<p>We study microscopic observables of the Discrete Gaussian model (i.e., the Gaussian free field restricted to take integer values) at high temperature using the renormalisation group method. In particular, we show the central limit theorem for the two-point function of the Discrete Gaussian model by computing the asymptotic of the moment generating function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5396_Article_IEq1.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$\big \langle e^{{\mathcalligra {z}}(\sigma (0) - \sigma (y))} \big \rangle _{\beta , {\mathbb {Z}}^2}^{\operatorname {DG}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">〈</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi mathvariant="script">z</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>-</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> <msubsup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">〉</mo> </mrow> <mrow> <mi>β</mi> <mo>,</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </mrow> <mo>DG</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5396_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcalligra {z}}\in {\mathbb {C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> sufficiently small. The method we use has direct connection with the multi-scale polymer expansion used in Bauerschmidt et al. (Ann Probab 52(4):1253–1359, 2024, Ann Probab 52(4):1360–1398, 2024), where it was used to study the scaling limit of the Discrete Gaussian model. The method also applies to multi-point functions and lattice models of sine-Gordon type studied in Fröhlich and Spencer (Commun Math Phys 81(4): 527–602, 1981).</p>

Central Limit Theorem for Multi-Point Functions of the 2D Discrete Gaussian Model at High Temperature

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发表日期：

2025年9月1日

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Jiwoon Park

摘要

We study microscopic observables of the Discrete Gaussian model (i.e., the Gaussian free field restricted to take integer values) at high temperature using the renormalisation group method. In particular, we show the central limit theorem for the two-point function of the Discrete Gaussian model by computing the asymptotic of the moment generating function $\big \langle e^{{\mathcalligra {z}}(\sigma (0) - \sigma (y))} \big \rangle _{\beta , {\mathbb {Z}}^2}^{\operatorname {DG}}$ $〈 e^{z (σ (0) - σ (y))} 〉_{β, Z^{2}}^{DG}$ for ${\mathcalligra {z}}\in {\mathbb {C}}$ $z \in C$ sufficiently small. The method we use has direct connection with the multi-scale polymer expansion used in Bauerschmidt et al. (Ann Probab 52(4):1253–1359, 2024, Ann Probab 52(4):1360–1398, 2024), where it was used to study the scaling limit of the Discrete Gaussian model. The method also applies to multi-point functions and lattice models of sine-Gordon type studied in Fröhlich and Spencer (Commun Math Phys 81(4): 527–602, 1981).