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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.22760 |
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| _version_ | 1866917520590503936 |
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| author | Novikov, Svyatoslav |
| author_facet | Novikov, Svyatoslav |
| contents | We study the high excursion probability of a centered Gaussian field on a square.
Writing \(σ\) and \(r\) for its standard deviation and correlation function,
we assume that \(σ\) has a unique maximum at the corner
\(\boldsymbol{0}=(0,0)\) and
\[
1-σ(\boldsymbol{t}) \sim t_1^β+t_2^β+t_1^a t_2^a ,
\qquad \boldsymbol{t}=(t_1,t_2)\to\boldsymbol{0}
\]
in \(\mathbb R_+^2\). The local correlation is assumed to satisfy
\[
1-r(\boldsymbol{t},\boldsymbol{s})\sim |t_1-s_1|^α+|t_2-s_2|^α,
\qquad 0<α<β.
\]
This product form of the standard-deviation loss is not covered by the usual
locally additive assumptions. In the range \(a<β/2\), the classical essential rectangle
at the variance-loss scale no longer captures the leading contribution; the
relevant localization becomes side-attached and, in one regime, effectively
one-dimensional. We determine the corresponding high-level asymptotics,
including the logarithmic and side-dominated regimes which do not arise in the
locally additive case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22760 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extremes of Gaussian fields with a product term in the variance Novikov, Svyatoslav Probability We study the high excursion probability of a centered Gaussian field on a square. Writing \(σ\) and \(r\) for its standard deviation and correlation function, we assume that \(σ\) has a unique maximum at the corner \(\boldsymbol{0}=(0,0)\) and \[ 1-σ(\boldsymbol{t}) \sim t_1^β+t_2^β+t_1^a t_2^a , \qquad \boldsymbol{t}=(t_1,t_2)\to\boldsymbol{0} \] in \(\mathbb R_+^2\). The local correlation is assumed to satisfy \[ 1-r(\boldsymbol{t},\boldsymbol{s})\sim |t_1-s_1|^α+|t_2-s_2|^α, \qquad 0<α<β. \] This product form of the standard-deviation loss is not covered by the usual locally additive assumptions. In the range \(a<β/2\), the classical essential rectangle at the variance-loss scale no longer captures the leading contribution; the relevant localization becomes side-attached and, in one regime, effectively one-dimensional. We determine the corresponding high-level asymptotics, including the logarithmic and side-dominated regimes which do not arise in the locally additive case. |
| title | Extremes of Gaussian fields with a product term in the variance |
| topic | Probability |
| url | https://arxiv.org/abs/2605.22760 |