A class of d-dimensional directed polymers in a Gaussian environment
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910043950022656 |
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| author | Chen, Le Ouyang, Cheng Tindel, Samy Xia, Panqiu |
| author_facet | Chen, Le Ouyang, Cheng Tindel, Samy Xia, Panqiu |
| contents | We introduce and analyze a broad class of continuous directed polymers in $\mathbb{R}^d$ driven by Gaussian environments that are white in time and spatially correlated, under Dalang's condition. Using an Itô-renormalized stochastic-heat-equation representation, we establish structural properties of the partition function, including positivity, stationarity, scaling, homogeneity, and a Chapman--Kolmogorov relation. On finite time intervals, we prove Brownian-type pathwise behavior, namely Hölder continuity and identification of the quadratic variation. We then obtain a sharp measure-theoretic dichotomy: the quenched polymer measure is singular with respect to Wiener measure if and only if $\widehat f(\mathbb{R}^d)=\infty$ (equivalently, the noise is non-trace-class), and it is equivalent otherwise. Finally, in dimension $d\ge 3$, we prove diffusive behavior at large times in the high-temperature regime. This extends the Alberts--Khanin--Quastel framework from the $1+1$ white-noise setting to higher-dimensional Gaussian environments with general spatial covariance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06574 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A class of d-dimensional directed polymers in a Gaussian environment Chen, Le Ouyang, Cheng Tindel, Samy Xia, Panqiu Probability Mathematical Physics 60H15 (primary), 60K37, 82D60, 60G15 (secondary) We introduce and analyze a broad class of continuous directed polymers in $\mathbb{R}^d$ driven by Gaussian environments that are white in time and spatially correlated, under Dalang's condition. Using an Itô-renormalized stochastic-heat-equation representation, we establish structural properties of the partition function, including positivity, stationarity, scaling, homogeneity, and a Chapman--Kolmogorov relation. On finite time intervals, we prove Brownian-type pathwise behavior, namely Hölder continuity and identification of the quadratic variation. We then obtain a sharp measure-theoretic dichotomy: the quenched polymer measure is singular with respect to Wiener measure if and only if $\widehat f(\mathbb{R}^d)=\infty$ (equivalently, the noise is non-trace-class), and it is equivalent otherwise. Finally, in dimension $d\ge 3$, we prove diffusive behavior at large times in the high-temperature regime. This extends the Alberts--Khanin--Quastel framework from the $1+1$ white-noise setting to higher-dimensional Gaussian environments with general spatial covariance. |
| title | A class of d-dimensional directed polymers in a Gaussian environment |
| topic | Probability Mathematical Physics 60H15 (primary), 60K37, 82D60, 60G15 (secondary) |
| url | https://arxiv.org/abs/2603.06574 |