Random-walk displacement on circle-packed doubling graphs

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 $G$ be an infinite connected locally finite planar graph, or a planar triangulation, with uniform polynomial volume growth of exponent $d>2$: there are $0<c<C<\infty$ such that \[c r^d\le |B_G(v,r)|\le C r^d\quad\text{for every vertex }v\text{ and integer }r\ge1.\] Let $(X_t)_{t\ge0}$ be discrete-time simple random walk starting at a root $o$; distance means unit-edge graph distance.

Statement recorded in the supplied source.

Prove that there exist $\alpha<1/2$ and $K<\infty$ such that $\mathbb E_o d_G(o,X_t)\le Kt^\alpha$ for all integers $t\ge1$. Is the bound valid with exponent $\alpha=1/d$? [S2]

Short English statement

Does random walk on a planar graph with uniformly more than quadratic polynomial growth move subdiffusively, with exponent at most the reciprocal of the growth dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000031 (AMR-099-0031). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Coarse Geometry and Randomness, Saint-Flour lecture notes, version dated 30 October 2013; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 7, Definition 7.8 and Conjecture 7.9, p. 55. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.