Linear support of harmonic measure in recurrent planar triangulations

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 bounded-degree recurrent planar triangulation rooted at $o$, circle-packed in the whole Euclidean plane. Graph balls use unit-edge distance. For a finite connected vertex domain $U\ni o$, write $\partial_VU$ for vertices outside $U$ adjacent to $U$, and define harmonic measure $\omega_U(v)=\mathbb P_o(X_{\tau_U}=v)$ on $\partial_VU$, where $\tau_U=\inf\{t:X_t\notin U\}$ for simple random walk. Each boundary vertex corresponds to one circle. A random rooted graph is unimodular if for every nonnegative measurable function $F$ of a graph with two distinguished vertices, \[\mathbb E\sum_vF(G,o,v)=\mathbb E\sum_vF(G,v,o).\]

Statement recorded in the supplied source.

Are there radii $r_j\to\infty$, finite domains $U_j\supset B_G(o,r_j)$, boundary sets $A_j\subset\partial_VU_j$, and $\varepsilon_j\downarrow0$ such that \[|A_j|\le r_j^{1+\varepsilon_j},\qquad\omega_{U_j}(A_j)\ge1-\varepsilon_j?\] If this fails for general recurrent triangulations, does it hold almost surely under unimodularity? In particular, does the corresponding harmonic-measure conclusion hold for the UIPT? [S2]

Short English statement

Can almost all exit probability be concentrated on a nearly linear number of boundary circles along arbitrarily large scales, at least for unimodular triangulations or the UIPT?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000081 (AMR-099-0081). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Random Planar Metrics; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 3.3, harmonic-measure question and following qualification, p. 8. https://arquivo.pt/noFrame/replay/20201231041544id_/http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf.