<p>In this paper we show that a Brownian Gibbsian line ensemble whose top curve approximates a parabola must be given by the parabolic Airy line ensemble. More specifically, we prove that if <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi mathvariant="bold-script">L</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">L</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="script">L</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\boldsymbol{\mathcal{L}} = (\mathcal{L}_{1}, \mathcal{L}_{2}, \ldots )$</EquationSource> </InlineEquation> is a line ensemble satisfying the Brownian Gibbs property, such that for any <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>ε</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\varepsilon > 0$</EquationSource> </InlineEquation> there exists a constant <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\mathfrak{K} (\varepsilon ) > 0$</EquationSource> </InlineEquation> with <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/222_2025_1381_Equa_HTML.png" Format="PNG" Height="75" Rendition="HTML" Resolution="300" Type="Linedraw" Width="1036" /> </MediaObject> </Equation> then <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi mathvariant="bold-script">L</mi> </math></EquationSource> <EquationSource Format="TEX">$\boldsymbol{\mathcal{L}}$</EquationSource> </InlineEquation> is the parabolic Airy line ensemble, up to an independent affine shift. Specializing this result to the case when <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi mathvariant="bold-script">L</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mn>2</mn> <mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$\boldsymbol{\mathcal{L}} (t) + 2^{-1/2} t^{2}$</EquationSource> </InlineEquation> is translation-invariant confirms a prediction of Okounkov and Sheffield from 2006 and Corwin–Hammond from 2014.</p>

Strong characterization for the Airy line ensemble

期刊论文
发表日期：

2025年10月20日

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Amol Aggarwal,
Jiaoyang Huang

摘要

In this paper we show that a Brownian Gibbsian line ensemble whose top curve approximates a parabola must be given by the parabolic Airy line ensemble. More specifically, we prove that if $L = (L_{1}, L_{2}, \dots)$ $\boldsymbol{\mathcal{L}} = (\mathcal{L}_{1}, \mathcal{L}_{2}, \ldots )$ is a line ensemble satisfying the Brownian Gibbs property, such that for any $ε > 0$ $\varepsilon > 0$ there exists a constant $K (ε) > 0$ $\mathfrak{K} (\varepsilon ) > 0$ with then $L$ $\boldsymbol{\mathcal{L}}$ is the parabolic Airy line ensemble, up to an independent affine shift. Specializing this result to the case when $L (t) + 2^{- 1 / 2} t^{2}$ $\boldsymbol{\mathcal{L}} (t) + 2^{-1/2} t^{2}$ is translation-invariant confirms a prediction of Okounkov and Sheffield from 2006 and Corwin–Hammond from 2014.