Amir Dembo
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Amir Dembo has a total of 29 co-authors. Here are the top co-authors by number of shared publications:
- Ofer Zeitouni
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
4 shared publications - Li-Cheng Tsai
Departments of Mathematics, University of Utah, Salt Lake City, USA
3 shared publications - Andrey Sarantsev
Department of Mathematics and Statistics, University of Nevada, Reno, USA
2 shared publications - Eyal Lubetzky
Courant Institute, New York University, 251 Mercer Street, New York, New York 10012
2 shared publications - Nicholas A. Cook
Department of Mathematics Duke University a
2 shared publications - Reza Gheissari
Departments of Statistics and EECS, University of California, Berkeley, USA
2 shared publications
Latest Publications
- 13 August 2025
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 249 (5) , 1-70
kpz-Type Equation from Growth Driven by a Non-Markovian Diffusion
- 13 August 2025
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 249 (5) , 1-70
kpz-Type Equation from Growth Driven by a Non-Markovian Diffusion
- Open Access
Capacity of the range of random walk: The law of the iterated logarithm
We establish both the $\limsup$ and the $\liminf$ law of the iterated logarithm (LIL), for the capacity of the range of a simple random walk in any dimension $d\ge 3$. While for $d \ge 4$, the order of growth in $n$ of such LIL at dimension $d$ matches that for the volume of the random walk range in dimension $d-2$, somewhat surprisingly this correspondence breaks down for the capacity of the range at $d=3$. We further establish such LIL for the Brownian capacity of a $3$-dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a $4$-dimensional simple random walk.Open AccessCapacity of the range of random walk: The law of the iterated logarithm
We establish both the $\limsup$ and the $\liminf$ law of the iterated logarithm (LIL), for the capacity of the range of a simple random walk in any dimension $d\ge 3$. While for $d \ge 4$, the order of growth in $n$ of such LIL at dimension $d$ matches that for the volume of the random walk range in dimension $d-2$, somewhat surprisingly this correspondence breaks down for the capacity of the range at $d=3$. We further establish such LIL for the Brownian capacity of a $3$-dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a $4$-dimensional simple random walk. - Open Access
Regularity method and large deviation principles for the Erdős--Rényi hypergraph
We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails of homomorphism counts in the $r$-uniform Erd\H{o}s--R\'enyi hypergraph for any fixed $r\ge 2$, generalizing and improving on previous results for the Erd\H{o}s--R\'enyi graph ($r=2$). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields tail asymptotics for other nonlinear functionals, such as induced homomorphism counts.Open AccessRegularity method and large deviation principles for the Erdős--Rényi hypergraph
We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails of homomorphism counts in the $r$-uniform Erd\H{o}s--R\'enyi hypergraph for any fixed $r\ge 2$, generalizing and improving on previous results for the Erd\H{o}s--R\'enyi graph ($r=2$). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields tail asymptotics for other nonlinear functionals, such as induced homomorphism counts. - Open Access
On the limiting law of line ensembles of Brownian polymers with geometric area tilts
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 the law of the top $k$ paths, for any fixed $k$, was established under either zero or free boundary conditions, which in the former setting implied the existence of a limit via a monotonicity argument. Here we address the open problem of a limit under free boundary conditions: we prove that as the interval length, followed by the number of paths, go to $\infty$, the top $k$ paths converge to the same limit as in the free boundary case, as conjectured by Caputo, Ioffe and Wachtel.Open AccessOn the limiting law of line ensembles of Brownian polymers with geometric area tilts
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 the law of the top $k$ paths, for any fixed $k$, was established under either zero or free boundary conditions, which in the former setting implied the existence of a limit via a monotonicity argument. Here we address the open problem of a limit under free boundary conditions: we prove that as the interval length, followed by the number of paths, go to $\infty$, the top $k$ paths converge to the same limit as in the free boundary case, as conjectured by Caputo, Ioffe and Wachtel. - Open Access
- 4 February 2021
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 180 (3-4) , 1057-1097
Diffusions interacting through a random matrix: universality via stochastic Taylor expansion
Consider $$(X_{i}(t))$$ solving a system ofNstochastic differential equations interacting through a random matrix $${\mathbf {J}} = (J_{ij})$$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$ , initialized from some $$\mu $$ independent of $${\mathbf {J}}$$ , are universal, i.e., only depend on the choice of the distribution $$\mathbf {J}$$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.- 4 February 2021
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 180 (3-4) , 1057-1097
Open AccessDiffusions interacting through a random matrix: universality via stochastic Taylor expansion
Consider $$(X_{i}(t))$$ solving a system ofNstochastic differential equations interacting through a random matrix $${\mathbf {J}} = (J_{ij})$$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$ , initialized from some $$\mu $$ independent of $${\mathbf {J}}$$ , are universal, i.e., only depend on the choice of the distribution $$\mathbf {J}$$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks. - Open Access
Diffusions interacting through a random matrix: universality via stochastic Taylor expansion
Consider $(X_{i}(t))$ solving a system of $N$ stochastic differential equations interacting through a random matrix $\mathbf J = (J_{ij})$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $(X_i(t))$, initialized from some $\mu$ independent of $\mathbf J$, are universal, i.e., only depend on the choice of the distribution $\mathbf{J}$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.Open AccessDiffusions interacting through a random matrix: universality via stochastic Taylor expansion
Consider $(X_{i}(t))$ solving a system of $N$ stochastic differential equations interacting through a random matrix $\mathbf J = (J_{ij})$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $(X_i(t))$, initialized from some $\mu$ independent of $\mathbf J$, are universal, i.e., only depend on the choice of the distribution $\mathbf{J}$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks. - Open Access
Upper Tail For Homomorphism Counts In Constrained Sparse Random Graphs
Consider the upper tail probability that the homomorphism count of a fixed graph $H$ within a large sparse random graph $G_n$ exceeds its expected value by a fixed factor $1+\delta$. Going beyond the Erd\H{o}s-R\'enyi model, we establish here explicit, sharp upper tail decay rates for sparse random $d_n$-regular graphs (provided $H$ has a regular $2$-core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs $H_1,\ldots, H_k$ (extending the known results for $k=1$), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erd\H{o}s-R\'enyi graphs.Open AccessUpper Tail For Homomorphism Counts In Constrained Sparse Random Graphs
Consider the upper tail probability that the homomorphism count of a fixed graph $H$ within a large sparse random graph $G_n$ exceeds its expected value by a fixed factor $1+\delta$. Going beyond the Erd\H{o}s-R\'enyi model, we establish here explicit, sharp upper tail decay rates for sparse random $d_n$-regular graphs (provided $H$ has a regular $2$-core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs $H_1,\ldots, H_k$ (extending the known results for $k=1$), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erd\H{o}s-R\'enyi graphs. - 29 November 2020
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 74 (7) , 1453-1492
Averaging Principle and Shape Theorem for a Growth Model with Memory
We present a general approach to study a class of random growth models in n‐dimensional Euclidean space. These models are designed to capture basic growth features that are expected to manifest at the mesoscopic level for several classical self‐interacting processes originally defined at the microscopic scale. It includes once‐reinforced random walk with strong reinforcement, origin‐excited random walk, and a few others, for which the set of visited vertices is expected to form a limiting shape. We prove an averaging principle that leads to such a shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain. © 2020 Wiley Periodicals LLC
- 29 November 2020
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 74 (7) , 1453-1492
Averaging Principle and Shape Theorem for a Growth Model with Memory
We present a general approach to study a class of random growth models in n‐dimensional Euclidean space. These models are designed to capture basic growth features that are expected to manifest at the mesoscopic level for several classical self‐interacting processes originally defined at the microscopic scale. It includes once‐reinforced random walk with strong reinforcement, origin‐excited random walk, and a few others, for which the set of visited vertices is expected to form a limiting shape. We prove an averaging principle that leads to such a shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain. © 2020 Wiley Periodicals LLC
- Open Access
Averaging Principle and Shape Theorem for a Growth Model with Memory
We present a general approach to study a class of random growth models in $n$-dimensional Euclidean space. These models are designed to capture basic growth features which are expected to manifest at the mesoscopic level for several classical self-interacting processes originally defined at the microscopic scale. It includes once-reinforced random walk with strong reinforcement, origin-excited random walk, and few others, for which the set of visited vertices is expected to form a "limiting shape". We prove an averaging principle that leads to such shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain.
Open AccessAveraging Principle and Shape Theorem for a Growth Model with Memory
We present a general approach to study a class of random growth models in $n$-dimensional Euclidean space. These models are designed to capture basic growth features which are expected to manifest at the mesoscopic level for several classical self-interacting processes originally defined at the microscopic scale. It includes once-reinforced random walk with strong reinforcement, origin-excited random walk, and few others, for which the set of visited vertices is expected to form a "limiting shape". We prove an averaging principle that leads to such shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain.
- 6 July 2020
- Published by Springer Nature in Journal of Statistical Physics
- Vol. 181 (2) , 465-514
Dynamics for Spherical Spin Glasses: Disorder Dependent Initial Conditions
- 6 July 2020
- Published by Springer Nature in Journal of Statistical Physics
- Vol. 181 (2) , 465-514
Dynamics for Spherical Spin Glasses: Disorder Dependent Initial Conditions