We study the linear growth rate of the range of size-conditioned Branching Random Walks (BRW) when the offspring distribution $\mu$ is critical and attracted to an $\alpha$-stable law. This is done via the infinite invariant BRW introduced by Le Gall&Lin and a new criterion which relates this growth rate of the range to a notion of dimension of the underlying tree in a general way. Then, in the transient case (that is, when the range does grow linearly), we extend the notion of branching capacity to this $\alpha$-stable case. We show that it is still related to the asymptotic probability that a BRW (or its infinite version) reaches a distant set in $\mathbb Z^d$, and we estimate the $\alpha$-stable branching capacity of balls.
We study scaling limits of $\mathbb R^d$-valued branching random walks (BRWs for short) in the following setting: indexing-trees are Galton-Watson trees with critical offspring distribution which are assumed, when appropriately rescaled, to converge to a critical $\psi$-L\'evy tree; conditional on the indexing tree, th...
Let $(c_t)_{t\geq 0}$ be a deterministic family of edge conductances on a countable vertex set, monotone in $t$, and let $(X_t)$ be the random walk that takes its $t$-th step using the conductances $c_t$. We prove that if $c_t\uparrow c_\infty$ and $c_\infty$ is recurrent (respectively, $c_0$ is transient), then $(X_t)...
We study a class of random aggregation trees that generalizes the discrete stick-breaking construction of the uniform labelled tree (Aldous, 1991). To sample the tree of size $n$, vertices are added sequentially, with the $i$th vertex starting a new branch with a prescribed probability $f(n,i)$; otherwise, it extends t...
Serte Donderwinkel, Jeroen van Haastert· 0 citations
We consider a random walk on the integers that at each site is biased to step right until the first time it has stepped left from that site, thereafter stepping left or right from that site with equal chance, introduced by Pinsky. Pinsky calculated the speed of the walk and the probability that such a walk escapes to i...
We prove a shape theorem for a multidimensional periodic branching diffusion, which is a branching particle system where particle trajectories evolve according to a diffusion, drift, branching rate and offspring distribution which all depend periodically on space. More precisely, we show that almost surely as $t\to\inf...
We prove recurrence criteria for inhomogeneous long-range percolation in dimensions one and two. In dimension one, recurrence follows from a purely geometric scarcity condition: long edges eventually disappear on exponential scales. This applies to weight-dependent random connection models and related one-dimensional s...
Johannes Bäumler, Lukas Lüchtrath, C. Mönch· 0 citations
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