Local limit theorems for random isometries of the plane

TL;DR

This paper proves local limit theorems for random walks generated by random isometries of the plane, establishing precise distributional results at very small scales under various algebraic and Diophantine conditions.

Contribution

It introduces new local limit theorems for random isometries with explicit scale bounds, extending understanding of fine-scale distributions in geometric group actions.

Findings

LCLT holds down to super-polynomial scales for certain rotations

LCLT valid at exponential scales under algebraic conditions

Finer scale LCLT achieved for asymmetric measures

Abstract

We consider a random walk $(Y_N)_{N\geq 0}$ on $\mathbb{R}^2$ generated by successively applying independent random isometries, drawn from a fixed measure $\mu$, to the point $0$. When the support of $\mu$ is finite and includes an irrational rotation satisfying a Diophantine condition, we establish a local central limit theorem (LCLT) for $Y_N$ down to super-polynomially small scales. When $\mu$ includes rotations satisfying a further algebraic condition, we prove that a LCLT holds down to the scale $\exp(-cN^{1/3}/(\log N)^2)$. Due to group-theoretic obstructions, this is sharp for symmetric $\mu$, up to the $\log$ factor. Lastly for a special class of asymmetric $\mu$, we obtain an LCLT down to the much finer scale $\exp(-cN^{1/2})$. The proofs relate the fine-scale distribution of $Y_N$ to a question about the values of integer polynomials on the unit circle.