Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5366_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {M}_{g,n(g)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> be the moduli space of hyperbolic surfaces of genus g with n(g) punctures endowed with the Weil–Petersson metric. In this paper we study the asymptotic behavior of the Cheeger constants and spectral gaps of random hyperbolic surfaces in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5366_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {M}_{g,n(g)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation>, when n(g) grows slower than g as <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5366_Article_IEq3.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>.

Arbitrarily Small Spectral Gaps for Random Hyperbolic Surfaces with Many Cusps

期刊论文
发表日期：

2025年7月2日

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Yang Shen,
Yunhui Wu

摘要

Let $\mathcal {M}_{g,n(g)}$ $M_{g, n (g)}$ be the moduli space of hyperbolic surfaces of genus g with n(g) punctures endowed with the Weil–Petersson metric. In this paper we study the asymptotic behavior of the Cheeger constants and spectral gaps of random hyperbolic surfaces in $\mathcal {M}_{g,n(g)}$ $M_{g, n (g)}$ , when n(g) grows slower than g as $g\rightarrow \infty $ $g \to \infty$ .