Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime

Stefan Junk; Hubert Lacoin

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arXiv · 2611.02998

Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime

Stefan Junk·Hubert Lacoin

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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 $β_c$. 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 $β\in [0,β_c]$.

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Stefan Junk, Hubert Lacoin. 2026-11-02. Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime. https://arxiv.org/abs/2611.02998

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