The factorial growth of topological recursion

TL;DR

This paper establishes that the correlation functions in topological recursion grow factorially with genus and points, providing an upper bound that supports large genus curve counting and resurgence analysis.

Contribution

It proves that correlation functions in topological recursion grow at most factorially with genus, offering a key estimate for large genus asymptotics.

Findings

Correlation functions grow at most like (2g - 2 + n)!

Provides an upper bound for large genus curve counting

Supports resurgence analysis in topological recursion

Abstract

We show that the $n$-point, genus-$g$ correlation functions of topological recursion on any regular spectral curve with simple ramifications grow at most like $(2g - 2 + n)!$ as $g \rightarrow \infty$, which is the expected growth rate. This provides, in particular, an upper bound for many curve counting problems in large genus and serves as a preliminary step for a resurgence analysis.