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<p>For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we give a rigorous probabilistic construction on the cylinder <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {R}\times (\mathbb {R}/ (2\pi R \mathbb {Z}))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>R</mi> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the (massless) Sinh-Gordon model. In particular we define the <i>n</i>-point correlation functions of the model and show that these exhibit a scaling relation with respect to <i>R</i>. The construction, which relies on the massless Gaussian Free Field, is based on the spectral analysis of a quantum operator associated to the model. Using the theory of Gaussian multiplicative chaos, we prove that this operator has discrete spectrum and a strictly positive ground state.</p>

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2d Sinh-Gordon Model on the Infinite Cylinder

  • 期刊论文

  • 发表日期：

    2026年4月4日

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  • Colin Guillarmou,
  • Trishen S. Gunaratnam,
  • Vincent Vargas

摘要

For \(R>0\) R > 0 , we give a rigorous probabilistic construction on the cylinder \(\mathbb {R}\times (\mathbb {R}/ (2\pi R \mathbb {Z}))\) R × ( R / ( 2 π R Z ) ) of the (massless) Sinh-Gordon model. In particular we define the n-point correlation functions of the model and show that these exhibit a scaling relation with respect to R. The construction, which relies on the massless Gaussian Free Field, is based on the spectral analysis of a quantum operator associated to the model. Using the theory of Gaussian multiplicative chaos, we prove that this operator has discrete spectrum and a strictly positive ground state.