Local limit theorem for directed polymers beyond the $L^2$-phase
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908429574995968 |
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| author | Junk, Stefan |
| author_facet | Junk, Stefan |
| contents | We consider the directed polymer model in the weak disorder phase under the assumption that the partition function is $L^p$-bounded for some $p>1+\frac{2}d$. We prove that the point-to-point partition function can be approximated by two point-to-plane partition functions at the startpoint and endpoint, and in particular that it is $L^p$-bounded as well. Some consequences of this result are also discussed, the most important of which is a local limit theorem for the polymer measure. We furthermore show that the required $L^p$-boundedness holds for some range of $β$ beyond the $L^2$-critical point, and in the whole interior of the weak disorder phase for environments with finite support. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_05097 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local limit theorem for directed polymers beyond the $L^2$-phase Junk, Stefan Probability 60K37 We consider the directed polymer model in the weak disorder phase under the assumption that the partition function is $L^p$-bounded for some $p>1+\frac{2}d$. We prove that the point-to-point partition function can be approximated by two point-to-plane partition functions at the startpoint and endpoint, and in particular that it is $L^p$-bounded as well. Some consequences of this result are also discussed, the most important of which is a local limit theorem for the polymer measure. We furthermore show that the required $L^p$-boundedness holds for some range of $β$ beyond the $L^2$-critical point, and in the whole interior of the weak disorder phase for environments with finite support. |
| title | Local limit theorem for directed polymers beyond the $L^2$-phase |
| topic | Probability 60K37 |
| url | https://arxiv.org/abs/2307.05097 |