Some Questions — Question 33

Definitions and mathematical statement

Fix an integer $k>1$, and let $C$ range over transient simple $k$-regular Cayley graphs rooted at the identity $o$. Simple random walk starts at $o$ and at each discrete time chooses a neighbor uniformly. Define the first positive return time and eventual return probability by \[ \tau_o^+=\inf\{t\geq1:X_t=o\},\qquad q(C)=\mathbb P_o(\tau_o^+<\infty). \] An empty event gives infinite return time. Transience means $q(C)<1$, equivalently that the expected total number of root visits is finite.

Does there exist a number $c(k)<1$ such that $q(C)\leq c(k)$ for every transient $k$-regular Cayley graph? The constant may depend on the degree but not on the group, generating set, or any further geometric parameter. If there are no transient graphs of a particular degree, its assertion is vacuous.

By the Markov property, the Green value including time zero is $1/(1-q(C))$. Thus the question is equivalently a uniform finite bound on this expected visit count in each fixed degree. The probability here is that of ever returning, not a fixed-time return probability or the spectral radius of the random-walk operator. Transient graphs can be amenable, so a bound restricted to nonamenable groups is insufficient. A counterexample would be a sequence of transient graphs of one degree whose eventual return probabilities approach one.

Short English statement

In each fixed degree, is the eventual return probability uniformly bounded away from one for all transient Cayley graphs?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1200033 (AMR-011-0033), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Abert - Some questions (pdf) (2010), Question 33. Original source indexed by AMR-011-0033. https://www.renyi.hu/~abert/questions.pdf.