Lower tail large deviations of the stochastic six vertex model
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914001234952192 |
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| author | Das, Sayan Liao, Yuchen Mucciconi, Matteo |
| author_facet | Das, Sayan Liao, Yuchen Mucciconi, Matteo |
| contents | In this paper, we study lower tail probabilities of the height function $\mathfrak{h}(M,N)$ of the stochastic six-vertex model. We introduce a novel combinatorial approach to demonstrate that the tail probabilities $\mathbb{P}(\mathfrak{h}(M,N) \ge r)$ are log-concave in a certain weak sense. We prove further that for each $α>0$ the lower tail of $-\mathfrak{h}(\lfloor αN \rfloor, N)$ satisfies a Large Deviation Principle (LDP) with speed $N^2$ and a rate function $Φ_α^{(-)}$, which is given by the infimal deconvolution between a certain energy integral and a parabola.
Our analysis begins with a distributional identity from BO17 [arXiv:1608.01564], which relates the lower tail of the height function, after a random shift, with a multiplicative functional of the Schur measure. Tools from potential theory allow us to extract the LDP for the shifted height function. We then use our weak log-concavity result, along with a deconvolution scheme from our earlier paper [arXiv:2307.01179], to convert the LDP for the shifted height function to the LDP for the stochastic six-vertex model height function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08530 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower tail large deviations of the stochastic six vertex model Das, Sayan Liao, Yuchen Mucciconi, Matteo Probability In this paper, we study lower tail probabilities of the height function $\mathfrak{h}(M,N)$ of the stochastic six-vertex model. We introduce a novel combinatorial approach to demonstrate that the tail probabilities $\mathbb{P}(\mathfrak{h}(M,N) \ge r)$ are log-concave in a certain weak sense. We prove further that for each $α>0$ the lower tail of $-\mathfrak{h}(\lfloor αN \rfloor, N)$ satisfies a Large Deviation Principle (LDP) with speed $N^2$ and a rate function $Φ_α^{(-)}$, which is given by the infimal deconvolution between a certain energy integral and a parabola. Our analysis begins with a distributional identity from BO17 [arXiv:1608.01564], which relates the lower tail of the height function, after a random shift, with a multiplicative functional of the Schur measure. Tools from potential theory allow us to extract the LDP for the shifted height function. We then use our weak log-concavity result, along with a deconvolution scheme from our earlier paper [arXiv:2307.01179], to convert the LDP for the shifted height function to the LDP for the stochastic six-vertex model height function. |
| title | Lower tail large deviations of the stochastic six vertex model |
| topic | Probability |
| url | https://arxiv.org/abs/2407.08530 |