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The weak-strong survival transition on trees and nonamenable graphs · 2006-prob-197

Special processes · ICM 2006: The weak-strong survival transition on trees and nonamenable graphs

MSC 60 · Probability theory and stochastic processes

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

Let d≥3d\ge 3, and let Td\mathbb T^d be the infinite homogeneous tree of degree dd, rooted at a vertex oo. For λ>0\lambda>0, let (ξt)t≥0(\xi_t)_{t\ge0} be the isotropic contact process on Td\mathbb T^d, started from ξ0={o}\xi_0=\{o\}: each infected vertex becomes healthy at rate 11, and each healthy vertex becomes infected at rate λ/d\lambda/d times its number of infected neighbors.

Let 0<λc<λu<∞0<\lambda_c<\lambda_u<\infty be the critical intensities characterized as follows: the process dies out almost surely for λ≤λc\lambda\le\lambda_c, survives weakly with positive probability for λc<λ≤λu\lambda_c<\lambda\le\lambda_u, and survives strongly with positive probability for λ>λu\lambda>\lambda_u. Here weak survival means that ξt≠∅\xi_t\ne\varnothing for every t≥0t\ge0, but every finite set of vertices is eventually free of infection, whereas strong survival means that every nonempty finite set is infected at arbitrarily large times.

Identify the boundary ∂Td\partial\mathbb T^d with the set of infinite simple rays starting at oo. For α,β∈∂Td\alpha,\beta\in\partial\mathbb T^d, let N(α,β)N(\alpha,\beta) be the number of initial edges shared by the two rays and equip the boundary with the ultrametric

ρ(α,β)=2−N(α,β). \rho(\alpha,\beta)=2^{-N(\alpha,\beta)}.

For λc<λ≤λu\lambda_c<\lambda\le\lambda_u, on the event of survival let Λ⊆∂Td\Lambda\subseteq\partial\mathbb T^d be the set of ends α\alpha such that, for every n≥1n\ge1, some vertex lying beyond the first nn edges of α\alpha is infected at some time. The Hausdorff dimension of Λ\Lambda is almost surely constant on the event of survival; denote this constant by

δ(λ)=dim⁡H(Λ). \delta(\lambda)=\dim_H(\Lambda).

Define

λ∗=sup⁡{λ∈(λc,λu]:δ(λ)<12dim⁡H(∂Td)}. \lambda_*=\sup\left\{\lambda\in(\lambda_c,\lambda_u]: \delta(\lambda)<\frac12\dim_H(\partial\mathbb T^d)\right\}.

Determine whether

λ∗=λu. \lambda_*=\lambda_u.

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Question 1

Let d≥3d\ge3, let Td\mathbb T^d be the homogeneous tree of degree dd rooted at oo, and let (ξtλ)t≥0(\xi_t^\lambda)_{t\ge0} be the isotropic contact process started from {o}\{o\}, in which infected vertices recover at rate 11 and healthy vertices become infected at rate λ/d\lambda/d times their number of infected neighbors. For a vertex xx at distance nn from oo, set

un(λ)=P{x∈ξtλ for some t≥0},β(λ)=lim⁡n→∞un(λ)1/n, u_n(\lambda)=\mathbb P\{x\in\xi_t^\lambda\text{ for some }t\ge0\}, \qquad \beta(\lambda)=\lim_{n\to\infty}u_n(\lambda)^{1/n},

and define

λ∗=sup⁡{λ>0:β(λ)<1d−1}. \lambda_*=\sup\left\{\lambda>0:\beta(\lambda)<\frac1{\sqrt{d-1}}\right\}.

For each vertex xx, let

Jλ(x)=∫0∞1{x∈ξtλ} dt J_\lambda(x)=\int_0^\infty \mathbf 1_{\{x\in\xi_t^\lambda\}}\,dt

be its total infection time. Determine whether there exists a finite constant CdC_d, depending only on dd, such that

E ⁣[Jλ(x)∣Jλ(x)>0]≤Cd \mathbb E\!\left[J_\lambda(x)\mid J_\lambda(x)>0\right]\le C_d

for every vertex xx and every 0<λ≤λ∗0<\lambda\le\lambda_*.

Source reference: Steven P. Lalley, “The Weak/Strong Survival Transition on Trees and Nonamenable Graphs,” in Proceedings of the International Congress of Mathematicians, Madrid 2006, Vol. III, European Mathematical Society, 2007, pp. 637–647, Conjecture 10, DOI: 10.4171/022-3/33.

Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved. No citable result was found that rules out, for every degree d≥3d\ge3, a nontrivial interval immediately below λu\lambda_u on which the limit-set dimension is already 12dim⁡H(∂Td)\tfrac12\dim_H(\partial\mathbb T^d).

Steven P. Lalley and Tom Sellke, “Limit Set of a Weakly Supercritical Contact Process on a Homogeneous Tree,” The Annals of Probability 26(2) (1998), 644–657, DOI 10.1214/aop/1022855646, prove that, almost surely on survival,

δ(λ)=log⁡((d−1)β(λ))log⁡2,β(λ)=lim⁡n→∞un(λ)1/n, \delta(\lambda)=\frac{\log((d-1)\beta(\lambda))}{\log 2}, \qquad \beta(\lambda)=\lim_{n\to\infty}u_n(\lambda)^{1/n},

where un(λ)u_n(\lambda) is the probability of ever infecting a specified vertex at distance nn. They also prove δ(λ)≤12dim⁡H(∂Td)\delta(\lambda)\le \tfrac12\dim_H(\partial\mathbb T^d). Thus the exact question is whether β(λ)\beta(\lambda) can attain its ceiling 1/d−11/\sqrt{d-1} strictly before λu\lambda_u.

Steven P. Lalley, “Growth Profile and Invariant Measures for the Weakly Supercritical Contact Process on a Homogeneous Tree,” The Annals of Probability 27(1) (1999), 206–225, Project Euclid, originally asserted strict increase of β\beta throughout the weak-survival phase. Its correction, “Correction: Growth Profile and Invariant Measures for the Weakly Supercritical Contact Process on a Homogeneous Tree,” The Annals of Probability 30(4) (2002), 2108–2112, Project Euclid, states that the proof was incorrect. The corrected Theorem 3′3' proves strict increase only while β(λ)<1/d−1\beta(\lambda)<1/\sqrt{d-1} and explicitly leaves open a plateau β(λ)=1/d−1\beta(\lambda)=1/\sqrt{d-1} on an interval ending at λu\lambda_u. This is precisely the remaining gap in the present problem.

Steven P. Lalley and Thomas M. Sellke, “The Weak Survival/Strong Survival Phase Transition for the Contact Process on a Homogeneous Tree,” Bulletin of the Brazilian Mathematical Society 33(3) (2002), 341–350, DOI 10.1007/s005740200018, establish conditional critical-exponent progress: assuming a uniform bound on the conditional expected total infection time at a vertex, they obtain a square-root upper bound on the dimension deficit. This does not exclude the plateau or prove λ∗=λu\lambda_*=\lambda_u.

Steven P. Lalley, “The Weak/Strong Survival Transition on Trees and Nonamenable Graphs,” in Proceedings of the International Congress of Mathematicians, Madrid 2006, Vol. III (European Mathematical Society, 2007), 637–647, DOI 10.4171/022-3/33, records the exact assertion as Conjecture 8 and summarizes the established continuity, below-ceiling strictness, and equality at λu\lambda_u.

Irene Hueter, “Anisotropic Contact Process on Homogeneous Trees” (2001), arXiv:math/0109047, is an unrefereed, single-version preprint whose Theorem 5 claims the required strictness for the isotropic specialization on homogeneous trees of even degree. It does not cover odd degrees, and the later correction and ICM survey above continue to treat the general assertion as open. Hence it cannot resolve the exact all-degree problem.

As a later check on progress, Stein Andreas Bethuelsen, Gabriel Baptista da Silva, and Daniel Valesin, “Graph Constructions for the Contact Process with a Prescribed Critical Rate,” Journal of Theoretical Probability 35 (2022), 863–893, DOI 10.1007/s10959-020-01063-4, uses precisely the corrected result that β\beta is strictly increasing only below its ceiling; it supplies no theorem excluding the unresolved plateau.

References
  1. [1]ICM Conjectures · 2006-prob-197 link
  2. [2]ICM 2006: The weak-strong survival transition on trees and nonamenable graphs link

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