Maximum of the Gaussian interface model in random external fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911817236742144 |
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| author | Sakagawa, Hironobu |
| author_facet | Sakagawa, Hironobu |
| contents | We consider the Gaussian interface model in the presence of random external fields, that is the finite volume (random) Gibbs measure on $\mathbb{R}^{Λ_N}$, $Λ_N=[-N, N]^d\cap \mathbb{Z}^d$ with Hamiltonian $H_N(ϕ)= \frac{1}{4d}\sum\limits_{x\sim y}(ϕ(x)-ϕ(y))^2-\sum\limits_{x\in Λ_N}η(x)ϕ(x)$ and $0$-boundary conditions. $\{η(x)\}_{x\in \mathbb{Z}^d}$ is a family of i.i.d. symmetric random variables. We study how the typical maximal height of a random interface is modified by the addition of quenched bulk disorder. We show that the asymptotic behavior of the maximum changes depending on the tail behavior of the random variable $η(x)$ when $d\geq 5$. In particular, we identify the leading order asymptotics of the maximum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_12583 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Maximum of the Gaussian interface model in random external fields Sakagawa, Hironobu Probability Mathematical Physics 82B24, 82B44, 60K35 We consider the Gaussian interface model in the presence of random external fields, that is the finite volume (random) Gibbs measure on $\mathbb{R}^{Λ_N}$, $Λ_N=[-N, N]^d\cap \mathbb{Z}^d$ with Hamiltonian $H_N(ϕ)= \frac{1}{4d}\sum\limits_{x\sim y}(ϕ(x)-ϕ(y))^2-\sum\limits_{x\in Λ_N}η(x)ϕ(x)$ and $0$-boundary conditions. $\{η(x)\}_{x\in \mathbb{Z}^d}$ is a family of i.i.d. symmetric random variables. We study how the typical maximal height of a random interface is modified by the addition of quenched bulk disorder. We show that the asymptotic behavior of the maximum changes depending on the tail behavior of the random variable $η(x)$ when $d\geq 5$. In particular, we identify the leading order asymptotics of the maximum. |
| title | Maximum of the Gaussian interface model in random external fields |
| topic | Probability Mathematical Physics 82B24, 82B44, 60K35 |
| url | https://arxiv.org/abs/2307.12583 |