Strong Characterization for the Airy Line Ensemble
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917000023900160 |
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| author | Aggarwal, Amol Huang, Jiaoyang |
| author_facet | Aggarwal, Amol Huang, Jiaoyang |
| contents | 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 $\boldsymbol{\mathcal{L}} = (\mathcal{L}_1, \mathcal{L}_2, \ldots )$ is a line ensemble satisfying the Brownian Gibbs property, such that for any $\varepsilon > 0$ there exists a constant $\mathfrak{K} (\varepsilon) > 0$ with $$\mathbb{P} \Big[ \big| \mathcal{L}_1 (t) + 2^{-1/2} t^2 \big| \le \varepsilon t^2 + \mathfrak{K} (\varepsilon) \Big] \ge 1 - \varepsilon, \qquad \text{for all $t \in \mathbb{R}$},$$ then $\boldsymbol{\mathcal{L}}$ is the parabolic Airy line ensemble, up to an independent affine shift. Specializing this result to the case when $\boldsymbol{\mathcal{L}} (t) + 2^{-1/2} t^2$ is translation-invariant confirms a prediction of Okounkov and Sheffield from 2006 and Corwin-Hammond from 2014. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11908 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strong Characterization for the Airy Line Ensemble Aggarwal, Amol Huang, Jiaoyang Probability Statistical Mechanics Mathematical Physics 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 $\boldsymbol{\mathcal{L}} = (\mathcal{L}_1, \mathcal{L}_2, \ldots )$ is a line ensemble satisfying the Brownian Gibbs property, such that for any $\varepsilon > 0$ there exists a constant $\mathfrak{K} (\varepsilon) > 0$ with $$\mathbb{P} \Big[ \big| \mathcal{L}_1 (t) + 2^{-1/2} t^2 \big| \le \varepsilon t^2 + \mathfrak{K} (\varepsilon) \Big] \ge 1 - \varepsilon, \qquad \text{for all $t \in \mathbb{R}$},$$ then $\boldsymbol{\mathcal{L}}$ is the parabolic Airy line ensemble, up to an independent affine shift. Specializing this result to the case when $\boldsymbol{\mathcal{L}} (t) + 2^{-1/2} t^2$ is translation-invariant confirms a prediction of Okounkov and Sheffield from 2006 and Corwin-Hammond from 2014. |
| title | Strong Characterization for the Airy Line Ensemble |
| topic | Probability Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2308.11908 |