Related papers: Small value probabilities via the branching tree h…

We consider the simple random walk on Galton-Watson trees with supercritical offspring distribution, conditioned on non-extinction. In case the offspring distribution has finite support, we prove an upper bound for the annealed return…

This work is a numerical experiment of stochastic motion of conservative Hamiltonian system or weakly damped Brownian particles. The objective is to prove the existence of path probability and to compute its values. By observing a large…

In this note, we introduce a unified analytic framework that connects simple varieties of trees, Bienayme-Galton-Watson processes and Khinchin families. Using Lagrange's inversion formula, we derive new coefficient-based expressions for…

In this paper we study several aspects of the growth of a supercritical Galton-Watson process {Z_n:n\ge1}, and bring out some criticality phenomena determined by the Schroder constant. We develop the local limit theory of Z_n, that is, the…

The paper deals with the asymptotic laws of functional of standard random variables. These classes of statistics are closely related to estimators of the extreme value index when the underlying distribution function is in the Weibull domain…

We consider a branching random walk on a multi($Q$)-type, supercritical Galton-Watson tree which satisfies Kesten-Stigum condition. We assume that the displacements associated with the particles of type $Q$ have regularly varying tails of…

We present a new algorithm for computing the quasi-stationary distribution of subcritical Galton--Watson branching processes. This algorithm is based on a particular discretization of a well-known functional equation that characterizes the…

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can…

A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed "locally perturbating set") are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the…

Let $(W_n(\theta))_{n\in\mathbb N_0}$ be the Biggins martingale associated with a supercritical branching random walk and denote by $W_\infty(\theta)$ its limit. Assuming essentially that the martingale $(W_n(2\theta))_{n\in\mathbb N_0}$ is…

We identify the local scaling limit of multiple boundary-to-boundary branches in a uniform spanning tree (UST) as a local multiple SLE(2), i.e., an SLE(2) process weighted by a suitable partition function. By recent results, this also…

We study random typical minimal factorizations of the $n$-cycle, which are factorizations of $(1, \ldots,n)$ as a product of $n-1$ transpositions, chosen uniformly at random. Our main result is, roughly speaking, a local convergence theorem…

The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a…

In this paper, we study a new discrete tree and the resulting branching process, which we call the \textbf{E}rlang \textbf{W}eighted \textbf{T}ree(\textbf{EWT}). The EWT appears as the local weak limit of a random graph model proposed…

We develop a method based on martingales to study first-passage problems of time-additive observables exiting an interval of finite width in a Markov process. In the limit that the interval width is large, we derive generic expressions for…

Given a super-critical branching random walk on $\mathbb R$ started from the origin, let $M_n$ be the maximal position of individuals at the $n$-th generation. Under some mild conditions, it is known from \cite{A13} that as…

We consider Galton-Watson trees associated with a critical offspring distribution and conditioned to have exactly $n$ vertices. These trees are embedded in the real line by affecting spatial positions to the vertices, in such a way that the…

We develop a numerical method for the martingale analogue of the Benamou--Brenier optimal transport problem, which seeks a martingale interpolating two prescribed marginals which is closest to the Brownian motion. Recent contributions have…

We prove that the rescaled costs of partial match queries in a random two-dimensional quadtree converge almost surely towards a random limit which is identified as the terminal value of a martingale. Our approach shares many similarities…

Consider the first exit time of one-dimensional Brownian motion $\{B_s\}_{s\geq 0}$ from a random passageway. We discuss a Brownian motion with two time-dependent random boundaries in quenched sense. Let $\{W_s\}_{s\geq 0}$ be an other…