arXiv subjects

Etienne Bellin

Publications and source records attributed to Etienne Bellin.

4 recordsLinked to original sources

arXiv2022

Random monotone factorisations of the cycle

In this article we study decreasing and increasing factorisations of the cycle, which are decompositions of the cycle $(1~2\dots n)$ into a product of $n-1$ transpositions satisfying monotonicity conditions. We explicit a bijection between such factorisations and plane trees with $n$ vertices. This will allow us to study some of their combinatorial properties, as well as a geometric representation in terms of laminations, which are non-crossing line segments in the unit disk.

math.PR↗

arXiv2022

On the independence number of random trees via tricolourations

We are interested in the independence number of large random simply generated trees and related parameters, such as their matching number or the kernel dimension of their adjacency matrix. We express these quantities using a canonical tricolouration, which is a way to colour the vertices of a tree with three colours. As an application we obtain limit theorems in $L^p$ for the renormalised independence number in large simply generated trees (including large size-conditioned Bienaymé-Galton-Watson trees).

math.PR↗

arXiv2021

Asymptotic behaviour of the first positions of uniform parking functions

In this paper we study the asymptotic behavior of a random uniform parking function $π_n$ of size $n$. We show that the first $k_n$ places $π_n(1),\dots,π_n(k_n)$ of $π_n$ are asymptotically i.i.d. and uniform on $\{1,2,\dots,n\}$, for the total variation distance when $k_n = o(\sqrt{n})$, and for the Kolmogorov distance when $k_n=o(n)$, improving results of Diaconis & Hicks. Moreover we give bounds for the rate of convergence, as well as limit theorems for some statistics like the sum or the maximum of the first $k_n$ parking places. The main tool is a reformulation using conditioned random walks.

math.PR↗

arXiv2020

Degrees in random uniform minimal factorizations

We are interested in random uniform minimal factorizations of the $n$-cycle which are factorizations of $(1~2\dots n)$ into a product of $n-1$ transpositions. Our main result is an explicit formula for the joint probability that 1 and 2 appear a given number of times in a uniform minimal factorization. For this purpose, we combine bijections with Cayley trees together with explicit computations of multivariate generating functions.

math.CO↗