The contact-process inequality λ1<λp on Zd

For every integer $d\ge 2$ , consider the nearest-neighbor contact process on $\mathbb{Z}^d$ with infection rate $\lambda$ and recovery rate $1$ . Let $\lambda_1$ be the critical value for global survival, and let $\lambda_p$ be the critical value at which the infected vertices under the upper invariant measure contain an infinite connected component with positive probability. The conjecture is that $\lambda_1<\lambda_p$ for every $d\ge 2$ .

References

Primary source

arXiv

Additional references

Percolation of the contact process on the regular tree — arXiv

Posted 4 weeks ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A new paper advances the analogous contact-process inequality on regular trees, but the original problem in two or more dimensions remains open.

The problem asks whether the strict ordering between the two contact-process thresholds holds on $\mathbb{Z}^d$ for $d \ge 2$ . The retrieved literature continues to describe this lattice case as unresolved.

Known results

Pemantle proved $\lambda_1 < \lambda_2$ on regular trees of degree at least $4$ ; Liggett handled the remaining regular-tree cases, and Stacey gave a more general proof.

September 2026 regular-tree advance

A recent paper, Percolation of the contact process on the regular tree, reports the desired ordering on regular trees and separation of all four thresholds for sufficiently high degree. This is substantial progress for a nonamenable class, but it does not settle the inequality on $\mathbb{Z}^d$ for $d \ge 2$ ; the reported advance is unverified here.

Current status (as of September 2026): The inequality remains open on $\mathbb{Z}^d$ for $d \ge 2$ ; regular-tree analogues are reported as settled, but the latest advance does not resolve the lattice problem.

Sources

No solutions have been posted yet.