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The weak-strong survival transition on trees and nonamenable graphs · 2006-prob-198

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

Fix an integer d≥3d\ge 3, and let Td\mathbb T^d be the infinite homogeneous tree in which every vertex has degree dd. Choose a root oo. Consider the isotropic contact process (ξt)t≥0(\xi_t)_{t\ge 0}, started from ξ0={o}\xi_0=\{o\}, in which each infected vertex becomes healthy at rate 11, while a healthy vertex with kk infected neighbors becomes infected at rate λk/d\lambda k/d, where λ>0\lambda>0.

Let

λu:=inf⁡{λ>0:Pλ(o∈ξt for arbitrarily large t)>0} \lambda_u:=\inf\bigl\{\lambda>0:\mathbb P_\lambda(o\in\xi_t\text{ for arbitrarily large }t)>0\bigr\}

be the upper critical parameter separating weak survival from strong (local) survival. An end of Td\mathbb T^d is an equivalence class of infinite simple rays, where two rays are equivalent if they eventually coincide. For ends α,β\alpha,\beta, let N(α,β)N(\alpha,\beta) be the number of common edges in their rays starting at oo, and equip the boundary ∂Td\partial\mathbb T^d with the metric

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

For λ≤λu\lambda\le \lambda_u, let Λ⊆∂Td\Lambda\subseteq\partial\mathbb T^d be the boundary accumulation set of the set ⋃t≥0ξt\bigcup_{t\ge0}\xi_t of vertices ever infected; equivalently, α∈Λ\alpha\in\Lambda if every cone neighborhood of α\alpha contains a vertex infected at some time. Let δ(λ)\delta(\lambda) denote the almost-sure value, conditional on global survival, of the Hausdorff dimension of Λ\Lambda in the metric ρ\rho; when global survival has probability zero, set δ(λ)=0\delta(\lambda)=0. Define

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

For each vertex x∈Tdx\in\mathbb T^d, define its total infection time by

J(x):=∫0∞1{x∈ξt} dt. J(x):=\int_0^\infty \mathbf 1_{\{x\in\xi_t\}}\,dt.

Determine whether there exists a finite constant CdC_d, depending only on dd, such that for every λ≤λ∗\lambda\le\lambda_* and every vertex x∈Tdx\in\mathbb T^d,

Eλ ⁣[J(x)∣J(x)>0]≤Cd. \mathbb E_\lambda\!\left[J(x)\mid J(x)>0\right]\le C_d.

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Question 1

Fix an integer d≥3d\ge 3, and let Td\mathbb T^d be the infinite homogeneous tree of degree dd, rooted at a vertex oo. Consider the isotropic contact process (ξt)t≥0(\xi_t)_{t\ge0}, started from ξ0={o}\xi_0=\{o\}, in which infected vertices recover at rate 11 and a healthy vertex with kk infected neighbors becomes infected at rate λk/d\lambda k/d. Define

λu:=inf⁡{λ>0:Pλ(o∈ξt for arbitrarily large t)>0}. \lambda_u:=\inf\bigl\{\lambda>0:\mathbb P_\lambda(o\in\xi_t\text{ for arbitrarily large }t)>0\bigr\}.

Equip the space ∂Td\partial\mathbb T^d of ends with the metric ρ(α,β)=2−N(α,β)\rho(\alpha,\beta)=2^{-N(\alpha,\beta)}, where N(α,β)N(\alpha,\beta) is the number of common initial edges in the rays from oo representing α\alpha and β\beta. For λ≤λu\lambda\le\lambda_u, let Λ\Lambda be the set of ends whose every vertex is infected at some time, and let δ(λ)\delta(\lambda) be the almost-sure value, conditional on global survival, of dim⁡H(Λ)\dim_H(\Lambda), with δ(λ)=0\delta(\lambda)=0 when global survival has probability zero. Set

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

Prove that λ∗=λu\lambda_*=\lambda_u.

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 8, DOI: https://doi.org/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 proof or counterexample was found for the uniform bound, including at λ=λ∗\lambda=\lambda_*. The distinction between known results and the conjecture is important: weak survival implies J(x)<∞J(x)<\infty almost surely, and exponential temporal decay gives EλJ(x)<∞\mathbb E_\lambda J(x)<\infty for each fixed λ<λ∗\lambda<\lambda_*, but neither statement provides a constant uniform over both xx and λ\lambda, nor does it settle the endpoint λ∗\lambda_*.

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 (2002), 341–350, DOI 10.1007/s005740200018. In the equivalent parametrization with infection rate fixed and recovery rate varied, this paper states the requested occupation-time estimate as Conjecture 2 (in fact over the full range up to the upper survival transition). Lemma 3 compares it with conditional expected counts of recovery marks and infection arrows. The paper proves only a conditional consequence: if the conjecture holds, then the gap between the critical Malthusian parameter and its subcritical value is at most a constant times the square root of the distance to the transition. Thus it does not resolve the conjecture.

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, DOI 10.4171/022-3/33. This survey restates the exact bound as Conjecture 10, with λ∗\lambda_* defined through the Hausdorff dimension, and gives the same square-root estimate only conditionally as Theorem 11. This is the latest source found that directly formulates the exact problem.

Steven P. Lalley, “Growth Profile and Invariant Measures for the Weakly Supercritical Contact Process on a Homogeneous Tree,” The Annals of Probability 27 (1999), 206–225, DOI 10.1214/aop/1022677259, establishes exponential decay in time in the strict subcritical-dimension regime used to deduce pointwise integrability of J(x)J(x). Steven P. Lalley and Tom Sellke, “Limit Set of a Weakly Supercritical Contact Process on a Homogeneous Tree,” The Annals of Probability 26 (1998), 644–657, DOI 10.1214/aop/1022855646, identify the Hausdorff dimension through the exponential decay rate of vertex-hitting probabilities and prove that it is at most half the boundary dimension in weak survival. These are substantive partial ingredients, but they do not control the conditional first moment uniformly as required.

References
  1. [1]ICM Conjectures · 2006-prob-198 link
  2. [2]ICM 2006: The weak-strong survival transition on trees and nonamenable graphs link

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