Related papers: Permutations without long decreasing subsequences …

The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to…

A random $n$-permutation may be generated by sequentially removing random cards $C_1,...,C_n$ from an $n$-card deck $D = \{1,...,n\}$. The permutation $\sigma$ is simply the sequence of cards in the order they are removed. This permutation…

The limiting shape of the random Young diagrams associated with an inhomogeneous random word is identified as a multidimensional Brownian functional. This functional is identical in law to the spectrum of a random matrix. The Poissonized…

We prove a new CLT for the difference of linear eigenvalue statistics of a Wigner random matrix $H$ and its minor $\hat H$ and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a…

We consider the probability measures on Young diagrams in the $n \times k$ rectangle obtained by piecewise-continuously differentiable specializations of Schur polynomials in the dual Cauchy identity. We use a free fermionic representation…

We study numerically the distributions of the length $L$ of the longest increasing subsequence (LIS) for the two cases of random permutations and of one-dimensional random walks. Using sophisticated large-deviation algorithms, we are able…

We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We establish…

We address a question and a conjecture on the expected length of the longest common subsequences of two i.i.d.$\ $random permutations of $[n]:=\{1,2,...,n\}$. The question is resolved by showing that the minimal expectation is not attained…

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…

We consider $N\times N$ symmetric random matrices where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. We prove that the eigenvalue spacing statistics in the bulk of the…

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…

We consider random matrices of the form $H = W + \lambda V$, $\lambda\in\mathbb{R}^+$, where $W$ is a real symmetric or complex Hermitian Wigner matrix of size $N$ and $V$ is a real bounded diagonal random matrix of size $N$ with i.i.d.\…

We consider Ewens random permutations of length $n$ conditioned to have no cycle longer than $n^\beta$ with $0<\beta<1$ and to study the asymptotic behaviour as $n\to\infty$. We obtain very precise information on the joint distribution of…

We explore the probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r. Asymptotic formulas valid in specified regions for the ratio n/r are obtained using the…

We analyze statistical properties of the complex system with conditions which manifests through specific constraints on the column/row sum of the matrix elements. The presence of additional constraints besides symmetry leads to new…

In this work, we study some statistical properties of the extreme eigenstates of the randomly-weighted adjacency matrices of random graphs. We focus on two random graph models: Erd\H{o}s-R\'{e}nyi (ER) graphs and random geometric graphs…

Let $n$ be a positive integer and $m$ be a positive even integer. Let ${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetric tensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensor of the same size.…

We study higher order convexity properties of random point sets in the unit square. Given $n$ uniform i.i.d random points, we derive asymptotic estimates for the maximal number of them which are in $k$-monotone position, subject to mild…

The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph…

We consider asymptotics behavior of Poissonized Plancharel measures as the poissonization parameter $N$ goes to infinity. Recently Moll proved a convergent series expansion for statistics of a measure $\mu_\lambda$ which is the…