Branching Random Walks on Free Products of Groups
Elisabetta Candellero, Lorenz A. Gilch, Sebastian M\"uller

TL;DR
This paper analyzes phase transitions of branching random walks on Cayley graphs of free products, focusing on the boundary dimensions of the trace and the structural changes at critical growth parameters.
Contribution
It establishes the existence and properties of a boundary dimension function and identifies phase transitions in the structure of the trace and end boundary.
Findings
The boundary dimension function $\
(\
Finite and infinite word boundaries exhibit distinct Hausdorff dimensions at phase transitions.
Abstract
We study certain phase transitions of branching random walks (BRW) on Cayley graphs of free products. The aim of this paper is to compare the size and structural properties of the trace, i.e., the subgraph that consists of all edges and vertices that were visited by some particle, with those of the original Cayley graph. We investigate the phase when the growth parameter is small enough such that the process survives but the trace is not the original graph. A first result is that the box-counting dimension of the boundary of the trace exists, is almost surely constant and equals the Hausdorff dimension which we denote by . The main result states that the function has only one point of discontinuity which is at where is the radius of convergence of the Green function of the underlying random walk. Furthermore, is…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
