arXiv subjects

Jean-Francois Marckert

Publications and source records attributed to Jean-Francois Marckert.

4 recordsLinked to original sources

arXiv2023

Models of random subtrees of a graph

Consider a connected graph $G=(E,V)$ with $N=|V|$ vertices. The main purpose of this paper is to explore the question of uniform sampling of a subtree of $G$ with $n$ nodes, for some $n\leq N$ (the spanning tree case correspond to $n=N$, and is already deeply studied in the literature). We provide new asymptotically exact simulation methods using Markov chains for general connected graphs $G$, and any $n\leq N$. We highlight the case of the uniform subtree of $\mathbb{Z}^2$ with $n$ nodes, containing the origin $(0,0)$ for which Schramm asked several questions. We produce pictures, statistics, and some conjectures. A second aim of the paper is devoted to surveying other models of random subtrees of a graph, among them, DLA models, the first passage percolation, the uniform spanning tree and the minimum spanning tree. We also provide new models, some statistics, and some conjectures.

math.PR↗

arXiv2021

Almost triangular Markov chains on $\mathbb{N}$

A transition matrix $[U_{i,j}]_{i,j\geq 0}$ on $\mathbb{N}$ is said to be almost upper triangular if $U_{i,j}\geq 0\Rightarrow j\geq i-1$, so that the increments of the corresponding Markov chains are at least $-1$; a transition matrix $[L_{i,j}]_{i,j\geq 0}$ is said to be almost lower triangular if $L_{i,j}\geq 0\Rightarrow j\leq i+1$, and then, the increments of the corresponding Markov chains are at most $+1$. In the present paper, we characterize the recurrence, positive recurrence and invariant distribution for the class of almost triangular transition matrices. The upper case appears to be the simplest in many ways, with existence and uniqueness of invariant measures, when in the lower case, existence as well as uniqueness are not guaranteed. We present the time-reversal connection between upper and lower almost triangular transition matrices, which provides classes of integrable lower triangular transition matrices. These results encompass the case of birth and death processes (BDP) that are famous Markov chains (or processes) taking their values in $\mathbb{N}$, which are simultaneously almost upper and almost lower triangular, and whose study has been initiated by Karlin & McGregor in the 1950's. They found invariant measures, criteria for recurrence, null recurrence, among others; their approach relies on some profound connections they discovered between the theory of BDP, the spectral properties of their transition matrices, the moment problem, and the theory of orthogonal polynomials. Our approach is mainly combinatorial and uses elementary algebraic methods; it is somehow more direct and does not use the same tools.

math.PR↗

arXiv2021

Growing random uniform d-ary trees

Let T d (n) be the set of d-ary rooted trees with n internal nodes. We give a method to construct a sequence (t n , n $\ge$ 0) where, for any n $\ge$ 1, t n has the uniform distribution in T d (n), and t n is constructed from t n--1 by the addition of a new node, and a rearrangement of the structure of t n--1. This method is inspired by R{é}my's algorithm which does this job in the binary case, but it is different from it. This provides a method for the random generation of a uniform d-ary tree in T d (n) with a cost linear in n.

math.CO↗

arXiv2004

Martingales and Profile of Binary Search Trees

We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.

math.PR↗