Abstract:We consider a slight modification of the frog model. For a given graph, each vertex has $\mathrm{Poisson}(\lambda)$ particles (or frogs). At time zero, only the particles at the origin are active, and all the other particles are sleeping. Each active particle performs an independent, continuous-time simple random walk up to a fixed lifetime $t$, after which the particle dies and is removed from the system. Once an active frog jumps to a vertex, it activates all of its particles. The survival of active particles can be studied as a dependent percolation model with two parameters $\lambda$ and $t$. In the present work, we establish the existence of a phase transition with respect to each parameter for non-amenable graphs of bounded degree and quasi-transitive graphs of superlinear polynomial growth, as well as prove the sharpness of the phase transition for transitive graphs.
From: Daniel De La Riva Massaad [view email]
[v1]
Sun, 7 Dec 2025 03:17:31 UTC (62 KB)
[v2]
Sun, 25 Jan 2026 04:05:44 UTC (74 KB)
[v3]
Fri, 11 Sep 2026 11:25:28 UTC (80 KB)
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