Related papers: Spectra of large random trees

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral…

We study the spectral measure of large Euclidean random matrices. The entries of these matrices are determined by the relative position of $n$ random points in a compact set $\Omega_n$ of $\R^d$. Under various assumptions we establish the…

We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of…

We study the asymptotic distribution of the eigenvalues of random Hermitian periodic band matrices, focusing on the spectral edges. The eigenvalues close to the edges converge in distribution to the Airy point process if (and only if) the…

When a randomness is introduced at the level of real matrix elements, depending on its particular realization, a pair of eigenvalues can appear as real or form a complex conjugate pair. We show that in the limit of large matrix size the…

We consider the sum of two large Hermitian matrices $A$ and $B$ with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is…

Let ${\mathbf T}_n$ be a uniformly random tree with vertex set $[n]=\{1,\ldots,n\}$, let $\Delta_{{\mathbf T}_n}$ be the largest vertex degree in ${\mathbf T}_n$, and let $\lambda_1({\mathbf T}_n),\ldots,\lambda_n({\mathbf T}_n)$ be the…

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$…

We introduce a model of evolving preferential attachment trees where vertices are assigned weights, and the evolution of a vertex depends not only on its own weight, but also on the weights of its neighbours. We study the distribution of…

There are several common ways to encode a tree as a matrix, such as the adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of the natural random walk), and the matrix of pairwise distances between leaves. Such…

A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum…

We consider growing random recursive trees in random environment, in which at each step a new vertex is attached (by an edge of a random length) to an existing tree vertex according to a probability distribution that assigns the tree…

We extend the results of Zhang et al. to show that $\lambda$ is an eigenvalue of a $k$-uniform hypertree $(k \geq 3)$ if and only if it is a root of a particular matching polynomial for a connected induced subtree. We then use this to…

We study the height of the delta peak at 0 in the spectrum of random tree incidence matrices. We show that the average fraction of the spectrum occupied by the eigenvalue 0 in a large random tree is asymptotic to 2x-1 =…

We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit…

We prove almost sure convergence of the maximum degree in an evolving tree model combining local choice and preferential attachment. At each step in the growth of the graph, a new vertex is introduced. A fixed, finite number of possible…

We consider the empirical eigenvalue distribution of random real symmetric matrices with stochastically independent skew-diagonals and study its limit if the matrix size tends to infinity. We allow correlations between entries on the same…

We consider $n\times n$ non-Hermitian random matrices with independent entries and a variance profile, as well as an additive deterministic diagonal deformation. We show that their empirical eigenvalue distribution converges to a limiting…

We study asymptotics of the spectral measure of a randomly chosen partial automorphism of a rooted tree. To every partial automorphism $x$ we assign its action matrix $A_x$. It is shown that the uniform distribution on eigenvalues of $A_x$…

Let $T$ be a $k$-tree equipped with a weighting function $\w: V(T)\cup E(T)\rightarrow \C$, where $k \geq 3$. The weighted matching polynomial of the weighted $k$-tree $(T,\w)$ is defined to be $$ \mu(T,\w,x)= \sum_{M \in…