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The Scaling Limit of the Directed Polymer with Power-Law Tail Disorder
- Source :
- Communications in Mathematical Physics
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- In this paper, we study the so-called intermediate disorder regime for a directed polymer in a random environment with heavy-tail. Consider a simple symmetric random walk $(S_n)_{n\geq 0}$ on $\mathbb{Z}^d$, with $d\geq 1$, and modify its law using Gibbs weights in the product form $\prod_{n=1}^{N} (1+\beta\eta_{n,S_n})$, where $(\eta_{n,x})_{n\ge 0, x\in \mathbb{Z}^d}$ is a field of i.i.d. random variables whose distribution satisfies $\mathbb{P}(\eta>z) \sim z^{-\alpha}$ as $z\to\infty$, for some $\alpha\in(0,2)$. We prove that if $\alpha< \min(1+\frac{2}{d},2)$, when sending $N$ to infinity and rescaling the disorder intensity by taking $\beta=\beta_N \sim N^{-\gamma}$ with $\gamma =\frac{d}{2\alpha}(1+\frac{2}{d}-\alpha)$, the distribution of the trajectory under diffusive scaling converges in law towards a random limit, which is the continuum polymer with L\'evy $\alpha$-stable noise constructed in the companion paper arXiv:2007.06484.<br />Comment: 48 pages, comments are welcome
- Subjects :
- Physics
Probability (math.PR)
010102 general mathematics
FOS: Physical sciences
Statistical and Nonlinear Physics
Field (mathematics)
Mathematical Physics (math-ph)
Random walk
01 natural sciences
60K35, 82B44, 60G57
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Combinatorics
Distribution (mathematics)
Scaling limit
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
Product (mathematics)
0103 physical sciences
FOS: Mathematics
010307 mathematical physics
Continuum (set theory)
0101 mathematics
Random variable
Mathematics - Probability
Mathematical Physics
Intensity (heat transfer)
Subjects
Details
- ISSN :
- 14320916 and 00103616
- Volume :
- 386
- Database :
- OpenAIRE
- Journal :
- Communications in Mathematical Physics
- Accession number :
- edsair.doi.dedup.....136cb9c84f0fd1d71171e4c90d0859ba