The factorial growth of 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 $$(2g - 2 + n)!$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>!</mml:mo> </mml:mrow> </mml:math> as $$g \rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , 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.