Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.3902

On the number of collisions in $Λ$-coalescents

Alexander Gnedin·Yuri Yakubovich
View original source ↗Read source document ↗Export metadata ↓

Abstract

We examine the total number of collisions $C_n$ in the $Λ$-coalescent process which starts with $n$ particles. A linear growth and a stable limit law for $C_n$ are shown under the assumption of a power-like behaviour of the measure $Λ$ near 0 with exponent $0<α<1$.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexander Gnedin, Yuri Yakubovich. 2007-04-30. On the number of collisions in $Λ$-coalescents. https://arxiv.org/abs/0704.3902

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Additive processes on the real line and Loewner chains

This paper investigates additive processes with respect to several different independences in non-commutative probability in terms of the convolution hemigroups of the distributions of the increments of the processes. In particular, we focus on the relation of monotone convolution hemigroups and Loewner chains, a special kind of family of conformal mappings, on the upper half-plane. Generalizing the celebrated Loewner differential equation, we formulate an integral equation and the concept of ``generator'' for any Loewner chain of reciprocal Cauchy transforms. This generalization enables us to remove the assumption of the absolute continuity of Loewner chains which had been imposed in the literature. The locally uniform convergence of Loewner chains is then equivalent to a suitable convergence of generators. Using generators, we define homeomorphisms between the aforementioned class of Loewner chains, the set of monotone convolution hemigroups, and the set of classical convolution hemigroups on the real line. We also discuss similar homeomorphisms to free, boolean and anti-monotone convolution hemigroups on the real line.

math.PR↗
arXiv2026

Shape Theorem for the Contact Process in a Dynamical Random Environment

We study the contact process in a dynamical random environment defined on the vertices and edges of the d-dimensional integer lattice $\mathbb{Z}^d$. For a broad class of processes, we establish an asymptotic shape theorem for the set $\mathbf{H}_t$, which represents the vertices that have been infected up to time $t$. More precisely, we show that, similarly to the basic contact process, this asymptotic shape is characterised by a cone spanned by a convex set $U$, provided certain growth conditions are satisfied. Notably, we find that the asymptotic shape is independent of the initial configuration of the environment. Furthermore, we verify the growth conditions for various types of random environments, such as the contact process on a dynamical graph and a system with switching vertex states, where the monotonicity of the entire process is not guaranteed.

math.PR↗
arXiv2026

Random burning of the Euclidean torus

The burning number of a graph is the minimal number of steps that are needed to burn all of its vertices, with the following burning procedure: at each step, one can choose a point to set on fire, and the fire propagates constantly at unit speed along the edges of the graph. In this paper, we consider two natural random burning procedures in the discrete Euclidean torus $\mathbb{T}_n^d$, in which the points that we set on fire at each step are random variables. Our main result deals with the case where at each step, the law of the new point that we set on fire conditionally on the past is the uniform distribution on the complement of the set of vertices burned by the previous points. In this case, we prove that as $n\to\infty$, the corresponding random burning number (i.e, the first step at which the whole torus is burned) is asymptotic to $T\cdot n^{d/(d+1)}$ in probability, where $T=T(d)\in(0,\infty)$ is the explosion time of a so-called generalised Blasius equation.

math.PR↗