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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.04429 |
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| _version_ | 1866912876877316096 |
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| author | Gomez, Gaspard |
| author_facet | Gomez, Gaspard |
| contents | We study discrete statistical mechanics systems perturbed by a random environment without a finite second moment. Specifically, we consider a random environment whose tail distribution satisfies $P[ω> x] \sim x^{-γ}$ as $x \to +\infty$ for some $γ\in (1,2)$. Inspired by the seminal work of Caravenna, Sun and Zygouras \cite{csz_2016}, we adopt a general framework that encompasses as key examples both the disordered pinning model and the long-range directed polymer model. We provide some subcriticality condition under which we prove that the discrete disordered system possesses a non-trivial scaling limit. We also interpret the subcriticality condition in terms of a generalized Harris criterion without second moment, which gives a prediction for disorder relevance depending on the parameters of the system. Our analysis relies on the study of multilinear polynomials of independent heavy-tailed random variables known as polynomial chaos and their continuous analogue, given by multiple integrals with respect to a $γ$-stable Lévy white noise. We develop precise and flexible moments estimates adapted to the heavy-tailed setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04429 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Disordered systems and (subcritical) polynomial chaos with heavy-tail disorder Gomez, Gaspard Probability We study discrete statistical mechanics systems perturbed by a random environment without a finite second moment. Specifically, we consider a random environment whose tail distribution satisfies $P[ω> x] \sim x^{-γ}$ as $x \to +\infty$ for some $γ\in (1,2)$. Inspired by the seminal work of Caravenna, Sun and Zygouras \cite{csz_2016}, we adopt a general framework that encompasses as key examples both the disordered pinning model and the long-range directed polymer model. We provide some subcriticality condition under which we prove that the discrete disordered system possesses a non-trivial scaling limit. We also interpret the subcriticality condition in terms of a generalized Harris criterion without second moment, which gives a prediction for disorder relevance depending on the parameters of the system. Our analysis relies on the study of multilinear polynomials of independent heavy-tailed random variables known as polynomial chaos and their continuous analogue, given by multiple integrals with respect to a $γ$-stable Lévy white noise. We develop precise and flexible moments estimates adapted to the heavy-tailed setting. |
| title | Disordered systems and (subcritical) polynomial chaos with heavy-tail disorder |
| topic | Probability |
| url | https://arxiv.org/abs/2602.04429 |