Dimension-dependent annealed Kolmogorov estimate in one dimension

Establish whether, for the branching random walk in Bernoulli site-percolation environment with \(\mu_\circ=\frac12\delta_0+\frac12\delta_2\) and \(\mu_\bullet=\delta_1\) in dimension \(d=1\), there exists a constant \(C>0\) such that the annealed survival probability satisfies \(\mathbb{P}(Z_n\ge1)=\frac{2}{pn}+\frac{C+o(1)}{n^{3/2}}\) as \(n\to\infty\), and more generally determine the dimension dependence of the annealed Kolmogorov estimate.

Background

The model assigns the critical offspring distribution 12δ0+12δ2\frac12\delta_0+\frac12\delta_2 to open sites and the deterministic offspring distribution δ1\delta_1 to closed sites. The annealed survival probability is denoted by P(Zn≥1)\mathbb{P}(Z_n\ge1).

The paper reports that Engländer and Sieben formulated the conjecture on the basis of numerical simulations. It predicts behavior different from the classical critical Galton–Watson process, with a dimension-dependent correction term; in dimension one, the proposed correction is of order n−3/2n^{-3/2}. The paper does not establish this annealed asymptotic.

References

Further, by numerical simulation, Engländer and Sieben conjecture that the annealed Kolmogorov estimate behaves differently from the classical model and depends on the dimension d. More precisely, in dimension d=1, Conjecture 6.1 in says that there exists some positive constant C>0 such that as n\to\infty, (Z_n \ge 1) = \frac{2}{pn } + \frac{C+o(1)}{ n{3/2}.

— Branching random walk in random environment  (2608.13084 - Chen et al., 13 Aug 2026) in Section 1, subsection "Main results" (discussion of prior conjectures before Assumption (A1))