Amir Dembo

Stanford University, Palo Alto, California, United States

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Publications (182)167.86 Total impact

  • Amir Dembo · Sumit Mukherjee
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    ABSTRACT: Suppose the auto-correlations of real-valued, centered Gaussian process $Z(\cdot)$ are non-negative and decay as $\rho(|s-t|)$ for some $\rho(\cdot)$ regularly varying at infinity of order $-\alpha \in [-1,0)$. With $I_\rho(t)=\int_0^t \rho(s)ds$ its primitive, we determine the precise persistence probabilities decay rate $$ -\log\mathbb{P}(\sup_{t \in [0,T]}\{Z(t)\}<0)=\Theta\Big(\frac{T\log I_\rho(T)}{I_\rho(T)}\Big) \,, $$ thereby closing the gap between the lower and upper bounds of {NR}, which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of {Sak} about the dependence on $d$ of such persistence decay for the Langevin dynamics of certain $\nabla \phi$-interface models on $\mathbb{Z}^d$.
  • Amir Dembo · Ruojun Huang · Ben Morris · Yuval Peres
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    ABSTRACT: 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, having non-decreasing in time vertex conductances (i.e. reversing measure), thereby affirming part of [ABGK, Conj. 7.1].
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    Amir Dembo · Li-Cheng Tsai
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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 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.
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    Amir Dembo · Andrea Montanari · Subhabrata Sen
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    ABSTRACT: For Erd\H{o}s-R\'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 Max-Cut 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 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.
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    Amir Dembo · Ofer Zeitouni
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    ABSTRACT: 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).
    Journal of Statistical Physics 09/2014; 159(6). DOI:10.1007/s10955-015-1228-7 · 1.28 Impact Factor
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    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.v19-3607 · 0.63 Impact Factor
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    Sourav Chatterjee · Amir Dembo
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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 three-term arithmetic progressions in random sets of integers.
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    ABSTRACT: For normally reflected Brownian motion and for simple random walk on growing in time d-dimensional 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.v19-3272 · 0.77 Impact Factor
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    Amir Dembo · Jian Ding · Jason Miller · Yuval Peres
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    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 (non-smoothness 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 ${\mathbb Z}^3$.
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    Sourav Chatterjee · Amir Dembo · Jian Ding
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    ABSTRACT: 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.
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    Amir Dembo · Li-Cheng Tsai
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    ABSTRACT: We analyze a class of non-simple 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 non-linear drift term of the microscopic dynamical equation and to convert the drift term into a quasi-linear 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 one-point 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.
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    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 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.
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    Anirban Basak · Amir Dembo
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    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.v18-2466 · 0.63 Impact Factor
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    Amir Dembo · Sumit Mukherjee
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    ABSTRACT: Consider random polynomial $\sum_{i=0}^n a_i x^i$ of independent mean-zero 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 $\alpha=d/2-1$.
    The Annals of Probability 08/2012; 43(1). DOI:10.1214/13-AOP852 · 1.43 Impact Factor
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    Amir Dembo · Andrea Montanari · Allan Sly · Nike Sun
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    ABSTRACT: 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.
    Communications in Mathematical Physics 07/2012; 327(2). DOI:10.1007/s00220-014-1956-6 · 1.90 Impact Factor
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    Anirban Basak · Amir Dembo
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    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 edge-expanders, 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 (multi-type) Galton Watson trees, and the edge-expander property for the corresponding configuration model graphs.
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    Amir Dembo · Jian Ding · Fuchang Gao
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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, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{(2)}:=\PP(\max_{1\le i \le n}S_i^{(2)}< 0) \le c\sqrt{\frac{\EE|S_{n+1}|}{(n+1)\EE|X_1|}},$$ with $c \le 6 \sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $\min(-X_1,0)$ 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 non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0 < \gamma < 1/4$ there exist integrable, zero-mean random variables for which the rate of decay of $p_n^{(2)}$ is $n^{-\gamma}$.
    Annales de l Institut Henri Poincaré Probabilités et Statistiques 05/2012; 49(3). DOI:10.1214/11-AIHP452 · 0.97 Impact Factor
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    Amir Dembo · Andrea Montanari · Nike Sun
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    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 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 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 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 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 Galton-Watson law, this formula coincides with the non-rigorous "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; 41(6). DOI:10.1214/12-AOP828 · 1.43 Impact Factor
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    Amir Dembo · Fuchang Gao
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    ABSTRACT: Let p_n denote the persistence probability that the first n iterated partial sums of integrable, zero-mean, 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 super-exponentially 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, zero-mean random variables for which the rate of decay of p_n is n^{-c}.
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    Amir Dembo · Nike Sun
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    ABSTRACT: Let T be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor 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 Perron-Frobenius 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 lambda-biased 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 Peres-Zeitouni (2008) for single-type Galton-Watson 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 single-type setting by Peres-Zeitouni), 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 transience-recurrence criterion for the RWRE generalizing a result of Faraud (2011) for Galton-Watson trees.
    ELECTRONIC JOURNAL OF PROBABILITY 11/2010; DOI:10.1214/EJP.v17-2294 · 0.77 Impact Factor

Publication Stats

6k Citations
167.86 Total Impact Points

Institutions

  • 1988–2013
    • Stanford University
      • • Department of Mathematics
      • • Department of Statistics
      • • Information Systems Laboratory
      Palo Alto, California, United States
  • 1984–1995
    • Technion - Israel Institute of Technology
      • Electrical Engineering Group
      H̱efa, Haifa District, Israel
  • 1987–1990
    • Brown University
      • Department of Applied Mathematics
      Providence, Rhode Island, United States
  • 1989
    • AT&T Labs
      Austin, Texas, United States