Recurrence under square-root separation limits

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G_k$ be finite connected graphs with a common degree bound. Assume that for every nonempty vertex set $S\subset V(G_k)$ one can delete at most $|S|^{1/2}$ vertices so that every connected component of the induced graph on the remaining vertices of $S$ has at most $|S|/2$ vertices. Root $G_k$ at a uniform random vertex $o_k$. Local convergence to a random rooted graph $(G,o)$ means convergence in law of every fixed-radius induced rooted neighborhood. A graph is recurrent if simple random walk returns to its starting vertex infinitely often almost surely.

Statement recorded in the supplied source.

If $(G_k,o_k)$ converges locally to $(G,o)$, is $G$ recurrent almost surely? [S2]

Short English statement

Does square-root separation, together with bounded degrees, force recurrence in a random local limit?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000056 (AMR-099-0056). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Euclidean vs. Graph Metric, version dated 16 July 2012; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 2, definition of square-root separation and Question 2.2, pp. 2--3. https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf.