Some Questions — Question 32

Definitions and mathematical statement

Fix an integer $k\geq2$. Consider rooted simple $k$-regular Cayley graphs, rooted at the identity and identified up to rooted graph isomorphism. A sequence converges locally if, for every fixed radius, its rooted balls eventually agree with that of the limiting graph. On each graph run simple random walk, choosing uniformly among the $k$ neighbors at every step.

For a rooted graph $C$ with root $o$, define its Green value \[ \mathcal G(C)=\sum_{t=0}^{\infty}\mathbb P_o(X_t=o). \] This is the expected number of visits to the root, including time zero. Call the graph transient when $\mathcal G(C)<\infty$, and let $T_k$ be the subspace of transient $k$-regular Cayley graphs.

Is $\mathcal G:T_k\to\mathbb R$ continuous? Explicitly, if $C_n,C\in T_k$ and $C_n\to C$ locally, must $\mathcal G(C_n)\to\mathcal G(C)$? Subtracting the time-zero contribution gives the same continuity question for the number of returns after departure.

Local convergence determines each fixed-time return probability, but continuity of the infinite sum requires uniform control of its tail along the sequence. The limiting graph is assumed transient; convergence toward a recurrent graph is not part of the continuity test on $T_k$. The formulation uses the subspace topology and does not need an additional assertion about whether $T_k$ is closed in the ambient graph space. Spectral radii and first-return probabilities are related quantities but are not substituted for this specified expected-visit statistic.

Short English statement

Is the expected number of visits to the root continuous under local convergence among transient Cayley graphs of fixed degree?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1200032 (AMR-011-0032), 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 32. Original source indexed by AMR-011-0032. https://www.renyi.hu/~abert/questions.pdf.