Ofer Zeitouni

We calculate the average number of critical points $\overline{\mathcal{N}}$ of the energy landscape in a spin system with disordered two-body interactions and an on-site potential. Without the on-site potential, $\overline{\mathcal{N}}$ is at most linear in the system size. Nonetheless, we find that introducing a weak on-site potential can increase...

We study the distribution of the maximum of a large class of Gaussian fields indexed by a box $V_N\subset Z^d$ and possessing logarithmic correlations up to local defects that are sufficiently rare. Under appropriate assumptions that generalize those in Ding, Roy and Zeitouni (Annals Probab. (45) 2017, 3886-3928), we show that asymptotically, the c...

We study the line ensembles of non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This model, which mimics the level lines of the $(2+1)$D SOS model above a hard wall, was studied in two works from 2019 by Caputo, Ioffe and Wachtel. In those works, the tightness of...

We consider a branching Brownian motion in $\mathbb{R}^d$ with $d \geq 1$ in which the position $X_t^{(u)}\in \mathbb{R}^d$ of a particle $u$ at time $t$ can be encoded by its direction $\theta^{(u)}_t \in \mathbb{S}^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the {\it extremal point process} $\sum \delta_{\theta^{(u)}_t, R^{(u)}_t - m_...

Let $W_N(\beta) = \mathrm{E}_0\left[e^{ \sum_{n=1}^N \beta\omega(n,S_n) - N\beta^2/2}\right]$ be the partition function of a two-dimensional directed polymer in a random environment, where $\omega(i,x), i\in \mathbb{Z}_+, x\in \mathbb{Z}^2$ are i.i.d.\ standard normal and $\{S_n\}$ is the path of a random walk. With $\beta=\beta_N=\hat\beta \sqrt{\...

We consider critical points of the spherical pure p-spin spin glass model with Hamiltonian HNσ=1Np−1/2∑i1,…,ip=1NJi1,…,ipσi1…σip, where σ=σ1,…,σN∈SN−1≔σ∈RN:σ2=N and Ji1,…,ip are i.i.d. standard normal variables. Using a second moment analysis, we prove that for p ≥ 32 and any E > −E⋆, where E⋆ is the (normalized) ground state, the ratio of the numb...

We consider the stochastic heat equation \(\partial _{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda )u\), with a smooth space-time-stationary Gaussian random field V(s, y), in dimensions \(d\ge 3\), with an initial condition \(u(0,x)=u_0(\varepsilon x)\) and a suitably chosen \(\lambda \in {\mathbb {R}}\). It is known that, for \(\beta \) small e...

We prove that the minimum of the modulus of a random trigonometric polynomial with Gaussian coefficients, properly normalized, has limiting exponential distribution.

We consider critical points of the spherical pure $p$-spin spin glass model with Hamiltonian $H_{N}\left(\boldsymbol{\sigma}\right)=\frac{1}{N^{\left(p-1\right)/2}}\sum_{i_{1},...,i_{p}=1}^{N}J_{i_{1},...,i_{p}}\sigma_{i_{1}}\cdots\sigma_{i_{p}}$, where $\boldsymbol{\sigma}=\left(\sigma_{1},...,\sigma_{N}\right)\in \mathbb{S}^{N-1}:=\left\{ \boldsy...

We study the directed polymer model for general graphs (beyond Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^d$$\end{document}) and random walks. We p...

We give a new proof of a recent resolution by Michelen and Sahasrabudhe of a conjecture of Shepp and Vanderbei that the moduli of roots of Gaussian Kac polynomials of degree $n$, centered at $1$ and rescaled by $n^2$, should form a Poisson point process. We use this new approach to verify a conjecture of Michelen and Sahasrabudhe that the Poisson s...

We show that in branching Brownian motion (BBM) in $\mathbb{R}^d$, $d\geq 2$, the law of $R_t^*$, the maximum distance of a particle from the origin at time $t$, converges as $t\to\infty$ to the law of a randomly shifted Gumbel random variable.

We prove localization with high probability on sets of size of order $N/\log N$ for the eigenvectors of non-Hermitian finitely banded $N\times N$ Toeplitz matrices $P_N$ subject to small random perturbations, in a very general setting. As perturbation we consider $N\times N$ random matrices with independent entries of zero mean, finite moments, and...

We study in this paper lower bounds for the generalization error of models derived from multi-layer neural networks, in the regime where the size of the layers is commensurate with the number of samples in the training data. We show that unbiased estimators have unacceptable performance for such nonlinear networks in this regime. We derive explicit...

Let \(\{S_k:k\ge 0\}\) be a symmetric and aperiodic random walk on \(\mathbb {Z}^d\), \(d\ge 3\), and \(\{\xi (z),z\in \mathbb {Z}^d\}\) a collection of independent and identically distributed random variables. Consider a random walk in random scenery defined by \(T_n=\sum _{k=0}^n\xi (S_k)=\sum _{z\in \mathbb {Z}^d}l_n(z)\xi (z)\), where \(l_n(z)=...

Consider an \(N\times N\) Toeplitz matrix \(T_N\) with symbol \({{\varvec{a}} }(\lambda ) := \sum _{\ell =-d_2}^{d_1} a_\ell \lambda ^\ell \), perturbed by an additive noise matrix \(N^{-\gamma } E_N\), where the entries of \(E_N\) are centered i.i.d. random variables of unit variance and \(\gamma >1/2\). It is known that the empirical measure of e...

We prove a central limit theorem for the logarithm of the characteristic polynomial of random Jacobi matrices. Our results cover the G$\beta$E models for $\beta>0$.

We study the directed polymer model for general graphs (beyond $\mathbb Z^d$) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an $L^2$ region, and of very strong disorder, in terms of properties of the graph and of the random walk. We study in some detail (biased) random walk on var...

We prove that the minimum of the modulus of a random trigonometric polynomial with Gaussian coefficients, properly normalized, has limiting exponential distribution.

We prove a local central limit theorem for fluctuations of linear statistics of smooth enough test functions under the canonical Gibbs measure of two-dimensional Coulomb gases at any positive temperature. The proof relies on the existing global central limit theorem and a new decay estimate for the characteristic function of such fluctuations.

Nakamoto invented the longest chain protocol, and claimed its security by analyzing the private double-spend attack, a race between the adversary and the honest nodes to grow a longer chain. But is it the worst attack? We answer the question in the affirmative for three classes of longest chain protocols, designed for different consensus models: 1)...

Let PN be a uniform random N × N permutation matrix and let χN(z) = det(zIN − PN) denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of χN on the unit circle, specifically, with probability tending to 1 as N → ∞, for a numerical constant x0 ≈ 0.652. The main idea of the proof is to uncover a logarithmic co...

We consider a one dimensional ballistic nearest-neighbor random walk in a random environment. We prove an Erd\H{o}s-R\'enyi strong law for the increments.

We prove, using probabilistic techniques and analysis on the Wiener space, that the large scale fluctuations of the KPZ equation in d≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \be...

Let Cϵ,S2∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}^*_{\epsilon ,\mathbf{S}^2}$$\end{document} denote the cover time of the two dimensional sphere...

Lemma 9.1 in the paper is incorrect as stated.

We prove a quantitative deterministic equivalence theorem for the logarithmic potentials of deterministic complex $N\times N$ matrices subject to small random perturbations. We show that with probability close to $1$ this log-potential is, up to a small error, determined by the singular values of the unperturbed matrix which are larger than some sm...

We show that the number of real roots of random trigonometric polynomials with i.i.d. coefficients, which are either bounded or satisfy the logarithmic Sobolev inequality, satisfies an exponential concentration of measure.

Let T N T_N denote an N × N N\times N Toeplitz matrix with finite, N N independent symbol a \mathbfit {a} . For E N E_N a noise matrix satisfying mild assumptions (ensuring, in particular, that N − 1 / 2 ‖ E N ‖ H S → N → ∞ 0 {N^{-1/2}\|E_N\|_{{\mathrm {HS}}}}\to _{N\to \infty } 0 at a polynomial rate), we prove that the empirical measure of eigenv...

The short-term empirical law of Langevin dynamics for the asymmetric Sherrington-Kirkpatrick (SK) model with soft spins was shown by Ben Arous and Guionnet (1995) to have an a.s. limit $\mu_\star$ as the system size grows, as predicted by Mezard, Parisi and Virasoro (1987). The SK model has since been shown to be universal, in that its free energy...

We study the Gibbs measure of mixed spherical p‐spin glass models at low temperature, in (part of) the 1‐RSB regime, including, in particular, models close to pure in an appropriate sense. We show that the Gibbs measure concentrates on spherical bands around deep critical points of the (extended) Hamiltonian restricted to the sphere of radius , whe...

We show the existence of the scaling exponent \(\chi = \chi (\gamma )\), with $$\begin{aligned} 0 < \chi \le \frac{4}{\gamma ^2} \left( \left( 1+ {\gamma ^2} / 4 \right) - \sqrt{1+ {\gamma ^4} / {16} } \right) , \end{aligned}$$of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater \(\gamma <2\) on \...

Let $T_n$ denote the binary tree of depth $n$ augmented by an extra edge connected to its root. Let $C_n$ denote the cover time of $T_n$ by simple random walk. We prove that $\sqrt{ \mathcal{C}_{n} 2^{-(n+1) } } - m_n$ converges in distribution as $n\to \infty$, where $m_n$ is an explicit constant, and identify the limit.

Consider an $N\times N$ Toeplitz matrix $T_N$ with symbol ${ a }(\lambda) := \sum_{\ell=-d_2}^{d_1} a_\ell \lambda^\ell$, perturbed by an additive noise matrix $N^{-\gamma} E_N$, where the entries of $E_N$ are centered i.i.d.~complex random variables of unit variance and $\gamma>1/2$. It is known that the empirical measure of eigenvalues of the per...

Let $T_N$ denote an $N\times N$ Toeplitz matrix with finite, $N$ independent symbol ${\bf a}$. For $E_N$ a noise matrix satisfying mild assumptions (ensuring, in particular, that $N^{-1/2}\|E_N\|_{{\rm HS}}\to_{N\to\infty} 0$ at a polynomial rate), we prove that the empirical measure of eigenvalues of $T_N+E_N$ converges to the law of ${\bf a}(U)$,...

We prove, using probabilistic techniques and analysis on the Wiener space, that the large scale fluctuations of the KPZ equation in $d\geq 3$ with a small coupling constant, driven by a white in time and colored in space noise, are given by the Edwards-Wilkinson model. This gives an alternative proof, that avoids perturbation expansions, to the res...

We study the maximum of Branching Brownian motion (BBM) with branching rates that vary in space, via a periodic function of a particle's location. This corresponds to variant of the F-KPP equation in a periodic medium, extensively studied in the last 15 years, admitting pulsating fronts as solutions. Recent progress on this PDE due to Hamel, Nolen,...

We consider the heat equation with a multiplicative Gaussian potential in dimensions $d\geq 3$. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that i...

We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u$, driven by a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq 3$, with an initial condition $u(0,x)=u_0(\varepsilon x)$. It is known that the diffusively rescaled solution $u^{\varepsilon}(t,x)=u(\varepsilon^{-2}t,...

We study the partition function and free energy of the Curie-Weiss model with complex temperature, and partially describe its phase transitions. As a consequence, we obtain information on the locations of zeros of the partition function.

We show the existence of the scaling exponent $\chi\in (0,4[(1+\gamma^2/4)- \sqrt{1+\gamma^4/16}]/\gamma^2]$ of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater $\gamma<2$ on $\mathbb V =[0,1]^2 $. We also show that the Liouville heat kernel satisfies, for any fixed $u,v\in \mathbb V^o$, the shor...

Let $P_N$ be a uniform random $N\times N$ permutation matrix and let $\chi_N(z)=\det(zI_N- P_N)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of $\chi_N$ on the unit circle, specifically, \[ \sup_{|z|=1}|\chi_N(z)|= N^{x_0 + o(1)} \] with probability tending to one as $N\to \infty$, for a numerical c...

We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e}^{-n\operatorname{Tr}V(M^*M)}$ and $V$ convex. We prove that for unit eigenvectors $\mathbf{v},\mathbf{v}'$ associated with distinct eigenvalues $\lambda,\lambda'$ closest to specified points $z,z'$ in the complex plane, the resca...

We study the Gibbs measure of mixed spherical $p$-spin glass models at low temperature, in (part of) the 1-RSB regime, including, in particular, models close to pure in an appropriate sense. We show that the Gibbs measure concentrates on spherical bands around deep critical points of the (extended) Hamiltonian restricted to the sphere of radius $\s...

We prove subsequential tightness of centered maxima of two-dimensional Ginzburg-Landau fields with bounded elliptic contrast.

Sathamangalam Ranga Iyengar Srinivasa (Raghu) Varadhan was born in Chennai (then Madras). He received his Bachelor's and Master's degree from Presidency College, Madras, and his PhD from the Indian Statistical Institute in Kolkata, in 1963. That same year he came to the Courant Institute, New York University as a postdoc, and remained there as facu...

We study the Liouville heat kernel (in the L² phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each ε > 0 there exists such a field, whose covariance is a bounded perturbation of that of the two dimensional Gaussian free field, and such that the associated Liouville heat ker...

We study the angles between the eigenvectors of a random n × n complex matrix M with density ∝ e−nTrV(M∗M) and x ↦→ V (x²) convex. We prove that for unit eigenvectors v, v′ associated with distinct eigenvalues λ, λ′ that are the closest to specified points z, z′ in the complex plane, the rescaled inner product (Formula presented) is uniformly sub-G...

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_N$ be a deterministic $N\times N$ matrix, and let $G_N$ be a complex Ginibre matrix. We consider the matrix $\mathcal{M}_N=M_N+N^{-\gamma}G_N$, where $\gamma>1/2$. With $L_N$ the empirical measure of eigenvalues of $\mathc...

Let $\mathcal{C}^*_{\epsilon,\mathbb{S}^2}$ denote the cover time of the two dimensional sphere $\mathbb{S}^2$ by a Wiener sausage of radius $\epsilon$. We prove that $\sqrt{\mathcal{C}^*_{\epsilon,\mathbb{S}^2} } -2\sqrt{2}(\log \epsilon^{-1}-\frac14\log\log \epsilon^{-1})$ is tight.

We prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motio...

We show that for any centered stationary Gaussian process of integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in $[0,T]$ is within $\eta T$ of its mean value, up to an exponentially small in $T$ probability.

Recently, sharp results concerning the critical points of the Hamiltonian of the $p$-spin spherical spin glass model have been obtained by means of moments computations. In particular, these moments computations allow for the evaluation of the leading term of the ground-state, i.e., of the global minimum. In this paper, we study the extremal point...

Let $P_n^1,\dots, P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum_{\ell=1}^d P_n^\ell$. We show that if $d = \omega(\log^{16} n)$ and $d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d}$ converges weakly to the circular law in probability as $n \to \infty$.

We consider a finite horizon stochastic optimal control problem for nearest-neighbor random walk $\{X_i\}$ on the set of integers. The cost function is the expectation of exponential of the path sum of a random stationary and ergodic bounded potential plus $\theta X_n$. The random walk policies are measurable with respect to the random potential, a...

We consider the "searching for a trail in a maze" composite hypothesis testing problem, in which one attempts to detect an anomalous directed path in a lattice 2D box of side n based on observations on the nodes of the box. Under the signal hypothesis, one observes independent Gaussian variables of unit variance at all nodes, with zero, mean off th...

We use the large deviation approach to sum rules pioneered by Gamboa, Nagel, and Rouault to prove higher-order sum rules for orthogonal polynomials on the unit circle. In particular, we prove one half of a conjectured sum rule of Lukic in the case of two singular points, one simple and one double. This is important because it is known that the conj...

For the critical Galton--Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two dimensional manifolds. As...

We derive a large deviations principle for the two-dimensional two-component plasma in a box. As a consequence, we obtain a variational representation for the free energy, and also show that the macroscopic empirical measure of either positive or negative charges converges to the uniform measure. An appendix, written by Wei Wu, discusses applicatio...

We study the Liouville heat kernel (in the $L^2$ phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each $\varepsilon>0$ there exists such a field, whose covariance is a bounded perturbation of that of the two dimensional Gaussian free field, and such that the associated Liouv...

This is a pedagogical exposition of the large deviation approach to sum rules pioneered by Gamboa, Nagel and Rouault. We'll explain how to use their ideas to recover the Szeg}o and Killip{ Simon Theorems. The primary audience is spectral theorists and people working on orthogonal polynomials who have limited familiarity with the theory of large dev...

We prove the universality of the large deviations principle for the empirical measures of zeros of random polynomials whose coefficients are i.i.d. random variables possessing a density with respect to the Lebesgue measure on C, R or R + , under the assumption that the density does not vanish too fast at zero and decays at least as exp --|x| $\rho$...

We analyze the behavior of the Barvinok estimator of the hafnian of even dimension, symmetric matrices with nonnegative entries. We introduce a condition under which the Barvinok estimator achieves subexponential errors, and show that this condition is almost optimal. Using that hafnians count the number of perfect matchings in graphs, we conclude...

Let $U_N$ denote a Haar Unitary matrix of dimension N, and consider the field \[ {\bf U}(z) = \log |\det(1-zU_N)| \] for z in the unit disk. Then, \[ \frac{\max_{|z|=1} {\bf U}(z) -\log N + \frac{3}{4} \log\log N} {\log\log N} \to 0 \] in probability. This provides a verification up to second order of a conjecture of Fyodorov, Hiary and Keating, im...

We consider the smoothed multiplicative noise stochastic heat equation $$d u_{\eps,t}= \frac 12 \Delta u_{\eps,t} d t+ \beta \eps^{\frac{d-2}{2}}\, \, u_{\eps, t} \, d B_{\eps,t} , \;\;u_{\eps,0}=1,$$ in dimension $d\geq 3$, where $B_{\eps,t}$ is a spatially smoothed (at scale $\eps$) space-time white noise, and $\beta>0$ is a parameter. We show th...

We consider the GUE minor process, where a sequence of GUE matrices is drawn from the corner of a doubly infinite array of i.i.d. standard normal variables subject to the symmetry constraint. From each matrix, we take its largest eigenvalue, appropriately rescaled to converge to the standard Tracy-Widom distribution. We show the analogue of the law...

We show that the centered maximum of a sequence of log-correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a seque...

We consider the following detection problem: given a realization of a symmetric matrix ${\mathbf{X}}$ of dimension $n$, distinguish between the hypothesis that all upper triangular variables are i.i.d. Gaussians variables with mean 0 and variance $1$ and the hypothesis where ${\mathbf{X}}$ is the sum of such matrix and an independent rank-one pertu...

We consider the quadratic optimization problem $$F_n^{W,h}:= \sup_{x \in S^{n-1}} ( x^T W x/2 + h^T x )\,, $$ with $W$ a (random) matrix and $h$ a random external field. We study the probabilities of large deviation of $F_n^{W,h}$ for $h$ a centered Gaussian vector with i.i.d. entries, both conditioned on $W$ (a general Wigner matrix), and uncondit...

We analyze the behavior of the Barvinok estimator of the hafnian of even dimension, symmetric matrices with non negative entries. We introduce a condition under which the Barvinok estimator achieves sub-exponential errors, and show that this condition is almost optimal. Using that hafnians count the number of perfect matchings in graphs, we conclud...

We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We prove that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and asymptotic to $\frac{8\sqrt{3}}{\pi n^2}$ otherwise. This result is a corollary of a more general theorem that we prove concerning...

We present estimates on the small singular values of a class of matrices with independent Gaussian entries and inhomogeneous variance profile, satisfying a strong-connectedness condition. Using these estimates and concentration of measure for the spectrum of Gaussian matrices with independent entries, we prove that for a large class of graphs satis...

We initiate in this paper the study of analytic properties of the Liouville heat kernel. In particular, we establish regularity estimates on the heat kernel and derive non trivial lower and upper bounds.

The limiting extremal processes of the branching Brownian motion (BBM), the two-speed BBM, and the branching random walk are known to be randomly shifted decorated Poisson point processes (SDPPP). In the proofs of those results, the Laplace functional of the limiting extremal process is shown to satisfy \({L\left[\theta_{y}f\right]=g\left(y-\tau_{f...

We consider the regularization of matrices MN in Jordan form by additive Gaussian noise N{-\gamma }GN, where GN is a matrix of i.i.d. standard Gaussians and \gamma >{\tfrac {1}{2}} so that the operator norm of the additive noise tends to 0 with N. Under mild conditions on the structure of MN, we evaluate the limit of the empirical measure of eigenv...

Let $\eta^*_n$ denote the maximum, at time $n$, of a nonlattice one-dimensional branching random walk $\eta_n$ possessing (enough) exponential moments. In a seminal paper, Aidekon demonstrated convergence of $\eta^*_n$ in law, after recentering, and gave a representation of the limit. We give here a shorter proof of this convergence by employing re...

We consider controlled martingales with bounded steps where the controller is allowed at each step to choose the distribution of the next step, and where the goal is to hit a fixed ball at the origin at time n. We show that the algebraic rate of decay (as n increases to infinity) of the value function in the discrete setup coincides with its contin...

The mean discount rate for a simple stochastic model behaves asymptotically roughly like \(1/\sqrt{n}\) in contrast to the usual geometric discounting in a deterministic model.

We derive a large deviation principle for the empirical measure of zeros of random polynomials with i.i.d. exponential coefficients.

We consider controlled random walks that are martingales with uniformly bounded increments and nontrivial jump probabilities and show that such walks can be constructed so that P(S_n^u=0) decays at polynomial rate n^{-\alpha} where \alpha>0 can be arbitrarily small. We also show, by means of a general delocalization lemma for martingales, which is...

We consider a model of Branching Brownian Motion with time-inhomogeneous variance of the form \sigma(t/T), where \sigma is a strictly decreasing function. Fang and Zeitouni (2012) showed that the maximal particle's position M_T is such that M_T-v_\sigma T is negative of order T^{-1/3}, where v_\sigma is the integral of the function \sigma over the...

We study the limiting distribution of the eigenvalues of the Ginibre ensemble conditioned on the event that a certain proportion lie in a given region of the complex plane. Using an equivalent formulation as an obstacle problem, we describe the optimal distribution and some of its monotonicity properties.

Consider a $d$-ary rooted tree ($d\geq 3$) where each edge $e$ is assigned an i.i.d. (bounded) random variable $X(e)$ of negative mean. Assign to each vertex $v$ the sum $S(v)$ of $X(e)$ over all edges connecting $v$ to the root, and assume that the maximum $S_n^*$ of $S(v)$ over all vertices $v$ at distance $n$ from the root tends to infinity (nec...

We provide conditions that ensure that the maximum of the Gaussian free field on a sequence of graphs fluctuates at the same order as the field at the point of maximal standard deviation; under these conditions, the expectation of the maximum is of the same order as the maximal standard deviation. In particular, on a sequence of such graphs the rec...

We consider the two-dimensional Gaussian Free Field on a box of side length $N$, with Dirichlet boundary data, and prove the convergence of the law of the recentered maximum of the field.

We prove a Large Deviations Principle (LDP) for systems of diffusions (particles) interacting through their ranks, when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the approriate McKean-Vlasov equation and that the corresponding cumulative distribution function evolves acc...

We consider in this paper the collection of near maxima of the discrete, two dimensional Gaussian free field in a box with Dirichlet boundary conditions. We provide a rough description of the geometry of the set of near maxima, estimates on the gap between the two largest maxima, and an estimate for the right tail up to a multiplicative constant on...

We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to $T^{1/3}$. We conjecture that this is the worse case correction possible.

Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M=2, N become large or M become large and N=2.

We consider the maximum of the discrete two dimensional Gaussian free field (GFF) in a box, and prove that its maximum, centered at its mean, is tight, settling a long-standing conjecture. The proof combines a recent observation of Bolthausen, Deuschel and Zeitouni with elements from (Bramson 1978) and comparison theorems for Gaussian fields. An es...