Related papers: Sparse Random Matrices and Statistics of Rooted Tr…
The coexistence of sparsity and clustering (non-vanishing average fraction of triangles per node) is one of the few structural features that, irrespective of finer details, are ubiquitously observed across large real-world networks. This…
It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ is the…
This paper concerns the large deviations of a system of interacting particles on a random graph. There is no stochasticity, and the only sources of disorder are the random graph connections, and the initial condition. The average number of…
Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has…
In this work we study symmetric random matrices with variance profile satisfying certain conditions. We establish the convergence of the operator norm of these matrices to the largest element of the support of the limiting empirical…
A degree sequence is a sequence ${\bf s}=(N_i,i\geq 0)$ of non-negative integers satisfying $1+\sum_i iN_i=\sum_i N_i<\infty$. We are interested in the uniform distribution $\mathbb{P}_{{\bf s}}$ on rooted plane trees whose degree sequence…
In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various…
Using the superstatistics method, we propose an extension of the random matrix theory to cover systems with mixed regular-chaotic dynamics. Unlike most of the other works in this direction, the ensembles of the proposed approach are basis…
Bounded infinite graphs are defined on the basis of natural physical requirements. When specialized to trees this definition leads to a natural conjecture that the average connectivity dimension of bounded trees cannot exceed two. We verify…
We study the asymptotics of sums of matricially free random variables called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint…
We consider a non-projective class of inhomogeneous random graph models with interpretable parameters and a number of interesting asymptotic properties. Using the results of Bollob\'as et al. [2007], we show that i) the class of models is…
The class of norm-dependent Random Matrix Ensembles is studied in the presence of an external field. The probability density in those ensembles depends on the trace of the squared random matrices, but is otherwise arbitrary. An exact…
We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by…
We place ourselves in the setting of high-dimensional statistical inference, where the number of variables $p$ in a data set of interest is of the same order of magnitude as the number of observations $n$. More formally, we study the…
An electrical network with the structure of a random tree is considered: starting from a root vertex, in one iteration each leaf (a vertex with zero or one adjacent edges) of the tree is extended by either a single edge with probability $p$…
We study a new random matrix ensemble $X$ which is constructed by an application of a two dimensional linear filter to a matrix of iid random variables with infinite fourth moments. Our result gives asymptotic lower and upper bounds for the…
This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of…
In this paper, we shall try to deduce asymptotic behaviour of component spectrum of random $n \times n$ magical squares with line sum $r \in \mathbb{N}$, which can also be identified as $r$-regular bipartite graphs on $2n$ vertices, chosen…
In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…
We study the asymptotic distribution of the number of matchings of size $\ell=\ell(n)$ in $G(n,p)$ for a wide range of $p=p(n)\in (0,1)$ and for every $1\le \ell\le \lfloor n/2\rfloor$. We prove that this distribution changes from normal to…