Weak survival of the contact process on nonamenable Cayley graphs
Let $\Gamma$ be an infinite finitely generated group with identity $e$, and $S \subset \Gamma \setminus \{e\}$ a finite symmetric generating set. Its Cayley graph $G$ has vertex set $\Gamma$ and an edge $\{x, xs\}$ for each $x \in \Gamma$ and $s \in S$. Assume $G$ is nonamenable: $$h_E(G) = \inf_{\varnothing \ne K \subset \Gamma,\ |K| < \infty} \frac{|\partial_E K|}{|K|} > 0,$$ where $\partial_E K$ is the set of edges with exactly one endpoint in $K$. In the contact process $(\xi_t)_{t \ge 0}$ on $G$ with infection rate $\lambda > 0$, each infected vertex becomes healthy at rate $1$ and each healthy vertex becomes infected at rate $\lambda$ times its number of infected neighbours; let $\xi_0 = \{e\}$ and write $\mathbb{P}_\lambda$ for its law. Let $$\lambda_{\mathrm{g}}(G) = \inf\{\lambda > 0 : \mathbb{P}_\lambda(\xi_t \ne \varnothing \text{ for all } t \ge 0) > 0\},$$ $$\lambda_{\mathrm{l}}(G) = \inf\{\lambda > 0 : \mathbb{P}_\lambda(e \in \xi_t \text{ for arbitrarily large } t) > 0\},$$ so that $\lambda_{\mathrm{g}}(G) \le \lambda_{\mathrm{l}}(G)$.
On regular trees of degree at least $3$, $\lambda_{\mathrm{g}} < \lambda_{\mathrm{l}}$ (Pemantle 1992 for degree at least $4$, Liggett 1996 for degree $3$, with a simpler proof by Stacey 1996). Schonmann (2001) proved $\lambda_{\mathrm{g}}(G) < \lambda_{\mathrm{l}}(G)$ for every transitive graph $G$ of degree $d$ with $h_E(G)/d \ge 1/\sqrt{2}$. Pemantle and Stacey (2001) constructed nonamenable trees of bounded degree with $\lambda_{\mathrm{g}} = \lambda_{\mathrm{l}}$.
Problem. Must $\lambda_{\mathrm{g}}(G) < \lambda_{\mathrm{l}}(G)$ for every such Cayley graph $G$, that is, for every infinite finitely generated group $\Gamma$ and every finite symmetric generating set $S$ whose Cayley graph is nonamenable?
Source: Lyons (2000), §8, question following Theorem 8.5, for nonamenable transitive graphs; Lalley (2006, Kyoto lecture), conjecture that the contact process on every nonamenable finitely generated group has a weak survival phase. Stated by AIM — Open Applied Problems for the isotropic contact process on Cayley graphs.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- GR, MP
- Keywords
- AIM Open Applied Problems, Cayley graphs, contact process, nonamenable groups
References
@article{lyons2000,
author = {Lyons, Russell},
title = {Phase transitions on nonamenable graphs},
journal = {J. Math. Phys.},
volume = {41},
number = {3},
year = {2000},
pages = {1099--1126},
doi = {10.1063/1.533179},
eprint = {math/9908177},
archivePrefix = {arXiv},
}
@misc{lalley2006,
author = {Lalley, Steven P.},
title = {The weak/strong survival transition on trees and nonamenable graphs},
howpublished = {Lecture slides, Kyoto},
year = {2006},
note = {Dated 4 September 2006},
url = {https://galton.uchicago.edu/~lalley/Talks/kyoto.pdf},
}
@article{pemantle1992,
author = {Pemantle, Robin},
title = {The contact process on trees},
journal = {Ann. Probab.},
volume = {20},
number = {4},
year = {1992},
pages = {2089--2116},
doi = {10.1214/aop/1176989541},
}
@article{liggett1996,
author = {Liggett, Thomas M.},
title = {Multiple transition points for the contact process on the binary tree},
journal = {Ann. Probab.},
volume = {24},
number = {4},
year = {1996},
pages = {1675--1710},
doi = {10.1214/aop/1041903202},
}
@article{stacey1996,
author = {Stacey, A. M.},
title = {The existence of an intermediate phase for the contact process on trees},
journal = {Ann. Probab.},
volume = {24},
number = {4},
year = {1996},
pages = {1711--1726},
doi = {10.1214/aop/1041903203},
}
@article{schonmann2001,
author = {Schonmann, Roberto H.},
title = {Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs},
journal = {Comm. Math. Phys.},
volume = {219},
number = {2},
year = {2001},
pages = {271--322},
doi = {10.1007/s002200100417},
}
@article{pemantle2001,
author = {Pemantle, Robin and Stacey, Alan M.},
title = {The branching random walk and contact process on {G}alton--{W}atson and nonhomogeneous trees},
journal = {Ann. Probab.},
volume = {29},
number = {4},
year = {2001},
pages = {1563--1590},
doi = {10.1214/aop/1015345762},
}
@misc{aim-contact-process-nonamenable-weak-survival,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Weak survival of the contact process on nonamenable Cayley graphs}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/314-contact-process-nonamenable-weak-survival.md},
note = {Entry 314 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00418, "Weak survival of the contact process on nonamenable Cayley graphs", clopen.solutions.