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<p>We show that the <i>n</i>-point, genus-<i>g</i> correlation functions of topological recursion on any regular spectral curve with simple ramifications grow at most like <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1950_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">\((2g - 2 + n)!\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>g</mi> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1950_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(g \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, 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.</p>

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The factorial growth of topological recursion

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月28日

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  • Gaëtan Borot,
  • Bertrand Eynard,
  • Alessandro Giacchetto

摘要

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 \((2g - 2 + n)!\) ( 2 g - 2 + n ) ! as \(g \rightarrow \infty \) g → ∞ , 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.