相关论文: On the spectrum of Random Simplicial Complexes in …
In this paper, we consider the multi-parameter random simplicial complex model, which generalizes the Linial-Meshulam model and random clique complexes by allowing simplices of different dimensions to be included with distinct…
The Linial-Meshulam complex model is a natural higher-dimensional analog of the Erd\H{o}s-R\'enyi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate…
We study a class of Hermitian random matrices which includes and generalizes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as the adjacency matrices of Erdos-Renyi random graphs with p ~ 1/N. Our NxN random…
The $m$-neighbor complex of a graph is the simplicial complex in which faces are sets of vertices with at least $m$ common neighbors. We consider these complexes for Erdos-Renyi random graphs and find that for certain explicit families of…
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)…
For a fixed dimension $k\ge 1$, let us consider the randomly growing simplical complex on the vertex set $\{1,2,\dots,n\}$ defined as follows: We start with the empty complex, and for each $k+1$-element subset $\sigma$ of $\{1,2,\dots,n\}$,…
We study the spectrum of a random multigraph with a degree sequence ${\bf D}_n=(D_i)_{i=1}^n$ and average degree $1 \ll \omega_n \ll n$, generated by the configuration model, and also the spectrum of the analogous random simple graph. We…
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…
A Hamiltonian cycle in a graph is a spanning subgraph that is homeomorphic to a circle. With this in mind, it is natural to define a Hamiltonian d-sphere in a d-dimensional simplicial complex as a spanning subcomplex that is homeomorphic to…
Using the random complexes of Linial and Meshulam, we exhibit a large family of simplicial complexes for which, whenever affinely embedded into Euclidean space, the filling areas of simplicial cycles is greatly distorted. This phenomenon…
We study the adjacency matrix of the Linial-Meshulam complex model, which is a higher-dimensional generalization of the Erd\H{o}s-R\'enyi graph model. Recently, Knowles and Rosenthal proved that the empirical spectral distribution of the…
In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special…
We study the spectral properties of a class of random matrices of the form $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ where $X_k = \Sigma^{1/2}Z_k$, for $k=1,2$, $Z_k$'s are independent $p\times n$ complex-valued random matrices, and…
We study the spectral properties of a class of random matrices of the form $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ where $X_k = \Sigma_k^{1/2}Z_k$, $Z_k$'s are independent $p\times n$ complex-valued random matrices, and $\Sigma_k$ are…
Given an $n \times n$ complex matrix $A$, let $$\mu_{A}(x,y):= \frac{1}{n} |\{1\le i \le n, \Re \lambda_i \le x, \Im \lambda_i \le y\}|$$ be the empirical spectral distribution (ESD) of its eigenvalues $\lambda_i \in \BBC, i=1, ... n$. We…
We study the Linial--Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for $p\ll n^{-1}$ a random 2-complex $Y$ collapses simplicially to a graph and, in particular, the fundamental group…
We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$…
We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the…
In this article, we study random graphs with a given degree sequence $d_1, d_2, \cdots, d_n$ from the configuration model. We show that under mild assumptions of the degree sequence, the spectral distribution of the normalized Laplacian…
We prove that the (non-symmetric) adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices is asymptotically almost surely invertible, assuming $\min(d,n-d)\ge C\log^2n$ for a sufficiently large constant $C>0$. The…