Local limit theorems for random isometries of the plane

Abstract

We consider a random walk $(Y_N{)}_{N\ge 0}$ 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 $\exp (-c{N}^{1/3}/(\log N{)}^2)$. 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 $\exp (-c{N}^{1/2})$. The proofs relate the fine-scale distribution of $Y_N$ to a question about the values of integer polynomials on the unit circle.