Short- and long-time path tightness of the continuum directed random polymer
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866915077353897984 |
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| author | Das, Sayan Zhu, Weitao |
| author_facet | Das, Sayan Zhu, Weitao |
| contents | We consider the point-to-point continuum directed random polymer ($\mathsf{CDRP}$) model that arises as a scaling limit from $1+1$ dimensional directed polymers in the intermediate disorder regime. We show that the annealed law of a point-to-point $\mathsf{CDRP}$ of length $t$ converges to the Brownian bridge under diffusive scaling when $t \downarrow 0$. In case that $t$ is large, we show that the transversal fluctuations of point-to-point $\mathsf{CDRP}$ are governed by the $2/3$ exponent. More precisely, as $t$ tends to infinity, we prove tightness of the annealed path measures of point-to-point $\mathsf{CDRP}$ of length $t$ upon scaling the length by $t$ and fluctuations of paths by $t^{2/3}$. The $2/3$ exponent is tight such that the one-point distribution of the rescaled paths converges to the geodesics of the directed landscape. This point-wise convergence can be enhanced to process-level modulo a conjecture. Our short and long-time tightness results also extend to point-to-line $\mathsf{CDRP}$. In the course of proving our main results, we establish quantitative versions of quenched modulus of continuity estimates for long-time $\mathsf{CDRP}$ which are of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_05670 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Short- and long-time path tightness of the continuum directed random polymer Das, Sayan Zhu, Weitao Probability Mathematical Physics Primary: 60K37, 82B21, Secondary: 82D60 We consider the point-to-point continuum directed random polymer ($\mathsf{CDRP}$) model that arises as a scaling limit from $1+1$ dimensional directed polymers in the intermediate disorder regime. We show that the annealed law of a point-to-point $\mathsf{CDRP}$ of length $t$ converges to the Brownian bridge under diffusive scaling when $t \downarrow 0$. In case that $t$ is large, we show that the transversal fluctuations of point-to-point $\mathsf{CDRP}$ are governed by the $2/3$ exponent. More precisely, as $t$ tends to infinity, we prove tightness of the annealed path measures of point-to-point $\mathsf{CDRP}$ of length $t$ upon scaling the length by $t$ and fluctuations of paths by $t^{2/3}$. The $2/3$ exponent is tight such that the one-point distribution of the rescaled paths converges to the geodesics of the directed landscape. This point-wise convergence can be enhanced to process-level modulo a conjecture. Our short and long-time tightness results also extend to point-to-line $\mathsf{CDRP}$. In the course of proving our main results, we establish quantitative versions of quenched modulus of continuity estimates for long-time $\mathsf{CDRP}$ which are of independent interest. |
| title | Short- and long-time path tightness of the continuum directed random polymer |
| topic | Probability Mathematical Physics Primary: 60K37, 82B21, Secondary: 82D60 |
| url | https://arxiv.org/abs/2205.05670 |