Publications (180)162.69 Total impact
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ABSTRACT: We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semiinfinite ($ \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.  [Show abstract] [Hide abstract]
ABSTRACT: For Erd\H{o}sR\'enyi random graphs with average degree $\gamma$, and uniformly random $\gamma$regular graph on $n$ vertices, we prove that with high probability the size of both the MaxCut and maximum bisection are $n\Big(\frac{\gamma}{4} + {{\sf P}}_* \sqrt{\frac{\gamma}{4}} + o(\sqrt{\gamma})\Big) + o(n)$ while the size of the minimum bisection is $n\Big(\frac{\gamma}{4}{{\sf P}}_*\sqrt{\frac{\gamma}{4}} + o(\sqrt{\gamma})\Big) + o(n)$. Our derivation relates the free energy of the antiferromagnetic Ising model on such graphs to that of the SherringtonKirkpatrick model, with ${{\sf P}}_* \approx 0.7632$ standing for the ground state energy of the latter, expressed analytically via Parisi's formula.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the quadratic optimization problem $$F_n^{W,h}:= \sup_{x \in S^{n1}} ( x^T W x/2 + h^T x )\,, $$ 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 nonrigorous replica method in Y. V. Fyodorov, P. Le Doussal, J. Stat. phys. 154 (2014).Journal of Statistical Physics 09/2014; 159(6). DOI:10.1007/s1095501512287 · 1.28 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider recurrence versus transience for models of random walks on domains of $\mathbb{Z}^d$, in which monotone interaction enforces domain growth as a result of visits by the walk (or probes it sent), to the neighborhood of domain boundary.Electronic communications in probability 06/2014; 19. DOI:10.1214/ECP.v193607 · 0.63 Impact Factor 
Article: Nonlinear large deviations
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ABSTRACT: 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 Szemer\'edi's regularity lemma, works only for dense graphs. Applications are also made to exponential random graphs and threeterm arithmetic progressions in random sets of integers.  [Show abstract] [Hide abstract]
ABSTRACT: For normally reflected Brownian motion and for simple random walk on growing in time ddimensional domains, d>=3, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.ELECTRONIC JOURNAL OF PROBABILITY 12/2013; 19. DOI:10.1214/EJP.v193272 · 0.77 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Suppose that ${\mathcal G}$ is a finite, connected graph and $X$ is a lazy random walk on ${\mathcal G}$. The lamplighter chain $X^\diamond$ associated with $X$ is the lazy random walk on the wreath product ${\mathcal G}^\diamond = {\mathbb Z}_2 \wr {\mathcal G}$, the graph whose vertices consist of pairs $(\underline{f},x)$ where $\underline{f}$ is a $\{0,1\}$labeling of the vertices of ${\mathcal G}$ and $x$ is a vertex in ${\mathcal G}$. In each step, $X^\diamond$ moves from a configuration $(\underline{f},x)$ by updating $x$ to $y$ using the transition rule of $X$, and if $x \ne y$, replacing $f_x$ and $f_y$ by two independent uniform random bits. The mixing time of the lamplighter chain on the discrete torus ${\mathbb Z}_n^d$ is known to have a cutoff at a time asymptotic to the cover time of ${\mathbb Z}_n^d$ if $d=2$, and to half the cover time if $d \geq 3$. We show that the mixing time of the lamplighter chain on ${\mathcal G}_n(a)={\mathbb Z}_n^2 \times {\mathbb Z}_{a \log n}$ has a cutoff at $\psi(a)$ times the cover time of ${\mathcal G}_n(a)$ as $n \to \infty$, where $\psi$ is a weakly decreasing map from $(0,\infty)$ onto $[1/2,1)$. In particular, as $a > 0$ varies, the threshold continuously interpolates between the known thresholds for ${\mathbb Z}_n^2$ and ${\mathbb Z}_n^3$. Perhaps surprisingly, we find a phase transition (nonsmoothness of $\psi$) at the point $a_*=\pi r_3 (1+\sqrt{2})$, where high dimensional behavior ($\psi(a)=1/2$ for all $a>a_*$) commences. Here $r_3$ is the effective resistance from $0$ to $\infty$ in ${\mathbb Z}^3$. 
Article: On level sets of Gaussian fields
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ABSTRACT: In this short note, we present a theorem concerning certain "additive structure" for the level sets of nondegenerate Gaussian fields, which yields the multiple valley phenomenon for extremal fields with exponentially many valleys.  [Show abstract] [Hide abstract]
ABSTRACT: We analyze a class of nonsimple exclusion processes and the corresponding growth models by generalizing the Gaertner transformation. While the original argument, which matches three identities with three parameters, applies only to simple exclusion process, our approach is to identify the major nonlinear drift term of the microscopic dynamical equation and to convert the drift term into a quasilinear second order differential, yielding the stochastic heat equation (SHE) in the continuum limit. Using the generalized transformation, we prove convergence toward the KPZ equation, which is the first universality result of this kind in the context of KPZ universality class. While this class of exclusion processes are not explicitly solvable, we obtain the exact onepoint limit distribution for the step and step Bernoulli initial conditions by using the previous results of Amir et al. (2011), Corwin & Quastel (2010), and our convergence result.  [Show abstract] [Hide abstract]
ABSTRACT: 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 McKeanVlasov 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.  [Show abstract] [Hide abstract]
ABSTRACT: 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].Electronic communications in probability 08/2012; DOI:10.1214/ECP.v182466 · 0.63 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Consider random polynomial $\sum_{i=0}^n a_i x^i$ of independent meanzero normal coefficients $a_i$, whose variance is a regularly varying function (in $i$) of order $\alpha$. 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 $n^{b_\alpha+o(1)}$, and no roots in $(1,\infty)$ with probability $n^{b_0+o(1)}$, hence for $n$ even, it has no real roots with probability $n^{2b_\alpha  2b_0+o(1)}$. Here $b_\alpha =0$ when $\alpha \le 1$ and otherwise $b_\alpha \in (0,\infty)$ 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 $\phi_d(x,t)$ to the $d$dimensional heat equation initiated by a Gaussian white noise $\phi_d(x,0)$, we confirm that the probability of $\phi_d(x,t)\neq 0$ for all $t\in [1,T]$, is $T^{b_{\alpha} + o(1)}$, for $\alpha=d/21$.The Annals of Probability 08/2012; 43(1). DOI:10.1214/13AOP852 · 1.43 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We provide an explicit formula for the limiting free energy density (logpartition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequences converging locally to the dregular 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 dregular tree, the socalled replica symmetric solution. For uniformly random dregular 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.Communications in Mathematical Physics 07/2012; 327(2). DOI:10.1007/s0022001419566 · 1.90 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: 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 $G_n$, of uniformly bounded degrees, are edgeexpanders, and ergodicity of the simple random walk on the limiting tree, 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. We confirm the "continuity" and ergodicity properties in case of limiting (multitype) Galton Watson trees, and the edgeexpander property for the corresponding configuration model graphs. 
Article: Persistence of iterated partial sums
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ABSTRACT: Let $S_n^{(2)}$ denote the iterated partial sums. That is, $S_n^{(2)}=S_1+S_2+ ... +S_n$, where $S_i=X_1+X_2+ ... s+X_i$. Assuming $X_1, X_2,....,X_n$ are integrable, zeromean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{(2)}:=\PP(\max_{1\le i \le n}S_i^{(2)}< 0) \le c\sqrt{\frac{\EES_{n+1}}{(n+1)\EEX_1}},$$ with $c \le 6 \sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the nonzero $\min(X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{(2)}\asymp n^{1/4}$ for any nondegenerate squared integrable, i.i.d., zeromean $X_i$. In contrast, we show that for any $0 < \gamma < 1/4$ there exist integrable, zeromean random variables for which the rate of decay of $p_n^{(2)}$ is $n^{\gamma}$.Annales de l Institut Henri PoincarÃ© ProbabilitÃ©s et Statistiques 05/2012; 49(3). DOI:10.1214/11AIHP452 · 0.97 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: 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 phi, the limit of the logpartition 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 phi subject to uniqueness of a relevant Gibbs measure for the factor model on T. By way of example we compute phi for the independent set (or hardcore) 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 phi. 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 GaltonWatson 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.The Annals of Probability 10/2011; DOI:10.1214/12AOP828 · 1.43 Impact Factor 
Article: Persistence of iterated partial sums
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ABSTRACT: Let p_n denote the persistence probability that the first n iterated partial sums of integrable, zeromean, i.i.d. random variables X_k, are negative. We show that p_n is bounded above up to universal constant by the square root of the expected absolute value of the empirical average of {X_k}. A converse bound holds whenever P(X_1>t) is up to constant exp(b t) for some b>0 or when P(X_1>t) decays superexponentially in t. Consequently, for such random variables we have that p_n decays as n^{1/4} if X_1 has finite second moment. In contrast, we show that for any 0 < c < 1/4 there exist integrable, zeromean random variables for which the rate of decay of p_n is n^{c}.  [Show abstract] [Hide abstract]
ABSTRACT: Let T be a rooted supercritical multitype GaltonWatson (MGW) tree with types coming from a finite alphabet, conditioned to nonextinction. The lambdabiased random walk (X_t, t>=0) on T is the nearestneighbor random walk which, when at a vertex v with d(v) offspring, moves closer to the root with probability lambda/[lambda+d(v)], and to each of the offspring with probability 1/[lambda+d(v)]. This walk is recurrent for lambda>=rho and transient for 0<lambda<rho, with rho the PerronFrobenius eigenvalue for the (assumed) irreducible matrix of expected offspring numbers. Subject to finite moments of order p>4 for the offspring distributions, we prove the following quenched CLT for lambdabiased random walk at the critical value lambda=rho: for almost every T, the process X_{floor(nt)}/sqrt{n} converges in law as n tends to infinity to a reflected Brownian motion rescaled by an explicit constant. This result was proved under some stronger assumptions by PeresZeitouni (2008) for singletype GaltonWatson trees. Following their approach, our proof is based on a new explicit description of a reversing measure for the walk from the point of view of the particle (generalizing the measure constructed in the singletype setting by PeresZeitouni), and the construction of appropriate harmonic coordinates. In carrying out this program we prove moment and conductance estimates for MGW trees, which may be of independent interest. In addition, we extend our construction of the reversing measure to a biased random walk with random environment (RWRE) on MGW trees, again at a critical value of the bias. We compare this result against a transiencerecurrence criterion for the RWRE generalizing a result of Faraud (2011) for GaltonWatson trees.ELECTRONIC JOURNAL OF PROBABILITY 11/2010; DOI:10.1214/EJP.v172294 · 0.77 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We review this approach and provide some results towards a rigorous treatment of these problems. Comment: 111 pages, 3 eps figures, Lecture notes for the 2008 Brazilian School of Probability (to appear in BJPS without Sections 1.3 and 6)Brazilian Journal of Probability and Statistics 10/2009; DOI:10.1214/09BJPS027 · 0.47 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $\mu$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N by N symmetric matrix $Y_N^\sigma$ whose (i,j) entry is $\sigma(i/N,j/N)X_{ij}$ where $(X_{ij}, 0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an $\alpha$stable law, $0<\alpha<2$, and $\sigma$ is a deterministic function. For a random diagonal $D_N$ independent of $Y_N^\sigma$ and with appropriate rescaling $a_N$, we prove that the distribution $\mu$ of $a_N^{1}Y_N^\sigma + D_N$ converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries.Communications in Mathematical Physics 12/2008; DOI:10.1007/s0022000908224 · 1.90 Impact Factor
Publication Stats
5k  Citations  
162.69  Total Impact Points  
Top Journals
Institutions

1988–2013

Stanford University
 • Department of Mathematics
 • Information Systems Laboratory
Palo Alto, California, United States


1984–1995

Technion  Israel Institute of Technology
 Electrical Engineering Group
HÌ±efa, Haifa District, Israel


1987–1990

Brown University
 Department of Applied Mathematics
Providence, Rhode Island, United States


1989

AT&T Labs
Austin, Texas, United States
