Free energy and phase transition for 2D directed polymers with critical spatial correlations
Abstract
We study the two-dimensional directed polymer model in a Gaussian environment which is independent in time and spatially correlated, with covariances h(x) either summable or with a critical decay, satisfying h(x)∼(log|x|)a/|x|2 as |x|→∞ for some a>−1. We determine the precise high-temperature asymptotics of the free energy, confirming a conjecture of Lacoin (Ann. Probab. 2011), later refined by Cosco, Cottini and Donadini (2025). We also establish a phase transition for the diffusively rescaled partition functions: below some critical point they converge to the Lebesgue measure, while above it they converge to zero. A key feature of our approach is that both results are obtained using only second-moment estimates and are based on a simplified change-of-measure argument in the supercritical regime, that may prove useful for other disordered models.
Related
- Disorder Thresholds and Free Energy of Brownian Directed Polymers with Product and Radial Spatial Correlations
- Strong disorder and very strong disorder are equivalent for directed polymers
- Invariance in half-space last passage percolation and directed polymers
- Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime
- The localization transition for the directed polymer in a random environment is smooth
BibTeX
@misc{berger2026free,
title = {{Free energy and phase transition for 2D directed polymers with critical spatial correlations}},
author = {Quentin Berger and Francesca Cottini},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30829},
eprint = {2608.30829},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05736323v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Free energy and phase transition for 2D directed polymers with critical spatial correlations
AU - Quentin Berger
AU - Francesca Cottini
PY - 2026
DA - 2026/09/01
PB - HAL
DO - 10.48550/arxiv.2608.30829
UR - https://hal.science/hal-05736323v1
AB - We study the two-dimensional directed polymer model in a Gaussian environment which is independent in time and spatially correlated, with covariances h(x) either summable or with a critical decay, satisfying h(x)∼(log|x|)a/|x|2 as |x|→∞ for some a>−1. We determine the precise high-temperature asymptotics of the free energy, confirming a conjecture of Lacoin (Ann. Probab. 2011), later refined by Cosco, Cottini and Donadini (2025). We also establish a phase transition for the diffusively rescaled partition functions: below some critical point they converge to the Lebesgue measure, while above it they converge to zero. A key feature of our approach is that both results are obtained using only second-moment estimates and are based on a simplified change-of-measure argument in the supercritical regime, that may prove useful for other disordered models.
ER -