Showing papers by "Amir Dembo published in 2021"
TL;DR: In this article, the spectrum of a random multigraph with a degree sequence and average degree 1 ≪ ωn ≪ n, generated by the configuration model, and also of an analogous random simple graph was studied.
Abstract: We study the spectrum of a random multigraph with a degree sequence \({\mathbf {D}}_n=(D_i)_{i=1}^n\) and average degree 1 ≪ ωn ≪ n, generated by the configuration model, and also the spectrum of the analogous random simple graph. We show that, when the empirical spectral distribution (ESD) of \(\omega _n^{-1} {\mathbf {D}}_n \) converges weakly to a limit ν, under mild moment assumptions (e.g., Di∕ωn are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to \(
u \boxtimes \sigma _{{\text{SC}}}\), the free multiplicative convolution of ν with the semicircle law. Relating this limit with a variant of the Marchenko–Pastur law yields the continuity of its density (away from zero), and an effective procedure for determining its support. Our proof of convergence is based on a coupling between the random simple graph and multigraph with the same degrees, which might be of independent interest. We further construct and rely on a coupling of the multigraph to an inhomogeneous Erdős-Renyi graph with the target ESD, using three intermediate random graphs, with a negligible fraction of edges modified in each step.
12 citations
TL;DR: In this article, the authors prove that the sequence of random variables Cn 2−(n+1)−mn, where mn is an explicit constant, converges in distribution as n→∞.
Abstract: Let Tn denote the binary tree of depth n augmented by an extra edge connected to its root. Let Cn denote the cover time of Tn by simple random walk. We prove that the sequence of random variables Cn 2−(n+1)−mn, where mn is an explicit constant, converges in distribution as n→∞, and identify the limit.
11 citations
TL;DR: In this article, 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 was established.
Abstract: 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 Erdős-Renyi 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 Erdős-Renyi graphs.
6 citations
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TL;DR: In this paper, a quantitative large deviations theory for random Bernoulli tensors is developed, based on a decomposition theorem for arbitrary tensors outside a set of tiny measure, in terms of a novel family of norms generalizing the cut norm.
Abstract: We develop a quantitative large deviations theory for random Bernoulli tensors. The large deviation principles rest on a decomposition theorem for arbitrary tensors outside a set of tiny measure, in terms of a novel family of norms generalizing the cut norm. Combined with associated counting lemmas, these yield sharp asymptotics for upper tails of homomorphism counts in the $r$-uniform Erdős--Renyi hypergraph for any fixed $r\ge 2$, generalizing and improving on previous results for the Erdős--Renyi graph ($r=2$). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields (joint) upper and lower tail asymptotics for other nonlinear functionals of interest.
3 citations
TL;DR: In this article, it was shown that the trajectories of averaged observables of dynamical systems with random coefficients are universal, i.e., only depend on the choice of the distribution through its first and second moments.
Abstract: 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.
2 citations
TL;DR: In this paper, a general approach to study a class of random growth models in $n$-dimensional Euclidean space is presented, which 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.
Abstract: 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.