Error term in the Cohen-Lenstra heuristic via random matrix approach

TL;DR

This paper investigates the error term in the Cohen-Lenstra heuristic using random matrix models and derives the asymptotic distribution of the corank of random matrices over finite fields, linking it to Markov chains.

Contribution

It extends the random matrix approach to analyze the error term in the Cohen-Lenstra heuristic and characterizes the corank distribution via Markov chain models.

Findings

Quantifies the error term in the Cohen-Lenstra heuristic.

Derives the asymptotic distribution of the corank of random matrices.

Connects the corank distribution to Markov chain behavior.

Abstract

The Cohen-Lenstra heuristic predicts the distribution of ideal class groups over number fields. Random matrix models provide a natural framework for explaining this heuristic, and recent results demonstrate the effectiveness of these tools. In this paper, we extend the analysis of the random matrix model to examine the error term in the Cohen-Lenstra heuristic. Additionally, we derive the asymptotic distribution of the corank of random matrices over finite fields, which can be modeled as a special class of Markov chains.