Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime
Abstract
For directed polymers in random environment, we provide a short and mostly self-contained proof of the local limit theorem in the full weak disorder phase, including the critical inverse temperature βc. That is, the point-to-point partition function is well approximated by the product of two independent point-to-line partition functions (one starting from each endpoint of the system) multiplied by the simple random walk transition kernel. The approximation is valid in the Lp sense for an optimal range of p. The result is new at β=βc and for p>2, and in all cases the proof is significantly simpler, technically and conceptually, than what can be found in the literature. As a consequence of the result we establish an invariance principle for the polymer measure with fixed end-point, valid for all β∈ [0,βc].
Related
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BibTeX
@misc{junk2026local,
title = {{Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime}},
author = {Stefan Junk and Hubert Lacoin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02998},
eprint = {2611.02998},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02998},
}
RIS
TY - UNPB
TI - Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime
AU - Stefan Junk
AU - Hubert Lacoin
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02998
UR - https://arxiv.org/abs/2611.02998
AB - For directed polymers in random environment, we provide a short and mostly self-contained proof of the local limit theorem in the full weak disorder phase, including the critical inverse temperature β_c. That is, the point-to-point partition function is well approximated by the product of two independent point-to-line partition functions (one starting from each endpoint of the system) multiplied by the simple random walk transition kernel. The approximation is valid in the L^p sense for an optimal range of p. The result is new at β=β_c and for p>2, and in all cases the proof is significantly simpler, technically and conceptually, than what can be found in the literature. As a consequence of the result we establish an invariance principle for the polymer measure with fixed end-point, valid for all β∈ [0,β_c].
ER -