Critical-curve regularity for finite-lifespan frog models via local-to-global comparisons

Technical Summary: Critical-curve regularity for finite-lifespan frog models via local-to-global comparisons

Problem Statement
The paper investigates the phase boundary of the finite-lifespan frog model on infinite, connected, locally finite, transitive graphs of superlinear growth. In this model, particles (frogs) are initially dormant at vertices according to a Poisson($\lambda$) distribution. An initial activation at a root vertex $o$ triggers a cascade: active frogs perform independent continuous-time simple random walks (rate-one) and wake sleeping frogs upon visiting their vertices. Each frog is removed $t$ units of time after activation.

The central object of study is the trade-off between particle density $\lambda$ and lifespan $t$. The authors define the critical density $\lambda_c(t)$ and the critical lifespan $t_c(\lambda)$ as the thresholds separating the subcritical regime (finite activated cluster) from the supercritical regime (infinite activated cluster). While previous work established the sharpness of this phase transition, the regularity of the critical curves—specifically whether they are continuous and strictly decreasing—remained an open conjecture by Angel, de la Riva, Hermon, and Shi. The paper addresses whether the exchange between density and lifespan is "genuine and stable," ruling out flat regions (where increased lifespan yields no density benefit) and jumps (where infinitesimal lifespan increases cause macroscopic density drops).

Methodology
The authors develop a "local-to-global" framework for activation processes generated by independent finite rooted ranges. The methodology relies on two primary comparison mechanisms that operate in opposite directions:

1. Local-to-Global Comparison (Theorem B): The authors establish a principle where a local ordering of "hit probabilities" (the probability that a random range from a source intersects a target set) implies a global ordering of reachability and survival probabilities. This theorem generalizes previous nearest-neighbor comparisons to settings with disconnected ranges and long-range jumps. It relies on revealing source ranges one at a time and showing that the event of reaching a target depends on the exposed range only through a "one-hit" question.
2. Forward Comparison (Theorem C): This compares the effect of increasing lifespan $t$ against a reduction in density $\lambda$. By analyzing the number of jumps a frog makes by time $t$ (modeled via Poisson thinning), the authors prove that increasing the lifespan from $s$ to $t$ compensates for a fixed fraction of the frogs, specifically a factor of $\rho(s, t) = (1-e^{-s})/(1-e^{-t})$. This establishes that longer lifespans strictly reduce the required critical density, ruling out flat pieces in the curve.
3. Reverse Comparison (Section 4): To rule out jumps and prove continuity, the authors derive a reverse bound. Using finite-volume sensitivity formulas (Russo-type derivatives) and geometric exit estimates, they bound the effect of extending the lifespan by the effect of adding density. This requires a "two-step condition" on the jump kernel $Q$: $Q^2(x, y) \le \kappa Q(x, y)$ for $x \neq y$. This condition ensures that reaching a vertex in two steps is not significantly more likely than in one step, allowing the control of jump counts weighted by exit probabilities.

Key Contributions and Results

• Regularity of Critical Curves (Theorem A): On transitive graphs of superlinear growth, the critical density curve $\lambda_c(t)$ is proven to be continuous and strictly decreasing on $(0, \infty)$. Furthermore, $-\log \lambda_c(t)$ is locally bi-Lipschitz, providing a quantitative bound on the rate of change. The inverse curve $t_c(\lambda)$ shares these regularity properties on the interval $(\lambda_\infty, \infty)$, where $\lambda_\infty = \lim_{t \to \infty} \lambda_c(t)$.
• Resolution of Conjecture: The paper resolves the conjecture of Angel et al. completely for nonamenable transitive graphs and transitive graphs of superlinear polynomial growth, as these classes satisfy $\lambda_\infty = 0$. For general superlinear growth graphs, the regularity of the finite part of the boundary is established, leaving only the question of whether $\lambda_\infty = 0$ (i.e., whether every positive density has a finite critical lifespan) open.
• Small-Lifespan Scaling Limit: The authors prove the existence of the limit $L_G = \lim_{t \downarrow 0} t \lambda_c(t)$ and provide explicit bounds involving the graph degree $\Delta$ and the bond-percolation threshold $p_c(G)$.
• Long-Range Extensions (Theorem 6.3): The results are extended to a class of long-range frog models where frogs perform random walks with jump kernels satisfying the "two-step condition." This includes symmetric, translation-invariant power-law kernels on polynomial-growth Cayley graphs. For these models, the critical curves are continuous and strictly decreasing on all of $(0, \infty)$.
• Sharpness: The paper adapts the sharpness argument (linear lower bound for survival probability above criticality and exponential tail for subcritical cluster size) to the long-range setting using the two-step condition.

Significance
The paper claims to provide the first quantitative resolution of the regularity problem for the finite-lifespan frog model on general transitive graphs. By isolating the "local-to-global" principle, the authors demonstrate that the regularity of the phase boundary does not rely on nearest-neighbor geometry but rather on the structural properties of the activation ranges and the jump kernel. The work confirms that the trade-off between density and lifespan is stable and continuous, transforming the qualitative understanding of the phase transition from a monotonic relationship to a strictly decreasing, continuous, and locally bi-Lipschitz curve. The introduction of the two-step condition allows these results to apply to heavy-tailed, long-range interactions, complementing existing work on activation times and finite-volume cover lifespans.