We study spatial permutations with cycle weights that are bounded or slowly diverging. We show that a phase transition occurs at an explicit critical density. The long cycles are macroscopic and their cycle lengths satisfy a Poisson-Dirichlet law.

In several scale free graph models the asymptotic degree distribution and the characteristic exponent change when only a smaller set of vertices is considered. Looking at the common properties of these models, we present sufficient conditions for the almost sure…

We treat branching random walks in random environment using the framework of Linear Stochastic Evolution. In spatial dimensions three or larger, we establish diusive behaviour in the entire growth phase. This can be seen through a Central Limit Theorem with…

We define an equation on a simple graph which is an extension of Tanaka's equation and the skew Brownian motion equation. We then apply the theory of transition kernels developed by Le Jan and Raimond and show that all the…

This paper concerns the first hitting time $T_0$ of the origin for random walks on $d$-dimensional integer lattice with zero mean and a finite $2+\delta$ absolute moment ($\delta\geq0$). We derive detailed asymptotic estimates of the probabilities $\mathbb{P}_x(T_0=n)$ as $n\to\infty$ that…

We consider the product of a finite number of non-Hermitian random matrices with i.i.d. centered entries of growing size. We assume that the entries have a finite moment of order bigger than two. We show that the empirical spectral distribution…

In the present paper, we prove that under the assumption of the finite sixth moment for elements of a Wigner matrix, the convergence rate of its empirical spectral distribution to the Wigner semicircular law in probability is $O(n^{-1/2})$ when the…

We study the moments of the k-th nearest neighbor distance for independent identically distributed points in $\mathbb{R}^n$. In the earlier literature, the case with power higher than n has been analyzed by assuming a bounded support for the underlying density.…

Suppose we are given the free product $V$ of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors.…

A new diffusion process taking values in the space of all probability measures over $[0,1]$ is constructed through Dirichlet form theory in this paper. This process is reversible with respect to the Ferguson-Dirichlet process (also called Poisson Dirichlet process), which…