Exponential intersection tails for loop-erased random walk

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.

For an integer $d\ge1$, let $\mathbb Z^d$ have nearest-neighbor edges and root $0$. Chronological loop erasure of a finite path successively deletes each loop as soon as it is completed. Let $\mu_d$ be the infinite loop-erased random-walk law from $0$, defined as the local weak limit of loop erasures of simple random walks stopped on first leaving successively larger boxes. For two independent paths $\gamma_1,\gamma_2$ with law $\mu_d$, let $|\gamma_1\cap\gamma_2|$ count distinct common edges.

Statement recorded in the supplied source.

For which dimensions $d$ does $\mu_d$ have exponential intersection tails: is there $0<\theta<1$ such that $\mathbb P_{\mu_d\otimes\mu_d}(|\gamma_1\cap\gamma_2|>n)\le\theta^n$ for every integer $n\ge1$? [S2]

Short English statement

Determine whether infinite loop-erased random walks have exponentially small probabilities of sharing many edges.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000020 (AMR-099-0020). 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, Definition 4.16 and Open Problem 4.22, pp. 35--36. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.