Universality results for random matrices over finite local rings
Nikita Lvov
TL;DR
The paper proves quantitative universality for the cokernel of random matrices with i.i.d. entries over finite local rings, showing convergence in total variation to the Haar limit (and analogous results for determinants and spans) under a non-degeneracy condition modulo the maximal ideal or a subring. It introduces a column-swapping estimate via a Lindeberg replacement approach, yielding exponential convergence rates and enabling universality for finer invariants beyond the cokernel. A detailed random-matrix inequality and a decomposition framework for measures on finite modules are developed, employing an orthogonal Fourier-like decomposition indexed by characters and dualizing modules. The combination of replacements and module-decomposition techniques extends universality results to a broad class of finite local rings, providing a robust toolkit for analyzing random matrices in p-adic and modular settings with potential Markovian interpretations. Overall, the work delivers a rigorous, quantitative universality theory for cokernels, determinants, and linear spans over finite local rings, broadening the scope of p-adic random matrix models.
Abstract
Let $R$ be a finite local ring. We prove a quantitative universality statement for the cokernel of random matrices with i.i.d. entries valued in $R$. Rather than use the moment method, we use the Lindeberg replacement technique. This approach also yields a universality result for several invariants that are finer than the cokernel, such as the span and the determinant.