Regeneration comparison for random walks on trees

Prove or disprove Conjecture A.1: for every fixed walk length n, if X_0, X_1, \ldots, X_n is a simple symmetric random walk on the 3-regular tree T_3 and Y_0, Y_1, \ldots, Y_n is a simple symmetric random walk on any tree T whose vertices all have degree at least 3, then the probability that Y has a regeneration is at least the probability that X has a regeneration; more generally, establish or refute the corresponding comparison for the numbers of regenerations of Y and X.

Background

The paper presents the lonely passenger problem as a toy model designed to isolate a key difficulty in an unresolved problem attributed to László Márton Tóth. The unresolved problem concerns finite simple symmetric random walks on trees and the occurrence of a regeneration, defined as the existence of an edge traversed exactly once.

The conjecture compares a simple symmetric random walk on the 3-regular tree with one on an arbitrary tree whose vertices all have degree at least 3. It asserts that the latter walk has at least as large a probability of producing a regeneration, and proposes the stronger analogous comparison for the number of regenerations. The paper notes that a positive answer would have implications for the theory of Ramanujan graphs, specifically for Bernoulli graphings of unimodular Galton–Watson trees.

References

The problem studied in this paper was designed as a toy example to feature a key difficulty in the following, still open problem of László Márton Tóth. Consider a discrete time random walk of finite length n on a tree. We say that a regeneration occurs if there is an edge which is traversed exactly once. Let T3 denote the 3-regular tree, and let T denote any tree where every vertex has degree at least 3. Let X0, X1, . . . , Xn be a simple symmetric random walk on T3, and let Y0, Y1, . . . , Yn be a simple symmetric random walk on T . Prove or disprove the following Conjecture A.1 (László Márton Tóth, 2023).For any fixed nas above,PpA regeneration occurs for Y q ě PpA regeneration occurs for Xq (A.1) or, more generally,pnumber of regenerations for Y q ě pnumber of regenerations for Xq. (A.2)

— Lonely passenger problem: the more buses there are, the more lonely passengers there will be  (2501.05758 - Tóth, 10 Jan 2025) in Appendix A.1, Conjecture A.1, p. 12