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Amir Dembo

Publications and source records attributed to Amir Dembo.

At least 55 records · Page 3Linked to original sources
arXiv2017

Brownian Particles with Rank-Dependent Drifts: Out-of-Equilibrium Behavior

We study the long-range asymptotic behavior for an out-of-equilibrium countable one-dimensional system of Brownian particles interacting through their rank-dependent drifts. Focusing on the semi-infinite case, where only the leftmost particle gets a constant drift to the right, we derive and solve the corresponding one- sided Stefan (free-boundary) equations. Via this solution we explicitly determine the limiting particle-density profile as well as the asymptotic trajectory of the leftmost particle. While doing so we further establish stochastic domination and convergence to equilibrium results for the vector of relative spacings among the leading particles.

math.PR↗
arXiv2017

Large deviations for diffusions interacting through their ranks

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 according to the porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of a LDP for interacting diffusions, where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for a certain tilted version of the porous medium equation.

math.PR↗
arXiv2017

Universal Persistence for Local Time of One-dimensional Random Walk

We prove the power law decay $p(t,x) \sim t^{-ϕ(x,b)/2}$ in which $p(t,x)$ is the probability that the fraction of time up to $t$ in which a random walk $S$ of i.i.d. zero-mean increments taking finitely many values, is non-negative, exceeds $x$ throughout $s \in [1,t]$. Here $ϕ(x,b)= \mathbb{P}(\text{Lévy}(1/2,κ(x,b))<0)$ for $κ(x,b) = \frac{\sqrt{1-x} b - \sqrt{1+x}}{\sqrt{1-x} b + \sqrt{1+x}}$ and $b=b_S \geq 0$ measuring the asymptotic asymmetry between positive and negative excursions of the walk (with $b_s=1$ for symmetric increments).

math.PR↗
arXiv2016

Persistence of Gaussian processes: non-summable correlations

Suppose the auto-correlations of real-valued, centered Gaussian process $Z(\cdot)$ are non-negative and decay as $ρ(|s-t|)$ for some $ρ(\cdot)$ regularly varying at infinity of order $-α\in [-1,0)$. With $I_ρ(t)=\int_0^t ρ(s)ds$ its primitive, we show that the persistence probabilities decay rate of $ -\log\mathbb{P}(\sup_{t \in [0,T]}\{Z(t)\}<0)$ is precisely of order $(T/I_ρ(T)) \log I_ρ(T)$, thereby closing the gap between the lower and upper bounds of \cite{NR}, which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of \cite{Sak} about the dependence on $d$ of such persistence decay for the Langevin dynamics of certain $\grad ϕ$-interface models on $\Z^d$.

math.PR↗
arXiv2016

Nonlinear large deviations

We present a general technique for computing large deviations of nonlinear functions of independent Bernoulli random variables. The method is applied to compute the large deviation rate functions for subgraph counts in sparse random graphs. Previous technology, based on Szemeredi's regularity lemma, works only for dense graphs. Applications are also made to exponential random graphs and three-term arithmetic progressions in random sets of integers.

math.PR↗
arXiv2016

Transience in growing subgraphs via evolving sets

We extend the use of random evolving sets to time-varying conductance models and utilize it to provide tight heat kernel upper bounds. It yields the transience of any uniformly lazy random walk, on Z^d, d>=3, equipped with uniformly bounded above and below, independently time-varying edge conductances, of (effectively) non-decreasing in time vertex conductances (i.e. reversing measure), thereby affirming part of [ABGK, Conj. 7.1].

math.PR↗
arXiv2015

Ferromagnetic Ising Measures on Large Locally Tree-Like Graphs

We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, Sly '11], under an appropriate "continuity" property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with $+$ and $-$ boundary conditions on that tree. Under the extra assumptions that $G_n$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization, is the Ising measure with $+$ boundary condition on the limiting tree. The "continuity" property holds except possibly for countably many choices of $β$, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton Watson trees.

math.PR↗
arXiv2015

Weakly Asymmetric Non-Simple Exclusion Process and the Kardar-Parisi-Zhang Equation

We analyze a class of non-simple exclusion processes and the corresponding growth models by generalizing Gaertners Cole-Hopf transformation. We identify the main non-linearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the Kardar-Parisi-Zhang (KPZ) equation. This is the first universality result concerning interacting particle systems in the context of KPZ universality class. While this class of exclusion processes are not explicitly solvable, we obtain the exact one-point limiting distribution for the step initial condition by using the previous result of Amir et al. (2011) and our convergence result.

math.PR↗
arXiv2015

Extremal Cuts of Sparse Random Graphs

For Erdős-Rényi random graphs with average degree $γ$, and uniformly random $γ$-regular graph on $n$ vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are $n\Big(\fracγ{4} + {\sf P}_* \sqrt{\fracγ{4}} + o(\sqrtγ)\Big) + o(n)$ while the size of the minimum bisection is $n\Big(\fracγ{4}-{\sf P}_*\sqrt{\fracγ{4}} + o(\sqrtγ)\Big) + o(n)$. Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington-Kirkpatrick model, with ${\sf P}_* \approx 0.7632$ standing for the ground state energy of the latter, expressed analytically via Parisi's formula.

math.PR↗
arXiv2015

Equilibrium Fluctuation of the Atlas Model

We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite ($ \mathbb{Z}_+ $-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on $ \mathbb{R}_+ $. In this context, we show that the joint law of ranked particles, after being centered and scaled by $t^{-1/4}$, converges as $t \to \infty$ to the Gaussian field corresponding to the solution of the additive stochastic heat equation on $\mathbb{R}_+$ with Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a $ \frac{1}{4} $-fractional Brownian motion. In particular, we prove a conjecture of Pal and Pitman (2008) about the asymptotic Gaussian fluctuation of the ranked particles.

math.PR↗
arXiv2015

No zero-crossings for random polynomials and the heat equation

Consider random polynomial $\sum_{i=0}^na_ix^i$ of independent mean-zero normal coefficients $a_i$, whose variance is a regularly varying function (in $i$) of order $α$. We derive general criteria for continuity of persistence exponents for centered Gaussian processes, and use these to show that such polynomial has no roots in $[0,1]$ with probability $n^{-b_α+o(1)}$, and no roots in $(1,\infty)$ with probability $n^{-b_0+o(1)}$, hence for $n$ even, it has no real roots with probability $n^{-2b_α-2b_0+o(1)}$. Here, $b_α=0$ when $α\le-1$ and otherwise $b_α\in(0,\infty)$ is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path. Further, making precise the solution $ϕ_d({\mathbf{x}},t)$ to the $d$-dimensional heat equation initiated by a Gaussian white noise $ϕ_d({\mathbf{x}},0)$, we confirm that the probability of $ϕ_d({\mathbf{x}},t)\neq0$ for all $t\in[1,T]$, is $T^{-b_α+o(1)}$, for $α=d/2-1$.

math.PR↗
arXiv2014

Matrix optimization under random external fields

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).

math.PR↗
arXiv2013

Factor models on locally tree-like graphs

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $ϕ$, the limit of the log-partition function divided by the number of vertices $n$ as $n$ tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity $ϕ$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $ϕ$ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on $ϕ$. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on $T$. In the special case that $T$ has a Galton-Watson law, this formula coincides with the nonrigorous "Bethe prediction" obtained by statistical physicists using the "replica" or "cavity" methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.

math.PR↗
arXiv2013

On level sets of Gaussian fields

In this short note, we present a theorem concerning certain "additive structure" for the level sets of non-degenerate Gaussian fields, which yields the multiple valley phenomenon for extremal fields with exponentially many valleys.

math.PR↗
arXiv2013

Limiting Spectral Distribution of Sum of Unitary and Orthogonal Matrices

We show that the empirical eigenvalue measure for sum of $d$ independent Haar distributed $n$-dimensional unitary matrices, converge for $n \to \infty$ to the Brown measure of the free sum of $d$ Haar unitary operators. The same applies for independent Haar distributed $n$-dimensional orthogonal matrices. As a byproduct of our approach, we relax the requirement of uniformly bounded imaginary part of Stieltjes transform of $T_n$ that is made in [Guionnet, Krishnapur, Zeitouni; Theorem 1].

math.PR↗
arXiv2012

The replica symmetric solution for Potts models on d-regular graphs

We provide an explicit formula for the limiting free energy density (log-partition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequences converging locally to the d-regular tree for d even, covering all temperature regimes. This formula coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d-regular tree, the so-called replica symmetric solution. For uniformly random d-regular graphs we further show that the replica symmetric Bethe formula is an upper bound for the asymptotic free energy for any model with permissive interactions.

math.PR↗