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<p>We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by this, we study asymptotics for the discrete curvature of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-mated CRT maps. More precisely, we prove that the discrete curvature integrated against a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C_c^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>C</mi> <mi>c</mi> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation> test function is of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\epsilon ^{o(1)},\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ϵ</mi> <mrow> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> which is consistent with our scaling limit conjecture. On the other hand, we prove the total discrete curvature on a fixed space-filling SLE segment scaled by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\epsilon ^{\frac{1}{4}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ϵ</mi> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </msup> </math></EquationSource> </InlineEquation> converges in distribution to an explicit random variable.</p>

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Gaussian curvature on random planar maps and Liouville quantum gravity

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月9日

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  • Andres A. Contreras Hip,
  • Ewain Gwynne

摘要

We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by this, we study asymptotics for the discrete curvature of \(\epsilon \) ϵ -mated CRT maps. More precisely, we prove that the discrete curvature integrated against a \(C_c^2\) C c 2 test function is of order \(\epsilon ^{o(1)},\) ϵ o ( 1 ) , which is consistent with our scaling limit conjecture. On the other hand, we prove the total discrete curvature on a fixed space-filling SLE segment scaled by \(\epsilon ^{\frac{1}{4}}\) ϵ 1 4 converges in distribution to an explicit random variable.