Spherical statistics and phase transitions in high-dimensional lattices

Abstract

We study the spherical statistics of all the points in a thin shell of a high-dimensional random lattice. The exact relations between lattice points make it unclear when predictions based only on spherical geometry should hold. We answer this question for several statistics, including the number of shell points, the balance of their directions, and the occurrence, repetition, and distribution of differences between them. We identify sharp thresholds as the shell radius grows and show that these statistics undergo several distinct phase transitions. A shell can already agree with one geometric prediction while still differing strongly from another. Our results hold for a single sampled lattice, with probability tending to one as the dimension grows. They include estimates that hold across a complete shell, bounds close to the transition thresholds, and extensions to randomly shifted lattices. A Lean formalization verifies the main results, assuming the classical formulas and probability model stated in the code.