Comparison of \(\lambda_p\) and \(\lambda_2\) for low-degree regular trees

Determine how the upper-invariant infection percolation threshold \(\lambda_p\) compares with the strong-survival threshold \(\lambda_2\) for the \(d\)-regular tree when \(d=3\) or \(d=4\).

Background

The paper proves an upper bound implying λp<λ2\lambda_p<\lambda_2 for regular trees with d≥5d\ge5. The authors explicitly state that their methods do not settle the comparison for the two remaining non-line cases d=3,4d=3,4.

Resolving this question would complete the comparison between the percolation threshold of the upper invariant infected set and the strong-survival threshold across all regular trees of degree at least three.

References

We do not necessarily expect a general result on the comparison between $\lambda_p$ and $\lambda_2$, and in particular it is unclear to us how they compare when $d$ is $3$ or $4$.

— Percolation of the contact process on the regular tree  (2609.09972 - Fernley et al., 9 Sep 2026) in Remark following Theorem 2.1, Section 2