Critical-curve regularity for finite-lifespan frog models via local-to-global comparisons
Abstract
We study the phase boundary of the finite-lifespan frog model. For rate-one continuous-time simple random walk on an infinite, connected, locally finite transitive graph of superlinear growth, we prove that the critical-density curve is continuous and strictly decreasing, with −logλc locally bi-Lipschitz. The inverse critical-lifespan curve has the corresponding regularity wherever it is finite. Thus the critical-curve conjecture of Angel, de la Riva, Hermon, and Shi reduces to finiteness of the critical lifespan at every positive density. We identify the exact short-lifespan limit tλc(t)→Δlog(1/(1−pc(G))), where Δ is the degree and pc(G) is the bond-percolation threshold, and extend critical-curve regularity and sharpness to a class of long-range frog models. The main tool is a local-to-global principle for activation processes generated by independent finite rooted ranges: local one-hit comparison implies comparison of global reachability and survival. The comparison allows disconnected ranges and source-dependent laws, and therefore also applies to activation models beyond random walks on a fixed graph.
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BibTeX
@misc{monch2026criticalcurve,
title = {{Critical-curve regularity for finite-lifespan frog models via local-to-global comparisons}},
author = {Christian Mönch and Andreas Klippel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29699},
eprint = {2609.29699},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29699},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical-curve regularity for finite-lifespan frog models via local-to-global comparisons
AU - Christian Mönch
AU - Andreas Klippel
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.29699
UR - https://arxiv.org/abs/2609.29699
AB - We study the phase boundary of the finite-lifespan frog model. For rate-one continuous-time simple random walk on an infinite, connected, locally finite transitive graph of superlinear growth, we prove that the critical-density curve is continuous and strictly decreasing, with −logλ_c locally bi-Lipschitz. The inverse critical-lifespan curve has the corresponding regularity wherever it is finite. Thus the critical-curve conjecture of Angel, de la Riva, Hermon, and Shi reduces to finiteness of the critical lifespan at every positive density. We identify the exact short-lifespan limit tλ_c(t)→Δlog(1/(1−p_c(G))), where Δ is the degree and p_c(G) is the bond-percolation threshold, and extend critical-curve regularity and sharpness to a class of long-range frog models. The main tool is a local-to-global principle for activation processes generated by independent finite rooted ranges: local one-hit comparison implies comparison of global reachability and survival. The comparison allows disconnected ranges and source-dependent laws, and therefore also applies to activation models beyond random walks on a fixed graph.
ER -