arXiv subjects

Ron Rosenthal

Publications and source records attributed to Ron Rosenthal.

At least 19 recordsLinked to original sources

arXiv2022

Eigenvalues and spectral gap in sparse random simplicial complexes

We consider the adjacency operator $A$ of the Linial-Meshulam model $X(d,n,p)$ for random $d-$dimensional simplicial complexes on $n$ vertices, where each $d-$cell is added independently with probability $p\in[0,1]$ to the complete $(d-1)$-skeleton. We consider sparse random matrices $H$, which are generalizations of the centered and normalized adjacency matrix $\mathcal{A}:=(np(1-p))^{-1/2}\cdot(A-\mathbb{E}\left[A\right])$, obtained by replacing the Bernoulli$(p)$ random variables used to construct $A$ with arbitrary bounded distribution $Z$. We obtain bounds on the expected Schatten norm of $H$, which allow us to prove results on eigenvalue confinement and in particular that $\left\Vert H\right\Vert _{2}$ converges to $2\sqrt{d}$ both in expectation and $\mathbb{P}-$almost surely as $n\to\infty$, provided that $\mathrm{Var}(Z)\gg\frac{\log n}{n}$. The main ingredient in the proof is a generalization of [LVHY18,Theorem 4.8] to the context of high-dimensional simplicial complexes, which may be regarded as sparse random matrix models with dependent entries.

math.PR↗

arXiv2021

The chemical distance in random interlacements in the low-intensity regime

In $\mathbb{Z}^d$ with $d\ge 5$, we consider the time constant $\rho_u$ associated to the chemical distance in random interlacements at low intensity $u \ll 1$. We prove an upper bound of order $u^{-1/2}$ and a lower bound of order $u^{-1/2+\varepsilon}$. The upper bound agrees with the conjectured scale in which $u^{1/2}\rho_u$ converges to a constant multiple of the Euclidean norm, as $u\to 0$. Along the proof, we obtain a local lower bound on the chemical distance between the boundaries of two concentric boxes, which might be of independent interest. For both upper and lower bounds, the paper employs probabilistic bounds holding as $u\to 0$; these bounds can be relevant in future studies of the low-intensity geometry.

math.PR↗

arXiv2020

Simplicial spanning trees in random Steiner complexes

A spanning tree $T$ in a graph $G$ is a sub-graph of $G$ with the same vertex set as $G$ which is a tree. In 1981, McKay proved an asymptotic result regarding the number of spanning trees in random $k$-regular graphs. In this paper we prove a high-dimensional generalization of McKay's result for random $d$-dimensional, $k$-regular simplicial complexes on $n$ vertices, showing that the weighted number of simplicial spanning trees is of order $(\xi_{d,k}+o(1))^{\binom{n}{d}}$ as $n\to\infty$, where $\xi_{d,k}$ is an explicit constant, provided $k> 4d^2+d+2$. A key ingredient in our proof is the local convergence of such random complexes to the $d$-dimensional, $k$-regular arboreal complex, which allows us to generalize McKay's result regarding the Kesten-McKay distribution.

math.CO↗

arXiv2020

On the Spectrum of Dense Random Geometric Graphs

In this paper we study the spectrum of the random geometric graph $G(n,r)$, in a regime where the graph is dense and highly connected. In the \erdren $G(n,p)$ random graph it is well known that upon connectivity the spectrum of the normalized graph Laplacian is concentrated around $1$. We show that such concentration does not occur in the $G(n,r)$ case, even when the graph is dense and almost a complete graph. In particular, we show that the limiting spectral gap is strictly smaller than $1$. In the special case where the vertices are distributed uniformly in the unit cube and $r=1$, we show that for every $0\le k \le d$ there are at least $\binom{d}{k}$ eigenvalues near $1-2^{-k}$, and the limiting spectral gap is exactly $1/2$. We also show that the corresponding eigenfunctions in this case are tightly related to the geometric configuration of the points.

math.PR↗

arXiv2018

Eigenvector correlations in the complex Ginibre ensemble

The complex Ginibre ensemble is an $N\times N$ non-Hermitian random matrix over $\mathbb{C}$ with i.i.d. complex Gaussian entries normalized to have mean zero and variance $1/N$. Unlike the Gaussian unitary ensemble, for which the eigenvectors are distributed according to Haar measure on the compact group $U(N)$, independently of the eigenvalues, the geometry of the eigenbases of the Ginibre ensemble are not particularly well understood. In this paper we systematically study properties of eigenvector correlations in this matrix ensemble. In particular, we uncover an extended algebraic structure which describes their asymptotic behavior (as $N$ goes to infinity). Our work extends previous results of Chalker and Mehlig [CM98], in which the correlation for pairs of eigenvectors was computed.

math.PR↗

arXiv2018

Stabilization of DLA in a wedge

We consider Diffusion Limited Aggregation (DLA) in a two-dimensional wedge. We prove that if the angle of the wedge is smaller than $π/4$, there is some $a>2$ such that almost surely, for all $R$ large enough, after time $R^a$ all new particles attached to the DLA will be at distance larger than $R$ from the origin. This means that DLA stabilizes in growing balls, thus allowing a definition of the infinite DLA in a wedge via a finite time process.

math.PR↗

arXiv2017

Isotropic self-consistent equations for mean-field random matrices

We present a simple and versatile method for deriving (an)isotropic local laws for general random matrices constructed from independent random variables. Our method is applicable to mean-field random matrices, where all independent variables have comparable variances. It is entirely insensitive to the expectation of the matrix. In this paper we focus on the probabilistic part of the proof -- the derivation of the self-consistent equations. As a concrete application, we settle in complete generality the local law for Wigner matrices with arbitrary expectation.

math.PR↗

arXiv2017

Non-metricity in the continuum limit of randomly-distributed point defects

We present a homogenization theorem for isotropically-distributed point defects, by considering a sequence of manifolds with increasingly dense point defects. The loci of the defects are chosen randomly according to a weighted Poisson point process, making it a continuous version of the first passage percolation model. We show that the sequence of manifolds converges to a smooth Riemannian manifold, while the Levi-Civita connections converge to a non-metric connection on the limit manifold. Thus, we obtain rigorously the emergence of a non-metricity tensor, which was postulated in the literature to represent continuous distribution of point defects.

math.PR↗

arXiv2016

Local limit theorem and equivalence of dynamic and static points of view for certain ballistic random walks in i.i.d. environments

In this work, we discuss certain ballistic random walks in random environments on $\mathbb{Z}^d$, and prove the equivalence between the static and dynamic points of view in dimension $d\geq4$. Using this equivalence, we also prove a version of a local limit theorem which relates the local behavior of the quenched and annealed measures of the random walk by a prefactor.

math.PR↗

arXiv2016

On groups and simplicial complexes

The theory of $k$-regular graphs is closely related to group theory. Every $k$-regular, bipartite graph is a Schreier graph with respect to some group $G$, a set of generators $S$ (depending only on $k$) and a subgroup $H$. The goal of this paper is to begin to develop such a framework for $k$-regular simplicial complexes of general dimension $d$. Our approach does not directly generalize the concept of a Schreier graph, but still presents an extensive family of $k$-regular simplicial complexes as quotients of one universal object: the $k$-regular $d$-dimensional arboreal complex, which is itself a simplicial complex originating in one specific group depending only on $d$ and $k$. Along the way we answer a question from [PR12] on the spectral gap of higher dimensional Laplacians and prove a high dimensional analogue of Leighton's graph covering theorem. This approach also suggests a random model for $k$-regular $d$-dimensional multicomplexes.

math.CO↗

arXiv2015

Random Steiner systems and bounded degree coboundary expanders of every dimension

We introduce a new model of random $d$-dimensional simplicial complexes, for $d\geq 2$, whose $(d-1)$-cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash's recent result on designs [Ke14], and the proof of the expansion uses techniques developed by Evra and Kaufman in [EK15]. This gives a full solution to a question raised in [DK12], which was solved in the two-dimensional case by Lubotzky and Meshulam [LM13].

math.CO↗

arXiv2015

Quenched invariance principle for simple random walk on clusters in correlated percolation models

We prove quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on Z^d, d>=2, with long-range correlations introduced in arXiv:1212.2885, solving one of the open problems from there. This gives new results for random interlacements in dimension d>=3 at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). An essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models.

math.PR↗

arXiv2015

Eigenvalue confinement and spectral gap for random simplicial complexes

We consider the adjacency operator of the Linial-Meshulam model for random simplicial complexes on $n$ vertices, where each $d$-cell is added independently with probability $p$ to the complete $(d-1)$-skeleton. Under the assumption $np(1-p) \gg \log^4 n$, we prove that the spectral gap between the $\binom{n-1}{d}$ smallest eigenvalues and the remaining $\binom{n-1}{d-1}$ eigenvalues is $np - 2\sqrt{dnp(1-p)} \, (1 + o(1))$ with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a Füredi-Komlós-type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries.

math.PR↗

arXiv2015

Isoperimetry in two-dimensional percolation

We consider the unique infinite connected component of supercritical bond percolation on the square lattice and study the geometric properties of isoperimetric sets, i.e., sets with minimal boundary for a given volume. For almost every realization of the infinite connected component we prove that, as the volume of the isoperimetric set tends to infinity, its asymptotic shape can be characterized by an isoperimetric problem in the plane with respect to a particular norm. As an application we then show that the anchored isoperimetric profile with respect to a given point as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the plane.

math.PR↗

arXiv2014

Simplicial branching random walks and their applications

We define a new stochastic process on general simplicial complexes which allows to study their spectral and homological properties. Some results for random walks on graphs are shown to hold in this general setting. As an application, the process is used to calculate the spectral measure of high-dimensional analogues of regular trees and to construct solutions to the high-dimensional Dirichlet problem for forms.

math.PR↗

arXiv2013

Behavior of random walk on discrete point processes

We consider a model for random walks on random environments (RWRE) with random subset of Z^d as the vertices, and uniform transition probabilities on 2d points (two "coordinate nearest points" in each of the d coordinate directions). We give partial characterization of transience and recurrence in the different dimensions. Finally we prove Central Limit Theorem (CLT) for such random walks, under a condition on the distance between coordinate nearest points.

math.PR↗

arXiv2013

Isoperimetric Inequalities in Simplicial Complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes.

math.CO↗

arXiv2013

Simplicial complexes: spectrum, homology and random walks

Random walks on a graph reflect many of its topological and spectral properties, such as connectedness, bipartiteness and spectral gap magnitude. In the first part of this paper we define a stochastic process on simplicial complexes of arbitrary dimension, which reflects in an analogue way the existence of higher dimensional homology, and the magnitude of the high-dimensional spectral gap originating in the works of Eckmann and Garland. The second part of the paper is devoted to infinite complexes. We present a generalization of Kesten's result on the spectrum of regular trees, and of the connection between return probabilities and spectral radius. We study the analogue of the Alon-Boppana theorem on spectral gaps, and exhibit a counterexample for its high-dimensional counterpart. We show, however, that under some assumptions the theorem does hold - for example, if the codimension-one skeletons of the complexes in question form a family of expanders. Our study suggests natural generalizations of many concepts from graph theory, such as amenability, recurrence/transience, and bipartiteness. We present some observations regarding these ideas, and several open questions.

math.CO↗