Related papers: GUE minors, maximal Brownian functionals and longe…
In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably…
We improve the best known lower bounds on the exponential behavior of the maximum of the number of connected sets, $N(G)$, and dominating connected sets, $N_{dom}(G)$, for regular graphs. These lower bounds are improved by constructing a…
This paper deals with two GUE-matrices, coupled together through some inequalities between the spectra of the first few (small) principal minors. The main results of the paper is to show that the spectra of the principal minors of these…
We investigate the probability distribution of the length of the second row of a Young diagram of size $N$ equipped with Plancherel measure. We obtain an expression for the generating function of the distribution in terms of a derivative of…
We define and study the Plancherel-Hecke probability measure on Young diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong '06] is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the…
We study the distribution of the length of longest increasing subsequences in random permutations of $n$ integers as $n$ grows large and establish an asymptotic expansion in powers of $n^{-1/3}$. Whilst the limit law was already shown by…
The discrete distribution of the length of longest increasing subsequences in random permutations of $n$ integers is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of…
Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and…
Consider an infinite random matrix $H=(h_{ij})_{0<i,j}$ picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by $H_i=(h_{rs})_{1\leq r,s\leq i}$ and let the $j$:th largest eigenvalue of $H_i$ be $\mu^i_j$. We show that…
Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We…
In 1962, Dyson introduced dynamics in random matrix models, in particular into GUE (also for beta=1 and 4), by letting the entries evolve according to independent Ornstein-Uhlenbeck processes. Dyson shows the spectral points of the matrix…
We study the adjacency matrices of random $d$-regular graphs with large but fixed degree $d$. In the bulk of the spectrum $[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$ down to the optimal spectral scale, we prove that the Green's…
Recent applications in queuing theory and statistical mechanics have isolated the process formed by the eigenvalues of successive minors of the GUE. Analogous eigenvalue processes, formed in general from the eigenvalues of nested sequences…
Nonintersecting motion of Brownian particles in one dimension is studied. The system is constructed as the diffusion scaling limit of Fisher's vicious random walk. N particles start from the origin at time t=0 and then undergo mutually…
The joint distribution of the maximum loss and the maximum gain is obtained for a spectrally negative Levy process until the passage time of a given level. Their marginal distributions up to an independent exponential time are also…
We give an explicit construction of the scaling limit of the minimum spanning tree of the complete graph. The limit object is described using a recursive construction involving the convex minorants of a Brownian motion with parabolic drift…
We establish high probability estimates on the eigenvalue locations of Brownian motion on the $N$-dimensional unitary group, as well as estimates on the number of eigenvalues lying in any interval on the unit circle. These estimates are…
Considered are the large $N$, or large intensity, forms of the distribution of the length of the longest increasing subsequences for various models. Earlier work has established that after centring and scaling, the limit laws for these…
We establish universality for the largest singular values of products of random matrices with right unitarily invariant distributions, in a regime where the number of matrix factors and size of the matrices tend to infinity simultaneously.…
We consider large uniform labeled random graphs in different classes with prescribed decorations in their modular decomposition. Our main result is the estimation of the number of copies of every graph as an induced subgraph. As a…