Existence and sharpness of the phase transition for the frog model on transitive graphs

TL;DR

This work analyzes a frog model with death on broad graph classes, proving the existence of phase transitions in both particle density $\lambda$ and lifetime $t$ for non-amenable graphs of bounded degree and quasi-transitive graphs of superlinear growth, and establishing sharpness of the transition on vertex-transitive graphs. The authors deploy a two-parameter percolation framework, Abelian property-based explorations, and local criteria via $\phi_{\lambda,t}(S)$ to connect finite-set behavior to global survival, plus a renormalization scheme on group nets for polynomial-growth graphs and an escape-probability analysis on non-amenable graphs. The results yield quantitative phase-transition descriptions, including explicit bounds in the non-amenable case, and demonstrate that the frog model undergoes a sharp, many-to-one transition similar to Bernoulli percolation despite dependencies. Collectively, the paper advances our understanding of spreading processes with regeneration on heterogeneous networks and provides tools that may extend to broader interacting particle systems with long-range connections.

Abstract

We consider a slight modification of the frog model. For a given graph, each vertex has $\mathrm{Poisson}(λ)$ 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, becoming inactive after time $t$. 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 $λ$ 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 degrees and quasi-transitive graphs of superlinear polynomial growth, as well as prove the sharpness of the phase transition for transitive graphs.

Figures (2)

• Figure 1: In this example, we consider an exploration of $\mathcal{Z}_{1}$. We start with two active particles at the origin, $\omega_{\mathbf{0}}^{1}$ (red) and $\omega_{\mathbf{0}}^{2}$ (blue). We first follow the trajectory of $\omega_{\mathbf{0}}^{1}$ until it exits the set $S$ for the first time or is deactivated. Since $\omega_{\mathbf{0}}^{1}$ left $S$, within its lifespan, we do not explore any of its vertices, and include $v_1, v_2,$ and $v_3$ as children of $\mathbf{0}$. Now, we follow the trajectory of the particle $\omega_{\mathbf{0}}^{2}$, which didn't exit the set $S$ within its lifespan. The vertices in its trajectory were labeled as $w_1$ and $w_2$ for convenience. The vertex $w_1$ has one particle, say $\omega_{w_1}^{1},$ which immediately left the set at the vertex $v_{4}$ represented by the green trajectory. Since the green trajectory exits $S$, we count $v_{4}$ as the fourth child of $\mathbf{0}$. Since $w_2$ has no particles, the exploration of $\mathbf{0}$ terminates. In this realization of the branching process, $\mathbf{0}$ has four children, so $\mathcal{Z}_{1}=4$.
• Figure 2: We now illustrate the iteration starting at $v_1$, which was one of the vertices counted as a child in the first generation. We shifted the set $S$ to $v_{1}$ and denoted that as $\Psi_{v_{1}}(S)$. Vertices that had their particles revealed at the previous iteration will be refreshed with $\text{Po}(\lambda)$ particles performing i.i.d. continuous time simple random walks, described in the drawing by the empty blue circles. In the example above, $v_{1}$ only had one child that left the set $S$. In this case, $z_{1},z_{2}$ and $z_{3}$ are counted as the children of $v_{1}$. The total size of $\mathcal{Z}_{2}$ will be the sum of the children of $v_{1},v_{2},v_{3}$ and $v_{4},$ each one of them having the same independent distributions with mean less than one.

Theorems & Definitions (59)

• Conjecture 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Conjecture 1.8
• Conjecture 1.9
• Proposition 2.1