Local limit theorems for random isometries of the plane

TL;DR

This work analyzes the fine-scale distribution of random walks on planar isometries $Y_N$ generated from a finite-support measure on ${\mathrm{Isom}}(\mathbb{R}^2)$. A core innovation is reducing high-frequency decay of the characteristic function to a dense polynomial-valued property on the unit circle, quantified by the dense polynomial-value (DPV) framework, which in turn is tied to the translation-translation structure in the group. The authors prove sharp local limit theorems at scales $\exp(-c(\log N)^2)$ for irrational rotations under a Diophantine condition, at scales $\exp(-c N^{1/3}/(\log N)^2)$ in the symmetric rational-cosine/advanced algebraic setting, and at scales $\exp(-c\sqrt{N}/\log N)$ for certain asymmetric cases, highlighting intrinsic group-theoretic obstructions to exponentially fine scales. The results illuminate how arithmetic structure of rotations interacts with translations to govern the maximal possible local limit scales in 2D, with broader implications for isometry-based random processes and polynomial-value methods on the unit circle.

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 $μ$, to the point $0$. When the support of $μ$ 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 $μ$ 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 $μ$, up to the $\log$ factor. Lastly for a special class of asymmetric $μ$, 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.

Figures (2)

• Figure 1: Plot of $\sum_{j=0}^{17} \eta_j z^j$ with $\eta_j\in\{\pm1\}$ uniformly and independently, with $z=e^{2\pi i/7}$ on the left and $z=e^{\sqrt{2} \pi i}$ on the right. This corresponds to a random isometry with $\mu$ uniform on $\{\tau_{\pm 1}\circ\rho_\theta\}$. The distribution of $Y_N$ resembles a Gaussian down to much finer scales when irrational rotations are included.
• Figure 2: The point distributions for $Y_N\cap [-1,1]^2$ when $\mu$ is uniformly supported on the symmetric set $\{\rho_\theta,\rho_{-\theta},\tau_1,\tau_{-1}\}$ (left) and the asymmetric set $\{\rho_\theta,\tau_1,\tau_{-1}\}$ (right). Darker dots indicate points with higher multiplicity. The image above was generated with $N=13$ and $\theta$ such that $e^{i\theta} = \frac{1}{5}(3+4i)$. For the symmetric case, $\mathbb{P}(Y^{\rm sym}_N=0)\approx 0.02$, whereas in the asymmetric case we have $\mathbb{P}(Y^{\rm asym}_N=0)\approx 0.0002$. The asymmetric $Y^{\rm asym}$ has a noticeably more uniform distribution.

Theorems & Definitions (41)

• theorem 1.1
• theorem 1.2
• theorem 1.3
• definition 1: Dense polynomial values
• proposition 1
• proposition 2
• proposition 3
• lemma 1
• proof
• lemma 2