Criticality of the recurrence–transience threshold

Determine the behavior of random walks in random environments on random spherically symmetric trees and branching processes in varying environments at the critical thresholds where the combined growth and environmental exponents satisfy \(\gamma+\alpha=1\) and \(\gamma_v+\alpha=1\), respectively, including the resistance-series behavior in these logarithmically divergent cases.

Background

The paper proves recurrence below and transience above the thresholds determined by the sum of the tree-growth exponent and the logarithmic environmental drift. The equality cases are not resolved because the associated resistance series diverge only logarithmically. The authors identify the critical regimes for both random spherically symmetric trees with random environments and branching processes in varying environments with random environments as unresolved.

References

Two natural problems remain unresolved: the behavior at criticality, $\gamma+\alpha=1$ and $\gamma_v+\alpha=1$, where the resistance series diverges only logarithmically; and the speed on $BPVETw$, for which we provide an upper bound and state conjecture \ref{conj_2}.

— Random walk in a non-homogeneous random environment on some random trees and the non-negative integers  (2609.30648 - Oraby et al., 25 Sep 2026) in Section Conclusion