arXiv subjects
Publications and source records attributed to Ofer Zeitouni.
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 applications to the supercritical complex Gaussian multiplicative chaos.
We consider a model of Branching Brownian Motion with time-inhomogeneous variance of the form σ(t/T), where σis a strictly decreasing function. Fang and Zeitouni (2012) showed that the maximal particle's position M_T is such that M_T-v_σT is negative of order T^{-1/3}, where v_σis the integral of the function σover the interval [0,1]. In this paper, we refine we refine this result and show the existence of a function m_T, such that M_T-m_T converges in law, as T\to\infty. Furthermore, m_T=v_σT - w_σT^{1/3} - σ(1)\log T + O(1) with w_σ= 2^{-1/3}α_1 \int_0^1 σ(s)^{1/3} |σ'(s)|^{2/3}\,\dd s. Here, -α_1=-2.33811... is the largest zero of the Airy function. The proof uses a mixture of probabilistic and analytic arguments.
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 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 of iterated logarithm for this sequence, i.e. we divide the normalized n-th eigenvalue by a logarithmic factor and show the limsup of this sequence is a constant almost surely. We also give almost sure bounds for the appropriately scaled liminf.
We derive a large deviation principle for the empirical measure of zeros of random polynomials with i.i.d. exponential coefficients.
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 sequence of random variables, reminiscent of the derivative martingale in the theory of Branching Random Walk and Gaussian Chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields, some additional structural assumptions of the type we make are needed for convergence of the centered maximum.
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 $\frac{8\sqrt{3}}{πn^2}$ otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on { -1, 0, 1} and whose largest atom is strictly less than 1/\sqrt{3}. In this general case, we prove that the probability of having a double root equals the probability that either -1, 0 or 1 are double roots up to an o(n^{-2}) factor and we find the asymptotics of the latter probability.
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 perturbation. This setup applies to the situation where under the alternative, there is a planted principal submatrix ${\mathbf{B}}$ of size $L$ for which all upper triangular variables are i.i.d. Gaussians with mean $1$ and variance $1$, whereas all other upper triangular elements of ${\mathbf{X}}$ not in ${\mathbf{B}}$ are i.i.d. Gaussians variables with mean 0 and variance $1$. We refer to this as the `Gaussian hidden clique problem.' When $L=(1+ε)\sqrt{n}$ ($ε>0$), it is possible to solve this detection problem with probability $1-o_n(1)$ by computing the spectrum of ${\mathbf{X}}$ and considering the largest eigenvalue of ${\mathbf{X}}$. We prove that this condition is tight in the following sense: when $L<(1-ε)\sqrt{n}$ no algorithm that examines only the eigenvalues of ${\mathbf{X}}$ can detect the existence of a hidden Gaussian clique, with error probability vanishing as $n\to\infty$. We prove this result as an immediate consequence of a more general result on rank-one perturbations of $k$-dimensional Gaussian tensors. In this context we establish a lower bound on the critical signal-to-noise ratio below which a rank-one signal cannot be detected.
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 unconditioned when $W$ is a GOE matrix. Our results validate (in a certain region) and correct (in another region), the prediction obtained by the mathematically non-rigorous replica method in Y. V. Fyodorov, P. Le Doussal, J. Stat. phys. 154 (2014).
We present estimates on the small singular values of a class of matrices with independent Gaussian entries and inhomogeneous variance profile, satisfying a broad-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 satisfying an appropriate expansion property, the Barvinok--Godsil-Gutman estimator for the permanent achieves sub-exponential errors with high probability.
Consider a $d$-ary rooted tree ($d\geq3$) 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 (necessarily, linearly) as $n$ tends to infinity. We analyze the Metropolis algorithm on the tree and show that under these assumptions there always exists a temperature $1/β$ of the algorithm so that it achieves a linear (positive) growth rate in linear time. This confirms a conjecture of Aldous [Algorithmica 22 (1998) 388-412]. The proof is obtained by establishing an Einstein relation for the Metropolis algorithm on the tree.
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 $L[θ_y f]=g(y-τ_f)$ for any nonzero, nonnegative, compactly supported, continuous function $f$, where $θ_y$ is the shift operator, $τ_f$ is a real number that depends on $f$, and $g$ is a real function that is independent of $f$. We show that, under some assumptions, this property characterizes the structure of SDPPP. Moreover, when it holds, we show that $g$ has to be a convolution of the Gumbel distribution with some measure. The above property of the Laplace functional is closely related to a `freezing phenomenon' that is expected by physicists to occur in a wide class of log-correlated fields, and which has played an important role in the analysis of various models. Our results shed light on this intriguing phenomenon and provide a natural tool for proving an SDPPP structure in these and other models.
We consider the regularization of matrices $M^N$ written in Jordan form by additive Gaussian noise $N^{-γ}G^N$, where $G^N$ is a matrix of i.i.d. standard Gaussians and $γ>1/2$ so that the operator norm of the additive noise tends to $0$ with $N$. Under mild conditions on the structure of $M^N$ we evaluate the limit of the empirical measure of eigenvalues of $M^N+N^{-γ} G^N$ and show that it depends on $γ$, in contrast with the case of a single Jordan block.
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^{-α} where α>0 can be arbitrarily small. We also show, by means of a general delocalization lemma for martingales, which is of independent interest, that slower than polynomial decay is not possible.
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 the law of the centered maximum.
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.
We provide conditions that ensure that the recentered maximum of the Gaussian free field on a sequence of graphs fluctuates at the same order as the field at the point of maximal variance. In particular, on a sequence of such graphs the recentered maximum is not tight, similarly to the situation in Z but in contrast with the situation in Z^2. We show that our conditions cover a large class of "fractal" graphs.
Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W(N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U(N)/log(N) converges almost surely to 2/log(2), while W(N)/log(N) does not converge almost surely (and in particular, limsup W(N)/log(N) is at least 3/log(2)).