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First page of Spherical statistics and phase transitions in high-dimensional lattices

Spherical statistics and phase transitions in high-dimensional lattices

Thijs Laarhoven

math.NT Sep 12, 2026 · v1 math.MG math.PR
formalization number-theory probability
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TL;DR
A Lean formalization verifies the paper's main results on high-dimensional lattice shell statistics, assuming the classical formulas and probability model stated in code.
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.
Problem

High-dimensional random lattices have points linked by exact arithmetic rules, yet some statistics resemble those of random point clouds. It is unclear when statistics of a complete thin spherical shell of one typical lattice agree with continuous spherical-geometry predictions, and at what shell radius each agreement holds.

Approach

The Haar-random unimodular lattice model on SL_n(R)/SL_n(Z) is used, studying a thin shell of radius R=cR*. A growing-order Rogers moment method with marked shell-incidence counts converts integrated moment bounds into simultaneous pointwise regularity for all lattice points and differences. Statistics analyzed include shell-point counts, directional balance, representation functions, additive energy, and total-variation distance to continuum comparators. A Lean formalization verifies the main results assuming classical mean-value formulas and the stated probability model.

Results

Sharp thresholds and several distinct phase transitions are identified as the shell radius grows, holding for a single sampled lattice with probability tending to one as dimension grows. A same-shell autocorrelation transition occurs at c=2/sqrt(3), and an additive-energy exponent transition at c_E=3sqrt(3)/4. Results include complete-shell estimates, near-threshold bounds, and extensions to randomly shifted lattices.