Local limit theorems for random isometries of the plane

2026/01/22 by Reuben Drogin, Felipe Hernández · 1 voice
Mathematics · #math.PR #math.GR #math.NT

paper · pdf

Abstract

We consider a random walk (Y_N)_{N≥ 0} on ℝ² 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.

Citations

• Cut-off phenomenon for the ax+b Markov chain over a finite field
• Recent progress on Bernoulli convolutions
• On Random Walks in large compact Lie groups
• Local universality of zeroes of random polynomials
• Optimal Inverse Littlewood-Offord theorems
• A finitary version of Gromov's polynomial growth theorem
• The Solovay-Kitaev algorithm
• Flows on homogeneous spaces and Diophantine approximation on manifolds

Discussions

• Local Limit Theorems for Random Isometries of the Plane [hn, 2 points, 0 comments]