Related papers: Increasing paths on N-ary trees
Given an autoregressive process X of order p (i.e. X_n = a_1 X_{n-1} + ...+ a_p X_{n_p} + Y_n where the random variables Y_1, Y_2, ... are i.i.d.), we study the asymptotic behaviour of the probability that the process does not exceed a…
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…
An infection spreads in a binary tree of height n as follows: initially, each leaf is either infected by one of k states or it is not infected at all. The infection state of each leaf is independently distributed according to a probability…
In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $\alpha(n)$ be the number of rarely visited edges up to time…
We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16].…
We consider a random walk among i.i.d. obstacles on the one dimensional integer lattice under the condition that the walk starts from the origin and reaches a remote location y. The obstacles are represented by a killing potential, which…
Let $b$ be an integer greater than 1 and let $W^{\ee}=(W^{\ee}_n; n\geq 0)$ be a random walk on the $b$-ary rooted tree $\U_b$, starting at the root, going up (resp. down) with probability $1/2+\epsilon$ (resp. $1/2 -\epsilon$), $\epsilon…
We give a necessary and sufficient condition for species coexistence in a parasite-host growth process on infinite $d$-ary trees. The novelty of this work is that the spreading and death rates for hosts depend on the distance to the nearest…
We introduce a one-parametric family of tree growth models, in which branching probabilities decrease with branch age $\tau$ as $\tau^{-\alpha}$. Depending on the exponent $\alpha$, the scaling of tree depth with tree size $n$ displays a…
We consider catalytic branching populations. They consist of a catalyst population evolving according to a critical binary branching process in continuous time with a constant branching rate and a reactant population with a branching rate…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total…
We extend classical results on simple varieties of trees (asymptotic enumeration, average behavior of tree parameters) to trees counted by their number of leaves. Motivated by genome comparison of related species, we then apply these…
Let n be an integer greater than 1. A tree T is an n-ary tree provided that every node in T has at most n immediate successors. A forcing notion P has the n-localization property if every function from omega to omega in an extension via P…
We consider a dynamic version of the Neyman contagious point process that can be used for modelling the spacial dynamics of biological populations, including species invasion scenarios. Starting with an arbitrary finite initial…
We study an asymptotical behavior of the maximal degree in the degree distribution in an evolving tree model combining the local choice and the Mori's preferential attachment. In the considered model, the random graph is constructed in the…
This thesis presents analysis of the properties and run-time of the Rapidly-exploring Random Tree (RRT) algorithm. It is shown that the time for the RRT with stepsize $\epsilon$ to grow close to every point in the $d$-dimensional unit cube…
A natural extension of a right-continuous integer-valued random walk is one which can jump to the right by one or two units. First passage times above a given fixed level then admit a tractable Laplace transform (probability generating…
We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, $a(x)$, of the random walk. Applying it, we…
A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum…
We consider a branching random walk for which the maximum position of a particle in the n'th generation, M_n, has zero speed on the linear scale: M_n/n --> 0 as n --> infinity. We further remove ("kill") any particle whose displacement is…