Related papers: GUE minors, maximal Brownian functionals and longe…
In this paper, we consider the drawdown and drawup of the fractional Brownian motion with trend, which corresponds to the logarithm of geometric fractional Brownian motion representing the stock price in financial market. We derive the…
We consider the local statistics of $H = V^* X V + U^* Y U$ where $V$ and $U$ are independent Haar-distributed unitary matrices, and $X$ and $Y$ are deterministic real diagonal matrices. In the bulk, we prove that the gap statistics and…
The authors consider the length, $l_N$, of the length of the longest increasing subsequence of a random permutation of $N$ numbers. The main result in this paper is a proof that the distribution function for $l_N$, suitably centered and…
Connections between longest increasing subsequences in random permutations and eigenvalues of random matrices with complex entries have been intensely studied. This note applies properties of random elements of the finite general linear…
Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of…
We construct the least superharmonic majorant of a continuous function $g$ on the $d$-dimensional unit ball ($d \geq 2$) via a canonical sequential scheme. While classical theory identifies this majorant with the value function of the…
In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian…
The behavior of a certain random growth process is analyzed on arbitrary regular and non-regular graphs. Our argument is based on the Expander Mixing Lemma, which entails that the results are strongest for Ramanujan graphs, which…
The celebrated L\'evy--Khintchine theorem is a fundamental limiting law that describes the growth rate of the denominators of the convergents in the continued fraction expansion of a Lebesgue-typical real number. In a recent breakthrough,…
We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times $t=o(1)$ with deterministic initial data V . Our main result states that if the density of states of $V$ is bounded both…
We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the…
A sequence of accompanying laws is suggested in the limit theorem of B. V. Gnedenko for maximums of independent random variables belonging to maximum domain of attraction of the Gumbel distribution. It is shown that this sequence gives an…
A {\em restraint} on a (finite undirected) graph $G = (V,E)$ is a function $r$ on $V$ such that $r(v)$ is a finite subset of ${\mathbb N}$; a proper vertex colouring $c$ of $G$ is {\em permitted} by $r$ if $c(v) \not\in r(v)$ for all…
We study a spatial branching model, where the underlying motion is $d$-dimensional ($d\ge1$) Brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result…
We investigate a class of Young diagrams growing via the addition of unit cells and satisfying the constraint that the height difference between adjacent columns $\geq r$. In the long time limit, appropriately re-scaled Young diagrams…
In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the…
We study the behaviour of a natural measure defined on the leaves of the genealogical tree of some branching processes, namely self-similar growth-fragmentation processes. Each particle, or cell, is attributed a positive mass that evolves…
We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of…
We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the colouring number,…
We construct the full edge scaling limit of the singular values of Brownian motion on the general linear group $\mathsf{GL}_N(\mathbb{C})$ starting from general conditions. We show that the limiting paths solve an infinite system of SDE…