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We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $\beta>0$ per edge. This is called the arboreal gas model, and the special case when $\beta=1$ is the uniform forest…

Probability · Mathematics 2021-07-06 Roland Bauerschmidt , Nicholas Crawford , Tyler Helmuth , Andrew Swan

We study an one-dimensional stochastic model of vehicular traffic on open segments of a single-lane road of finite size $L$. The vehicles obey a stochastic discrete-time dynamics which is a limiting case of the generalized Totally…

Statistical Mechanics · Physics 2018-01-17 J. G. Brankov , N. Zh. Bunzarova , N. C. Pesheva , V. B. Priezzhev

The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and…

Probability · Mathematics 2021-02-05 Giulio Iacobelli , Rodrigo Ribeiro , Glauco Valle , Leonel Zuaznabar

The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite $d$-ary tree. We prove the model undergoes a phase…

Probability · Mathematics 2018-02-08 Christopher Hoffman , Tobias Johnson , Matthew Junge

The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform…

Probability · Mathematics 2018-01-16 Ella Hiesmayr , Ümit Işlak

We study completely asymmetric 2-channel exclusion processes in 1 dimension. It describes a two-way traffic flow with cars moving in opposite directions. The interchannel interaction makes cars slow down in the vicinity of approaching cars…

Statistical Mechanics · Physics 2008-11-26 H. -W. Lee , V. Popkov , D. Kim

We consider navigation schemes on planar diluted lattices and semi lattices with one discrete and one continuous component. More precisely, nodes that survive inhomogeneous Bernoulli site percolation, or are placed as inhomogeneous Poisson…

Probability · Mathematics 2025-06-24 Partha Pratim Ghosh , Benedikt Jahnel , Yannic Steenbeck

We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random;…

Probability · Mathematics 2015-03-13 Gabriel Faraud , Yueyun Hu , Zhan Shi

A parking function is a sequence $(a_1,\dots, a_n)$ of positive integers such that if $b_1\leq\cdots\leq b_n$ is the increasing rearrangement of $a_1,\dots,a_n$, then $b_i\leq i$ for $1\leq i\leq n$. In this paper we obtain some new results…

Combinatorics · Mathematics 2023-06-16 Richard P. Stanley , Mei Yin

We study the random acceleration model, which is perhaps one of the simplest, yet nontrivial, non-Markov stochastic processes, and is key to many applications. For this non-Markov process, we present exact analytical results for the…

Statistical Mechanics · Physics 2019-09-04 Satya N. Majumdar , Alberto Rosso , Andrea Zoia

We consider Bernoulli bond percolation on a large scale-free tree in the supercritical regime, meaning informally that there exists a giant cluster with high probability. We obtain a weak limit theorem for the sizes of the next largest…

Probability · Mathematics 2016-03-04 Jean Bertoin , Geronimo Uribe Bravo

We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces…

Probability · Mathematics 2014-05-06 Rudolf Grübel

We study a spatial model of random permutations on trees with a time parameter $T>0$, a special case of which is the random stirring process. The model on trees was first analysed by Bj\"ornberg and Ueltschi[BU16], who established the…

Probability · Mathematics 2018-05-31 Alan Hammond , Milind Hegde

In this paper, we study the asymptotic behaviour of the number of subtrees and the subtree density for a sequence of trees that converges in the Benjamini-Schramm sense. Benjamini-Schramm convergence, also called local weak convergence,…

Combinatorics · Mathematics 2026-05-11 Stijn Cambie , Stephan Wagner , Ruoyu Wang

We study Bernoulli bond percolation on a random recursive tree of size $n$ with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove…

Probability · Mathematics 2016-12-28 Erich Baur

Consider the following simple parking process on $\Lambda_n := \{-n, \ldots, n\}^d,d\ge1$: at each step, a site $i$ is chosen at random in $\Lambda_n$ and if $i$ and all its nearest neighbor sites are empty, $i$ is occupied. Once occupied,…

Probability · Mathematics 2024-05-24 Cristian F. Coletti , Sandro Gallo , Alejandro Roldán-Correa , León A. Valencia

We propose a generalized car parking problem where either a car of size $\sigma$ or of size $m\sigma$ ($m>1$) is sequentially parked on a line with probability $q$ and $(1-q)$, respectively. The free parameter $q$ interpolates between the…

Statistical Mechanics · Physics 2009-11-07 M. K. Hassan , J. Schmidt , B. Blasius , J. Kurths

Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton-Watson trees. For example, let $\mathcal{T}_1$ be the…

Probability · Mathematics 2022-08-05 Tobias Johnson

Inspired by Stufler's recent probabilistic proof of Otter's asymptotic number of unlabeled trees, we revisit work of Palmer and Schwenk, and study unlabeled forests from a probabilistic point of view. We show that the number of trees in a…

Probability · Mathematics 2025-07-23 Michal Bassan , Serte Donderwinkel , Brett Kolesnik

We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by…

Probability · Mathematics 2025-04-25 Conrado da Costa , Mikhail Menshikov , Andrew Wade