Related papers: Monotone Subsequences in Locally Uniform Random Pe…
These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex…
In this paper we study a mapping from permutations to Dyck paths. A Dyck path gives rise to a (Young) diagram and we give relationships between statistics on permutations and statistics on their corresponding diagrams. The distribution of…
We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a $k$-monotone density $g_0$ at a fixed point $x_0$ when $k>2$. We find that the $j$th derivative of the estimators at $x_0$ converges at the rate…
We study the lengths of monotone subsequences for permutations drawn from the Mallows measure. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a…
Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters…
Let $H_d(n,p)$ signify a random $d$-uniform hypergraph with $n$ vertices in which each of the ${n}\choose{d}$ possible edges is present with probability $p=p(n)$ independently, and let $H_d(n,m)$ denote a uniformly distributed with $n$…
We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one- and two-dimensional substrates focusing our analysis on the probability distribution function $P(M,L)$ of the number $M$ of maximal points (i.e., local…
We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation $\sigma$ is the order type of the restriction of $\sigma : [n] \to [n]$ to a subset $S…
In this article, we study a non-uniform distribution on permutations biased by their number of records that we call \emph{record-biased permutations}. We give several generative processes for record-biased permutations, explaining also how…
We introduce and study a new random permutation model that generalizes the $k$-card minimum model defined by Travers and the Mallows model. We calculate the permuton limit of such a sequence of random permutations. As a corollary, we deduce…
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…
Monotonic surfaces spanning finite regions of $Z^d$ arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. One method that has been used to uniformly generate these surfaces is a Markov chain that…
Uniform random intersection graphs have received much interest and been used in diverse applications. A uniform random intersection graph with $n$ nodes is constructed as follows: each node selects a set of $K_n$ different items uniformly…
We investigate the large deviations of the shape of the random RSK Young diagrams associated with a random word of size $n$ whose letters are independently drawn from an alphabet of size $m=m(n)$. When the letters are drawn uniformly and…
We study the optimal rectangular-discrepancy approximation of permutons by finite permutations. We transfer bounds from discrepancy theory to this more restricted setup. Moreover, we show that superlinear approximation can occur only for…
We consider the asymptotic behavior as $n\to\infty$ of the spectra of random matrices of the form \[\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_n ((k,k+1)),\] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian…
Spatial random permutations were originally studied due to their connections to Bose-Einstein condensation, but they possess many interesting properties of their own. For random permutations of a regular lattice with periodic boundary…
This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was posed as an open problem by Chatterjee and Diaconis in a recent publication. For that purpose, we generalize…
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 survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time…