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Exponential decay of correlations in the two-dimensional random field Ising model at positive temperaturesMay 14 2019We study random field Ising model on $\mathbb Z^2$ where the external field is given by i.i.d.\ Gaussian variables with mean zero and positive variance. We show that at any positive temperature the effect of boundary conditions on the magnetization in ... More

On the Ultrametricity Property in Random Field Ising ModelsApr 23 2019In this paper we show that the ultrametricity property remains valid in the Random Field Ising Model for any independent disorder whenever the field strength is a small perturbation.

The generalized TAP free energy IIMar 04 2019In a recent paper [14], we developed the generalized TAP approach for mixed $p$-spin models with Ising spins at positive temperature. Here we extend these results in two directions. We find a simplified representation for the energy of the generalized ... More

Exponential decay of correlations in the two-dimensional random field Ising model at zero temperatureFeb 08 2019We study random field Ising model on $\mathbb Z^2$ where the external field is given by i.i.d.\ Gaussian variables with mean zero and positive variance. We show that at zero temperature the effect of boundary conditions on the magnetization in a finite ... More

Exponential decay of correlations in the two-dimensional random field Ising model at zero temperatureFeb 08 2019Feb 17 2019We study random field Ising model on $\mathbb Z^2$ where the external field is given by i.i.d.\ Gaussian variables with mean zero and positive variance. We show that at zero temperature the effect of boundary conditions on the magnetization in a finite ... More

The meanfield limit of a network of Hopfield neurons with correlated synaptic weightsJan 29 2019May 13 2019We study the asymptotic behaviour for asymmetric neuronal dynamics in a network of Hopfield neurons. The randomness in the network is modelled by random couplings which are centered Gaussian correlated random variables. We prove that the annealed law ... More

The meanfield limit of a network of Hopfield neurons with correlated synaptic weightsJan 29 2019We study the asymptotic behaviour for asymmetric neuronal dynamics in a network of Hopfield neurons. The randomness in the network is modelled by random couplings which are centered Gaussian correlated random variables. We prove that the annealed law ... More

The generalized TAP free energyDec 12 2018In this paper, we define and compute the generalized TAP correction for the energy of the mixed $p$-spin models with Ising spins, and give the corresponding generalized TAP representation for the free energy. We study the generalized TAP states, which ... More

An exactly solvable continuous-time Derrida--Retaux~modelNov 21 2018To study the depinning transition in the limit of strong disorder, Derrida and Retaux (2014) introduced a discrete-time max-type recursive model. It is believed that for a large class of recursive models, including Derrida and Retaux' model, there is ... More

An exactly solvable continuous-time Derrida--Retaux modelNov 21 2018Apr 01 2019To study the depinning transition in the limit of strong disorder, Derrida and Retaux (2014) introduced a discrete-time max-type recursive model. It is believed that for a large class of recursive models, including Derrida and Retaux' model, there is ... More

On the Absence of Replica Symmetry Breaking for the Random Field Ising Model in the Presence of a Class of Non-Gaussian DisordersNov 16 2018Apr 23 2019This work is concerned with the theory of the Random Field Ising Model in the presence of disorders with non-Gaussian distribution on the hypercubic lattice. We showed the absence of replica symmetry in any dimensions, at any temperature and field strength, ... More

Extremal decomposition for random Gibbs measures: From general metastates to metastates on extremal random Gibbs measuresOct 17 2018Dec 18 2018The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling ... More

Replica symmetry breaking in multi-species Sherrington--Kirkpatrick modelOct 07 2018Jan 01 2019In the Sherrington--Kirkpatrick (SK) and related mixed $p$-spin models, there is interest in understanding replica symmetry breaking at low temperatures. For this reason, the so-called AT line proposed by de Almeida and Thouless as a sufficient (and conjecturally ... More

Equivalence of relative Gibbs and relative equilibrium measures for actions of countable amenable groupsAug 31 2018We formulate and prove a very general relative version of the Dobrushin-Lanford-Ruelle theorem which gives conditions on constraints of configuration spaces over a finite alphabet such that for every absolutely summable relative interaction, every translation-invariant ... More

A power-law upper bound on the correlations in the 2D random field Ising modelAug 25 2018As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through the addition of quenched random magnetic field of uniform ... More

On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical windowAug 10 2018The partition function of the directed polymer model on Z^{2+1} undergoes a phase transition in a suitable continuum and weak disorder limit. In this paper, we focus on a window around the critical point. Exploiting local renewal theorems, we compute ... More

Free Energy of Multiple Systems of Spherical Spin Glasses with Constrained OverlapsJun 26 2018The free energy of multiple systems of spherical spin glasses with constrained overlaps was first studied in arXiv:math/0604082. The authors proved an upper bound of the constrained free energy using Guerra's interpolation. In this paper, we prove this ... More

1-stable fluctuations in branching Brownian motion at critical temperature I: the derivative martingaleJun 13 2018Jun 19 2018Let $(Z_t)_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, i.e.\@ the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke [\textit{Ann. ... More

A note on recurrence of the Vertex reinforced jump process and fractional moments localizationApr 08 2018We give an alternative proof of the fact that, on Zd, the Vertex reinforced jump process is recurrent in strong reinforcement. Consequently, the edge reinforced random walk is recurrent in strong reinforcement, this is first proved separately in [ST15a] ... More

Beyond Hammersley's Last-Passage Percolation: a discussion on possible local and global constraintsFeb 12 2018May 31 2018Hammersley's Last-Passage Percolation (LPP), also known as Ulam's problem, is a well-studied model that can be described as follows: consider $m$ points chosen uniformly and independently in $[0,1]^2$, then what is the maximal number $\mathcal{L}_m$ of ... More

Continuum limit of random matrix products in statistical mechanics of disordered systemsDec 26 2017Sep 11 2018We consider a particular weak disorder limit ("continuum limit") of matrix products that arise in the analysis of disordered statistical mechanics systems, with a particular focus on random transfer matrices. The limit system is a diffusion model for ... More

Disorder and critical phenomena: the $α=0$ copolymer modelDec 06 2017The generalized copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent $1+ \alpha\geq 1$. It exhibits a localization transition which can be characterized ... More

Cadlag stability and approximation of reversible coalescing-fragmentating Wasserstein dynamicsNov 08 2017Jan 22 2018We construct a coalescing-fragmentating Wasserstein dynamics [arXiv:1709.02839] for any initial condition and interacting potential using a finite particle approximation. We show that such a model admits a description of each particle by a family of continuous ... More

Central limit theorem for the free energy of the random field Ising modelOct 24 2017A central limit theorem is proved for the free energy of the random field Ising model with all plus or all minus boundary condition, at any temperature (including zero temperature) and any dimension. This solves a problem posed by Wehr and Aizenman in ... More

Central limit theorem for the free energy of the random field Ising modelOct 24 2017Apr 16 2018A central limit theorem is proved for the free energy of the random field Ising model with all plus or all minus boundary condition, at any temperature (including zero temperature) and any dimension. This solves a problem posed by Wehr and Aizenman in ... More

Central limit theorem for the free energy of the random field Ising modelOct 24 2017Mar 28 2019A central limit theorem is proved for the free energy of the random field Ising model with all plus or all minus boundary condition, at any temperature (including zero temperature) and any dimension. This solves a problem posed by Wehr and Aizenman in ... More

On the decay of correlations in the random field Ising modelSep 13 2017Nov 30 2017In a celebrated 1990 paper, Aizenman and Wehr proved that the two-dimensional random field Ising model has a unique infinite volume Gibbs state at any temperature. The proof is ergodic-theoretic in nature and does not provide any quantitative information. ... More

On the TAP free energy in the mixed $p$-spin modelsSep 11 2017Jan 12 2019In [Physical Magazine, 35(3):593-601, 1977], Thouless, Anderson, and Palmer derived a representation for the free energy of the Sherrington-Kirkpatrick model, called the TAP free energy, written as the difference of the energy and entropy on the extended ... More

On concentration properties of disordered HamiltoniansJun 26 2017We present an elementary approach to concentration of disordered Hamiltonians. Assuming differentiability of the limiting free energy $F$ with respect to the inverse temperature $\beta$, we show that the Hamiltonian concentrates around the energy level ... More

Free Energy of the Cauchy Directed Polymer Model at High TemperatureJun 14 2017Jun 08 2018We study the Cauchy directed polymer model on $\mathbb{Z}^{1+1}$, where the underlying random walk is in the domain of attraction to the $1$-stable law. We show that, if the random walk satisfies certain regularity assumptions and its symmetrized version ... More

Localization and Eigenvalue Statistics for the Lattice Anderson model with Discrete DisorderMay 04 2017Jul 05 2017We prove localization and probabilistic bounds on the minimum level spacing for the Anderson tight-binding model on the lattice in any dimension, with single-site potential having a discrete distribution taking N values, with N large.

Many-Body Localization: Stability and InstabilityMay 02 2017Jul 13 2017Rare regions with weak disorder (Griffiths regions) have the potential to spoil localization. We describe a non-perturbative construction of local integrals of motion (LIOMs) for a weakly interacting spin chain in one dimension, under a physically reasonable ... More

The phase diagram of the complex branching Brownian motion energy modelApr 18 2017We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and ... More

Hitting times of interacting drifted Brownian motions and the vertex reinforced jump processApr 18 2017May 09 2018Consider a negatively drifted one dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law ... More

Disorder chaos in some diluted spin glass modelsMar 21 2017We prove disorder chaos at zero temperature for three types of diluted models with large connectivity parameter: $K$-spin antiferromagnetic Ising model for even $K\geq 2$, $K$-spin spin glass model for even $K\geq 2$, and random $K$-sat model for all ... More

Disorder relevance without Harris Criterion: the case of pinning model with $γ$-stable environmentOct 21 2016We investigate disorder relevance for the pinning of a renewal whose inter-arrival law has tail exponent $\alpha>0$ when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. We prove that ... More

On the energy landscape of the mixed even $p$-spin modelSep 14 2016Sep 19 2016We investigate the energy landscape of the mixed even $p$-spin model with Ising spin configurations. We show that for any given energy level between zero and the maximal energy, with overwhelming probability there exist exponentially many distinct spin ... More

On the energy landscape of the mixed even $p$-spin modelSep 14 2016Mar 24 2017We investigate the energy landscape of the mixed even $p$-spin model with Ising spin configurations. We show that for any given energy level between zero and the maximal energy, with overwhelming probability there exist exponentially many distinct spin ... More

On the Spectral Gap of Spherical Spin Glass DynamicsAug 23 2016We consider the time to equilibrium for the Langevin dynamics of the spherical $p$-spin glass model of system size $N$. We show that the log-Sobolev constant and spectral gap are order $1$ in $N$ at sufficiently high temperature whereas the spectral gap ... More

On the Spectral Gap of Spherical Spin Glass DynamicsAug 23 2016Jun 04 2018We consider the time to equilibrium for the Langevin dynamics of the spherical $p$-spin glass model of system size $N$. We show that the log-Sobolev constant and spectral gap are order $1$ in $N$ at sufficiently high temperature whereas the spectral gap ... More

Quenched large deviations for interacting diffusions in random mediaAug 23 2016The aim of the paper is to establish a large deviation principle (LDP) for the empirical measure of mean-field interacting diffusions in a random environment. The point is to derive such a result once the environment has been frozen (quenched model). ... More

On the $K$-sat model with large number of clausesAug 22 2016Aug 26 2016We show that in the $K$-sat model with $N$ variables and $\alpha N$ clauses, the expected ratio of the smallest number of unsatisfied clauses to the number of variables is $\alpha/2^K - \sqrt{\alpha} c_*(N)/2^K$ up to smaller order terms $o(\sqrt{\alpha})$ ... More

On the $K$-sat model with large number of clausesAug 22 2016Apr 01 2017We show that in the $K$-sat model with $N$ variables and $\alpha N$ clauses, the expected ratio of the smallest number of unsatisfied clauses to the number of variables is $\alpha/2^K - \sqrt{\alpha} c_*(N)/2^K$ up to smaller order terms $o(\sqrt{\alpha})$ ... More

Pinning of a renewal on a quenched renewalAug 10 2016We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $\sigma$, and $0$ elsewhere, ... More

Pinning of a renewal on a quenched renewalAug 10 2016Apr 26 2017We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $\sigma$, and $0$ elsewhere, ... More

Temperature chaos in some spherical mixed $p$-spin modelsAug 08 2016Dec 18 2016We give two types of examples of the spherical mixed even-$p$-spin models for which chaos in temperature holds. These complement some known results for the spherical pure $p$-spin models and for models with Ising spins. For example, in contrast to a recent ... More

Temperature chaos in some spherical mixed $p$-spin modelsAug 08 2016We give two types of examples of the spherical mixed even-$p$-spin models for which chaos in temperature holds. These complement some known results for the spherical pure $p$-spin models and for models with Ising spins. For example, in contrast to a recent ... More

Disorder and wetting transition: the pinned harmonic crystal in dimension three or largerJul 13 2016We consider the Lattice Gaussian free field in $d+1$ dimensions, $d=3$ or larger, on a large box (linear size $N$) with boundary conditions zero. On this field two potentials are acting: one, that models the presence of a wall, penalizes the field when ... More

Parisi formula for the ground state energy in the mixed p-spin modelJun 16 2016Jun 28 2016We show that the thermodynamic limit of the ground state energy in the mixed p-spin model can be identified as a variational problem. This gives a natural generalization of the Parisi formula at zero temperature.

Transversal fluctuations for a first passage percolation modelMay 19 2016We introduce a new first passage percolation model in a Poissonian environment on $\mathbb{R}^{2}$. In this model, the action of a path depends on the geometry of the path and the travel time. We prove that the transversal fluctuation exponent for point-to-line ... More

Diagonalization and Many-Body Localization for a Disordered Quantum Spin ChainMay 10 2016We consider a weakly interacting quantum spin chain with random local interactions. We prove that many-body localization follows from a physically reasonable assumption that limits the extent of level attraction in the statistics of eigenvalues. In a ... More

Singular behavior of the leading Lyapunov exponent of a product of random $2 \times 2$ matricesFeb 11 2016Dec 07 2016We consider a certain infinite product of random $2 \times 2$ matrices appearing in the solution of some $1$ and $1+1$ dimensional disordered models in statistical mechanics, which depends on a parameter $\varepsilon>0$ and on a real random variable with ... More

Singular behavior of the leading Lyapunov exponent of a product of random $2 \times 2$ matricesFeb 11 2016We consider a certain infinite product of random $2 \times 2$ matrices appearing in the solution of some $1$ and $1+1$ dimensional disordered models in statistical mechanics, which depends on a parameter $\varepsilon>0$ and on a real random variable with ... More

Critical Percolation and the Minimal Spanning Tree in SlabsDec 30 2015The minimal spanning forest on $\mathbb{Z}^{d}$ is known to consist of a single tree for $d \leq 2$ and is conjectured to consist of infinitely many trees for large $d$. In this paper, we prove that there is a single tree for quasi-planar graphs such ... More

Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin modelsDec 28 2015Oct 28 2016We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an ... More

Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin modelsDec 28 2015We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an ... More

Pinning and disorder relevance for the lattice Gaussian Free Field II: the two dimensional caseDec 16 2015Feb 16 2016This paper continues a study initiated in [34], on the localization transition of a lattice free field on $\mathbb Z^d$ interacting with a quenched disordered substrate that acts on the interface when its height is close to zero. The substrate has the ... More

On the replica symmetry phase of the independent set problemDec 14 2015Dec 16 2015The independent set problem, ISP for short, asks for the maximal number of vertices in a (large) graph which can be occupied such that none of them are neighbors. We address the question from a statistical mechanics perspective, in the case of Erdoes-Renyi ... More

Free energy in the mixed p-spin models with vector spinsDec 14 2015Dec 27 2015Using the synchronization mechanism developed in the previous work on the Potts spin glass model, arXiv:1512.00370, we obtain the analogue of the Parisi formula for the free energy in the mixed even $p$-spin models with vector spins, which include the ... More

Two-Sample Tests Based on Geometric Graphs: Asymptotic Distribution and Detection ThresholdsDec 01 2015May 18 2018In this paper we consider the problem of testing the equality of two multivariate distributions based on geometric graphs, constructed using the inter-point distances between the observations. These include the test based on the minimum spanning tree ... More

Distribution of Two-Sample Tests Based on Geometric Graphs and ApplicationsDec 01 2015Jun 22 2016In this paper central limit theorems are derived for multivariate two-sample tests based on stabilizing geometric graphs under general alternatives, in the Poissonized setting. For tests based on stabilizing graphs, such as the Friedman-Rafsky test (1979) ... More

Free energy in the Potts spin glassDec 01 2015Dec 28 2015We study the Potts spin glass model, which generalizes the Sherrington-Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational ... More

Free energy in the Potts spin glassDec 01 2015Nov 11 2016We study the Potts spin glass model, which generalizes the Sherrington-Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational ... More

Universality in marginally relevant disordered systemsOct 21 2015Jan 08 2017We consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant. These include the usual (short-range) directed polymer model in dimension (2+1), the long-range directed polymer model with Cauchy tails in dimension ... More

Universality in marginally relevant disordered systemsOct 21 2015Jan 26 2016We consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant. These include the usual (short-range) directed polymer model in dimension (2+1), the long-range directed polymer model with Cauchy tails in dimension ... More

The Legendre structure of the Parisi formulaOct 12 2015Oct 26 2015We show that the Parisi formula of the mixed $p$-spin model is a concave function of the squared inverse temperature. This allows us to derive a new expression for the Parisi formula that involves the inverse temperature and the Parisi measure as Legendre ... More

On the Long-range Directed PolymerOct 09 2015Sep 13 2016We study the long-range directed polymer model on $\mathbbm{Z}$ in a random environment, where the underlying random walk lies in the domain of attraction of an $\alpha$-stable process for some $\alpha\in(0,2]$. Similar to the more classic nearest-neighbor ... More

On the Long-range Directed Polymer ModelOct 09 2015Nov 23 2016We study the long-range directed polymer model on $\mathbbm{Z}$ in a random environment, where the underlying random walk lies in the domain of attraction of an $\alpha$-stable process for some $\alpha\in(0,2]$. Similar to the more classic nearest-neighbor ... More

Fluctuations of the free energy in the mixed $p$-spin models with external fieldSep 23 2015Nov 03 2015We show that the free energy in the mixed $p$-spin models of spin glasses does not superconcentrate in the presence of external field, which means that its variance is of the order suggested by the Poincar\'e inequality. This complements the result of ... More

Universality of the mean-field for the Potts modelAug 17 2015May 05 2016We consider the Potts model with $q$ colors on a sequence of weighted graphs with adjacency matrices $A_n$, allowing for both positive and negative weights. Under a mild regularity condition the mean-field prediction for the log partition function of ... More

Stochastic Ising model with flipping sets of spins and fast decreasing temperatureAug 04 2015Jan 19 2017This paper deals with the stochastic Ising model with a temperature shrinking to zero as time goes to infinity. A generalization of the Glauber dynamics is considered, on the basis of the existence of simultaneous flips of some spins. Such dynamics act ... More

A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphsJul 28 2015Jan 29 2016This paper concerns the Vertex reinforced jump process (VRJP), the Edge reinforced random walk (ERRW) and their link with a random Schr\"odinger operator. On infinite graphs, we define a 1-dependent random potential $\beta$ extending that defined in [16] ... More

A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphsJul 28 2015Jul 19 2018This paper concerns the Vertex reinforced jump process (VRJP), the Edge reinforced random walk (ERRW) and their link with a random Schr\"odinger operator. On infinite graphs, we define a 1-dependent random potential $\beta$ extending that defined in [20] ... More

Parabolic Anderson model in a dynamic random environment: random conductancesJul 21 2015The parabolic Anderson model is defined as the partial differential equation \partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where \kappa \in [0,\infty) is the diffusion constant, \Delta is the discrete Laplacian, ... More

The Vertex Reinforced Jump Process and a Random Schrödinger operator on finite graphsJul 16 2015Jan 21 2016We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the Inverse Gaussian distribution. Considered as the potential of a random Schr\"odinger operator, this exponential family is related ... More

Level Spacing for Non-Monotone Anderson ModelsJun 22 2015Feb 03 2016We prove localization and probabilistic bounds on the minimum level spacing for a random block Anderson model without monotonicity. Using a sequence of narrowing energy windows and associated Schur complements, we obtain detailed probabilistic information ... More

Universality for the pinning model in the weak coupling regimeMay 19 2015We consider disordered pinning models, when the return time distribution of the underlying renewal process has a polynomial tail with exponent $\alpha \in (1/2,1)$. This corresponds to a regime where disorder is known to be relevant, i.e. to change the ... More

Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto modelMay 04 2015Jun 22 2015The aim of the paper is to address the long time behavior of the Kuramoto model of mean-field coupled phase rotators, subject to white noise and quenched frequencies. We analyse the influence of the fluctuations of both thermal noise and frequencies (seen ... More

The glassy phase of the complex branching Brownian motion energy modelApr 20 2015Oct 27 2015We identify the fluctuations of the partition function for a class of random energy models, where the energies are given by the positions of the particles of the complex-valued branching Brownian motion (BBM). Specifically, we provide the weak limit theorems ... More

Some Properties of the Phase Diagram for Mixed $p$-Spin GlassesApr 10 2015Dec 22 2015In this paper we study the Parisi variational problem for mixed $p$-spin glasses with Ising spins. Our starting point is a characterization of Parisi measures whose origin lies in the first order optimality conditions for the Parisi functional, which ... More

Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shiftMar 25 2015Apr 08 2015The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point ... More

A Dynamic Programming Approach to the Parisi FunctionalFeb 16 2015Nov 30 2015G.Parisi predicted an important variational formula for the thermodynamic limit of the intensive free energy for a class of mean field spin glasses. In this paper, we present an elementary approach to the study of the Parisi functional using stochastic ... More

Chaos in temperature in generic 2p-spin modelsFeb 12 2015Dec 18 2015We prove chaos in temperature for even $p$-spin models which include sufficiently many $p$-spin interaction terms. Our approach is based on a new invariance property for coupled asymptotic Gibbs measures, similar in spirit to the invariance property that ... More

Pinning and disorder relevance for the lattice Gaussian free fieldJan 30 2015Jul 22 2015This paper provides a rigorous study of the localization transition for a Gaussian free field on $\mathbb{Z}^d$ interacting with a quenched disordered substrate that acts on the interface when the interface height is close to zero. The substrate has the ... More

Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand boundJan 27 2015May 13 2016The validity of the Parisi formula in the Sherrington-Kirkpatrick model (SK) was initially proved by Talagrand [18]. The central argument therein relied on a very dedicated study of the coupled free energy via the two-dimensional Guerra-Talagrand (GT) ... More

Disorder chaos in the spherical mean-field modelJan 09 2015We consider the problem of disorder chaos in the spherical mean-field model. It is concerned about the behavior of the overlap between two independently sampled spin configurations from two Gibbs measures with the same external parameters. The prediction ... More

Universality of chaos and ultrametricity in mixed p-spin modelsOct 29 2014We prove disorder universality of chaos phenomena and ultrametricity in the mixed p-spin model under mild moment assumptions on the environment. This establishes the long-standing belief among physicists that the Parisi solution in mean-field models is ... More

Some examples of quenched self-averaging in models with Gaussian disorderSep 07 2014In this paper we give an elementary approach to several results of Chatterjee in arXiv:0907.3381 and arXiv:1404.7178, as well as some generalizations. First, we prove quenched disorder chaos for the bond overlap in the Edwards-Anderson type models with ... More

Hierarchical pinning model: low disorder relevance in the $b=s$ caseAug 14 2014We consider a hierarchical pinning model introduced by B.Derrida, V.Hakim and J.Vannimenus which undergoes a localization/delocalization phase transition. This model depends on two parameters $b$ and $s$. We show that in the particular case where $b=s$, ... More

The continuum disordered pinning modelJun 19 2014Dec 02 2014Any renewal processes on $\mathbb{N}$ with a polynomial tail, with exponent $\alpha \in (0,1)$, has a non-trivial scaling limit, known as the $\alpha$-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in ... More

Structure of finite-RSB asymptotic Gibbs measures in the diluted spin glass modelsJun 18 2014Feb 25 2015We suggest a possible approach to proving the M\'ezard-Parisi formula for the free energy in the diluted spin glass models, such as diluted K-spin or random K-sat model at any positive temperature. In the main contribution of the paper, we show that a ... More

Multi-Scale Jacobi Method for Anderson LocalizationJun 11 2014Sep 30 2015A new KAM-style proof of Anderson localization is obtained. A sequence of local rotations is defined, such that off-diagonal matrix elements of the Hamiltonian are driven rapidly to zero. This leads to the first proof via multi-scale analysis of exponential ... More

On the critical curves of the Pinning and Copolymer models in Correlated Gaussian environmentApr 23 2014We investigate the disordered copolymer and pinning models, in the case of a correlated Gaussian environment with summable correlations, and when the return distribution of the underlying renewal process has a polynomial tail. As far as the copolymer ... More

On Many-Body Localization for Quantum Spin ChainsMar 30 2014Mar 30 2016For a one-dimensional spin chain with random local interactions, we prove that many-body localization follows from a physically reasonable assumption that limits the amount of level attraction in the system. The construction uses a sequence of local unitary ... More

Directed polymers in a random environment with a defect lineFeb 26 2014Jan 21 2015We study the depinning transition of the $1+1$ dimensional directed polymer in a random environment with a defect line. The random environment consists of i.i.d. potential values assigned to each site of $\mathbb{Z}^2$; sites on the positive axis have ... More

The Parisi formula has a unique minimizerFeb 20 2014Sep 05 2014In 1979, G. Parisi predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand and later ... More

Generalized Random Energy Model at Complex TemperaturesFeb 10 2014Motivated by the Lee--Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature $\beta$. We compute the limiting log-partition function and describe the fluctuations ... More

Phase Transitions in Nonlinear FilteringJan 24 2014Dec 08 2014It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information. While the existing theory provides a rather complete picture of classical ... More

Random Walk on Random WalksJan 18 2014Sep 10 2015In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density $\rho \in (0,\infty)$. At each step ... More

Polynomial chaos and scaling limits of disordered systemsDec 11 2013Aug 12 2014Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, ... More