N↑+⬁ −

_冑

AN共h兲 2␲共1 − ql−1兲

兺

k=0 共1−xl−1兲N

冉

冉

共1 − xl−1兲N k

冊

⫻

冕

R 共1 − e−␸共y兲_兲exp

冋

₋ 1 2共1 − q_l−1兲YN共h,y,v,tk,共1−xl−1兲N,l兲 2

册

_dy

冊

_{. 共4.11兲} We shall prove共4.11兲by a decreasing induction in l starting from l = n.

共2兲 The base of induction is a minor modification of the proof of Theorem 1.1. By the definition共4.8兲and independence, we have

LN共n,v兲 =

兿

k=0 N_n 共E exp关− 共␸ⴰ uh,N −1_兲共a nX + h⌬xn

冑

Ntk,N_n+v兲兴兲

冉

N_n k

冊

. 共4.12兲 For fixed k苸关0;Nn兴艚Z, E关exp共− 共␸ⴰ uN,h −1_兲共a nX + h⌬xn

冑

Ntk,N+v兲兲兴 =共2␲兲−1/2

冕

R dx exp关− x2/2 − 共␸ⴰ u_N,h−1兲共anx + h⌬xn

冑

Ntk,N+v兲兴. 共4.13兲

We introduce in共4.13兲the new integration variable,

y⬅ u_N,h−1共anx + h⌬xn

冑

Ntk,N_n+v兲. 共4.14兲

Using the change of variables共4.14兲, we get that the right hand side of共4.13兲is equal to AN共h兲

冑

2␲an

冕

R

dy exp

冋

− 1 2an

2YN共h,y,v,tk,N_n,n兲2−␸共y兲

册

. 共4.15兲

Combining共4.12兲and共4.15兲, we get

LN共n,v兲 =

兿

k=0 N_n

冉

AN共h兲

冑

2␲an

冕

R dy exp

冋

− 1 2an 2YN共h,y,v,tk,N_n,n兲2−␸共y兲

册

冊

冉

N_n k

冊

=

兿

k=0 N_n

冉

1 −

_冑

AN共h兲 2␲an

冕

R dy共1 − e−␸共y兲兲exp

冋

− 1 2an 2YN共h,y,v,tk,N_n,n兲2

册

冊

冉

N_n k

冊

. Define VN共h,v,t,n兲 ⬅ AN共h兲

冑

2␲an

冕

R dy共1 − e−␸共y兲兲exp

冋

− 1 2an 2YN共h,y,v,t,n兲2

册

.

Given any small enough␦⬎0, it straightforward to show that uniformly in v苸R such that vⱕ

冑

N共M共h兲 − h⌬xntⴱ共h␪n−1,n−1/2兲 −␦兲, we have LN共n,v兲 = N↑+⬁

兿

_k=0 N_n

冉

1 −

冉

Nn k

冊

VN共h,v,tk,Nn,n兲

冊

共1 + O共e −CN_{兲兲. 共4.16兲} Indeed, we have

exp

冋

− 1 2an

2YN共h,y,v,tk,N_n,n兲2

册

ⱕ exp关− Nn共log 2 − I共tn兲兲兴.

Next, using the fact that共1−e−␸共·兲兲苸C_K+共R兲, we get for some C⬎0,

冕

R

dy共1 − e−␸共y兲兲exp

冋

− 1 2an2

YN共h,y,v,tk,N_n,n兲2

册

ⱕ C exp关− Nn共log 2 − I共tn兲兲兴.

Applying the elementary bounds

x − x2_{ⱕ log共1 + x兲 ⱕ x for 兩x兩 ⬍}1 2 共4.17兲 to x⬅ −

_冑

AN共h兲 2␲an

冕

R dy共1 − e−␸共y兲兲exp

冋

− 1 2an 2YN共h,y,v,tk,N_n,n兲2

册

,

and using the fact that, due to共3.4兲, there exists C⬎0 such that uniformly in k苸关1;Nn兴艚N,

x2ⱕ exp关− 2Nn共log 2 − I共tn兲兲兴

冉

Nn

k

冊

ⱕ e −CN_,

we get共4.16兲and, consequently,共4.11兲holds for l = n.

共3兲 For simplicity of presentation, we prove only the induction step l=n
l=n−1. Due to

共4.10兲, we have LN共n − 1,v兲 =

兿

k_n−1=0 N_n−1 E关LN共n,v + an−1X + h⌬xn−1

冑

Ntk_n−1,N_n−1兲兴

冉

N_n−1 k_n−1

冊

. 共4.18兲 Define t共kn,kn−1兲 ⬅ 1 1 − x_l−2共⌬xntkn,Nn+⌬xn−1tkn−1,Nn−1兲.

Fix an arbitrary ␦⬎0 and ␧⬎0. Due to 共4.11兲 with l = n, there exists some C⬎0, such that uniformly for all kn, kn−1with

tk_n,k_n−1苸 兵t 苸 关− 1;1兴:兩tⴱ共␪n−1,n

−1/2 _{h兲 − t}

k_n,k_n−1兩 ⱕ ␧其, and uniformly for allv , x苸R satisfying

⌬xn共log 2 − I共tk_n,N_n兲兲 ⱕ 1 2an2 共M共h兲 −␦− an−1x − N−1/2v − h共⌬xntk_n,N_n+⌬xn−1tk_n−1,N_n−1兲兲2, 共4.19兲 we obtain 兩log LN共n,v + an−1x + h⌬xn−1

冑

Ntk,N_n−1兲兩 ⱕ CN exp共− N/C兲. 共4.20兲 Define xN共v兲 ⬅

冑

N an−1 共M共h兲 −␦−vN−1/2− an共2⌬xn共log 2 − I共tk_n,N_n兲兲兲1/2− h共⌬xntk_n,N_n+⌬xn−1tk_n−1,N_n−1兲兲. 共4.21兲 Using the elementary bounds

1 + xⱕ exⱕ 1 + x + x2 for 兩x兩 ⬍ 1, 共4.22兲 and the bound共4.20兲, we obtain

E关1兵XⱕxN共v兲其LN共n,v + an−1X + h⌬xn−1

冑

Ntkn−1,Nn−1兲兴 =

N_↑+⬁P兵X ⱕ xN共v兲其 + E关1兵XⱕxN共v兲其

log LN共n,v + an−1X + h⌬xn−1

冑

Ntk_n−1,N_n−1兲兴 + O共N exp共− N/C兲兲. 共4.23兲 Given kn−1苸关1;Nn−1兴艚N, we have E

冋

1_{兵XⱕxN共v兲其}exp

冉

− 1 2an2 YN共h,y,v + an−1X + h⌬xn−1

冑

Ntk_n−1,N_n−1,tk_n,N_n,n兲2

冊

册

=

_冑

1 2␲

冕

−⬁ xN共v兲 dx exp

冋

−x 2 2 − 1 2an 2共uN,h共y兲 − an−1x − h

冑

N共⌬xntk_n,Nn+⌬xn−1tkn−1,Nn−1兲 − v兲 2

册

= exp

冉

− 1 1 − qn−2 YN共h,y,v,t共kn,kn−1兲,n − 1兲2

冊

⫻

_冑

1 2␲

冕

−⬁ xN共v兲 exp

冋

−an 2 + a_n−12 2an 2

冉

x − a_n−1 an 2 + a_n−12 YN共h,y,v,t共kn,kn−1兲,n − 1兲

冊

2

册

dx. 共4.24兲 We claim that due to the strict inequalities共4.1兲, we have

1

冑

2␲

冕

−⬁ xN共v兲 exp

冋

−an 2_{+ a} n−1 2 2an2

冉

x − an−1 an2+ an−12 YN共h,y,v,t共kn,kn−1兲,n − 1兲

冊

2

册

dx→ N↑+⬁ an 共an2+ an−12 兲1/2 , 共4.25兲 uniformly inv苸R such that

vⱕ

冑

N共M共h兲 +␦

⬘

− h共⌬xntk_n,N_n+⌬xn−1tk_n−1,N_n−1兲 − 共an

2

+ a_n−12 兲␮共h兲兲 ⬅ vN

max

, 共4.26兲 where 0⬍␦

⬘

exists due to strict inequality共4.1兲, for l = n. Indeed, due to the standard bounds on Gaussian tails, to show共4.25兲it is enough to check that

a_n−1 an

2

+ a_n−12 YN共h,y,v,t共kn,kn−1兲,n − 1兲 +␦

冑

Nⱕ xN共v兲, 共4.27兲 forv satisfying共4.26兲. Due to共4.1兲with l = n, there exists␦3⬎0 such that we have

共2⌬xn共log 2 − I共tk_n,N_n兲兲兲1/2ⱕ␮共h兲 −␦3. 共4.28兲 Choosing a small enough␦

⬘

⬎0, we have

xN共v兲 − an−1 an 2 + a_n−12 YN共h,y,v,t共kn,kn−1兲,n − 1兲 +␦

冑

N = an2共M共h兲 − vN−1/2− h共⌬xntk_n,N_n+⌬xn−1tk_n−1,N_n−1兲兲 −共an 2

+ a_n−12 兲共an共2⌬xn共log 2 − I共tk_n,N_n兲兲兲1/2−␦兲 ⱖ

4.26

an2共共an2+ a2n−1兲␮共h兲 −␦

⬘

兲 − 共an2+ an−12 兲共an共2⌬xn共log 2 − I共tk_n,N_n兲兲兲1/2−␦兲 ⱖ 4.28an 2_共共a n 2 + a_n−12 兲␮共h兲 −␦

⬘

兲 − 共an 2 + a_n−12 兲共an 2_{␮共h兲 −␦兲}

=共an 2 + a_n−12 兲共␦3an 2 +␦兲 − an 2_␦

_⬘

⬎ 0, which proves共4.27兲.

We claim that there exists C⬎0 such that uniformly in kn−1苸关1;Nn−1兴艚N and in v苸R

satisfying共4.26兲we have

冉

Nn−1

kn−1

冊

P兵X ⱖ xN共v兲其 ⱕ exp共− N/C兲. 共4.29兲

Indeed, in view of共3.4兲and due to the classical Gaussian tail asymptotics, to obtain共4.29兲it is enough to show that

Nn−1共log 2 − I共tk_n−1,N_n−1兲兲 ⱕ 1 2xN 2_共v N max_{兲. 共4.30兲}

Using共4.26兲and共4.21兲, we obtain

xN共vNmax兲 =

N1/2 an−1

共共an2+ an−12 兲␮共h兲 − an共2⌬xn共log 2 − I共tk_n,Nn兲兲兲

1/2_+␦

_⬘

_{−␦兲. 共4.31兲} If n⬎2, then due to strict inequality 共4.1兲, for l = n − 2, there exists␦

⬙

⬎0 such that we have

共an

2

+ a_n−22 兲␮共h兲 −␦

⬙

⬎ 共共log 2 − I共t_ⴱ共␪_l,n−1/2h兲兲兲共an

2

+ a_n−22 兲共⌬xn+⌬xn−1兲兲1/2

ⱖ 共2an−12 ⌬xn−1共log 2 − I共tk_n−1,Nn−1兲兲兲 1/2_+共2a n 2_⌬x n共log 2 − I共tk_n,Nn兲兲兲 1/2_, 共4.32兲 where the last inequality may be obtained as a consequence of Slepian’s lemma.25 If n = 2, then

共4.32兲follows directly from Slepian’s lemma. Combining 共4.31兲and共4.32兲, we get共4.30兲. Note that共4.29兲, in particular, implies that

P兵X ⱖ xN共v兲其 ⱕ exp共− N/C兲. 共4.33兲

Given kn−1苸关1;Nn−1兴艚N, denote

LN共n − 1,v,kn−1兲 ⬅ E关LN共n,v + an−1X + h⌬xn−1

冑

Ntk_n−1,N_n−1兲兴

冉

N_n−1 k_n−1

冊

.

Due to共4.33兲 and共4.23兲, we have

LN共n − 1,v,kn−1兲 = E关共1兵XⱕxN共v兲其+1兵X⬎xN共v兲其兲LN共n,v + an−1X + h⌬xn−1

冑

Ntk_n−1,N_n−1兲兴

冉

N_n−1 k_n−1

冊

=共1 + E关1_兵Xⱕx N共v兲其LN共n,v + an−1X + h⌬xn−1

冑

Ntkn−1,Nn−1兲兴 +O共P兵X ⱖ xN共v兲其 + N exp共− N/C兲兲兲

冉

N_n−1 k_n−1

冊

.

Using共4.29兲and the standard bounds共4.17兲and共4.22兲, we get

LN共n − 1,v,kn−1兲 = exp

再

冉

Nn−1

kn−1

冊

E关1兵XⱕxN共v兲其log LN共n,v + an−1X + h⌬xn−1

冑

Ntk_n−1,Nn−1兲兴 +O共N exp共− N/C兲兲

冎

.

log LN共n − 1,v,kn−1兲 = − AN共h兲

冑

2␲共an2+ an−12 兲k

兺

_n=0 N_n

冉

冉

Nn kn

冊冉

N_n−1 kn−1

冊

冕

R 共1 − e−␸共y兲_兲 ⫻exp

冋

− 1 2共an 2 + a_n−12 兲YN共h,y,v,tkn,kn−1,n − 1兲 2

册

_dy

冊

_{+O共N exp共− N/C兲兲.} Finally, we arrive at log LN共n − 1,v兲 =

兺

k_n−1=0 N_n−1 log LN共n − 1,v,kn−1兲 = −

_冑

AN共h兲 2␲共an2+ an−12 兲k

兺

_n=0 N_n

兺

k_n−1=0 N_n−1

冉

冉

Nn kn

冊冉

N_n−1 kn−1

冊

⫻

冕

R 共1 − e−␸共y兲_兲exp

冋

₋ 1 2共an2+ an−12 兲 YN共h,y,v,tk_n,k_n−1,n − 1兲2

册

dy

冊

+O共N2exp共− N/C兲兲 = −

_冑

AN共h兲 2␲共an2+ an−12 兲

兺

k=0 N_n+Nn−1

冉

冉

Nn+ Nn−1 k

冊

⫻

冕

R 共1 − e−␸共y兲_兲exp

冋

₋ 1 2共an2+ an−12 兲 YN共h,y,v,tk,N_n+N_n−1,n − 1兲2

册

dy

冊

+O共N2exp共− N/C兲兲.

共4兲 Combining共4.9兲and共4.11兲for l = 1, we obtain L_EN共h兲共␸兲 = exp

冉

−

冕

R

共1 − e−␸共y兲_兲S

N共h,y兲dy + o共1兲

冊

, 共4.34兲

where SN共h,y兲 is given by共3.24兲. Invoking the proof of Theorem 1.1, we get that

L_EN共h兲共␸兲 →

N_↑+⬁

exp

冉

−

冕

R

共1 − e−␸共y兲_兲e−y_dy

冊

_{= L}

P共e−x_兲共␸兲.

This establishes共4.2兲.

共5兲 The proof of共4.5兲is very similar to the above proof of共4.2兲. The main difference is that

共4.25兲does not hold. Instead, if共4.4兲holds for l₀= n, then we have 1

冑

2␲

冕

−⬁ xN共v兲 exp

冋

−an 2_{+ a} n−1 2 2an 2

冉

x − an−1 an 2 + a_n−12 YN共h,y,v,t共kn,kn−1兲,n − 1兲

冊

2

册

dx → N↑+⬁ an 共an2+ an−12 兲1/2 P

再

X⬍

冑

N an−1

冑

an2+ an−12 关M共h兲 − vN−1/2_{−共1 − x} n−2兲htⴱ共h␪n,n −1/2_兲 −共an 2 + a_n−12 兲␮共h兲兴

冎

, 共4.35兲 uniformly in