Asymptotic constant for the strong-survival threshold

Determine the constant \(c\) governing the large-degree asymptotic behavior of the strong-survival threshold, specifically establish whether \(\lambda_2\sqrt d\to c\) as \(d\to\infty\) and identify the value of \(c\).

Background

The paper recalls that $\lambda_2$ is of order $d^{-1/2}$ for the $d$-regular tree and gives a non-sharp lower bound on $\lambda_2\sqrt d$.

Although the order of magnitude is known, the limiting constant in the asymptotic scaling remains unresolved. Determining it would provide a sharper description of the separation between weak and strong survival at large degree.