Counting Functions for Random Objects in a Category

TL;DR

This paper develops a probabilistic framework to analyze the asymptotic growth of counting functions for objects in categories, generalizing results in arithmetic statistics and number theory.

Contribution

It introduces a Law of Large Numbers for counting functions in categories with product structures, linking growth rates to moments of probability measures.

Findings

Asymptotic growth rates are determined by finite moments of measures.

Results apply broadly to categories with product structures.

Formalizes heuristic predictions in arithmetic statistics.

Abstract

In arithmetic statistics and analytic number theory, the asymptotic growth rate of counting functions giving the number of objects with order below $X$ is studied as $X\to \infty$. We define general counting functions which count epimorphisms out of an object on a category under some ordering. Given a probability measure $\mu$ on the isomorphism classes of the category with sufficient respect for a product structure, we prove a version of the Law of Large Numbers to give the asymptotic growth rate as $X$ tends towards $\infty$ of such functions with probability $1$ in terms of the finite moments of $\mu$ and the ordering. Such counting functions are motivated by work in arithmetic statistics, including number field counting as in Malle's conjecture and point counting as in the Batyrev-Manin conjecture. Recent work of Sawin--Wood gives sufficient conditions to construct such a measure $\mu$ from a well-behaved sequence of finite moments in very broad contexts, and we prove our results in this broad context with the added assumption that a product structure in the category is respected. These results allow us to formalize vast heuristic predictions about counting functions in general settings.