A Class of Gaussian Fields on $\mathbb{Z}_q^d$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917287955529728 |
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| author | Griffiths, Robert Mano, Shuhei |
| author_facet | Griffiths, Robert Mano, Shuhei |
| contents | Gaussian fields $(g_x)$ on $\mathbb{Z}_q^d$ are constructed from a class of reversible long range random walks $(X_t)_{t\in \mathbb{N}}$ on $\mathbb{Z}_q^d$ in arXiv:2510.22554. The construction is from taking the covariance function of $(g_x)$ as $(1-α)G(x,y;α)$, where $G(x,y;α)$ is the Green function of a random walk with killing in each transition at rate $1-α$. A decomposition of the Gaussian field into a sum of independent Gaussian random variables is made. By letting $q\to \infty$ the Gaussian field becomes defined from an infinite-dimensional random walk on a torus. The random walk model is also extended to $d=\infty$ by considering a de Finetti random walk where entries in the increments of the random walk are exchangeable. A limit Gaussian field on $\mathbb{R}^d$ arises from a central limit theorem approach. The transform of this Gaussian field, which is again a Gaussian field, is calculated. It has a simpler covariance matrix than the original field. The Hamiltonian connected to the Gaussian field is calculated. A limit theorem for the partition function arising from the Hamiltonian is found. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19469 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Class of Gaussian Fields on $\mathbb{Z}_q^d$ Griffiths, Robert Mano, Shuhei Probability 60G07 Gaussian fields $(g_x)$ on $\mathbb{Z}_q^d$ are constructed from a class of reversible long range random walks $(X_t)_{t\in \mathbb{N}}$ on $\mathbb{Z}_q^d$ in arXiv:2510.22554. The construction is from taking the covariance function of $(g_x)$ as $(1-α)G(x,y;α)$, where $G(x,y;α)$ is the Green function of a random walk with killing in each transition at rate $1-α$. A decomposition of the Gaussian field into a sum of independent Gaussian random variables is made. By letting $q\to \infty$ the Gaussian field becomes defined from an infinite-dimensional random walk on a torus. The random walk model is also extended to $d=\infty$ by considering a de Finetti random walk where entries in the increments of the random walk are exchangeable. A limit Gaussian field on $\mathbb{R}^d$ arises from a central limit theorem approach. The transform of this Gaussian field, which is again a Gaussian field, is calculated. It has a simpler covariance matrix than the original field. The Hamiltonian connected to the Gaussian field is calculated. A limit theorem for the partition function arising from the Hamiltonian is found. |
| title | A Class of Gaussian Fields on $\mathbb{Z}_q^d$ |
| topic | Probability 60G07 |
| url | https://arxiv.org/abs/2602.19469 |