Disorder Thresholds and Free Energy of Brownian Directed Polymers with Product and Radial Spatial Correlations
Abstract
We study a Brownian directed polymer in a centered Gaussian environment that is white in time and colored in space having long-range spatial correlations. For product-type covariances Q(x)asymp∏j=1d(1+|xj|)−αj, with αj∈(0,1) and κ=∑jαj, we identify the disorder transition at the marginal value κ=2. For κ>2, weak disorder holds at sufficiently small inverse temperature; for κ<2, the quenched free energy p(β) satisfies −p(β)asympβ4/(2−κ) as β↓0. For κ=2, strong disorder holds for every β>0, while p(β)=0 for all sufficiently small β, so βc=0<β̅c. We also consider the radial covariance cases, where $ Q(x)\asymp(1+|x|)^{-\vartheta}$, when d≥3,ϑ=2 and d=2,ϑ≥2, which was left unanswered in Lacoin~\cite{Lacoin2011}. When d=2, we get ln(−p(β))asymp−β−2 for ϑ>2 and ln(−p(β))asymp−β−1 for ϑ=2. The proofs consist of replica coupling, Feynman--Kac variational formula, overlap methods, and continuous-space fractional moments with ordered Wiener-chaos changes of measure.
Related
- Strong disorder and very strong disorder are equivalent for directed polymers
- Free energy and phase transition for 2D directed polymers with critical spatial correlations
- Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime
- Invariance in half-space last passage percolation and directed polymers
- Ballistic aggregation for one-sided Brownian initial velocity
BibTeX
@misc{cao2026disorder,
title = {{Disorder Thresholds and Free Energy of Brownian Directed Polymers with Product and Radial Spatial Correlations}},
author = {Junjie Cao and Guanglin Rang and Jianglun Wu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29501},
eprint = {2608.29501},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29501},
}
RIS
TY - UNPB
TI - Disorder Thresholds and Free Energy of Brownian Directed Polymers with Product and Radial Spatial Correlations
AU - Junjie Cao
AU - Guanglin Rang
AU - Jianglun Wu
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29501
UR - https://arxiv.org/abs/2608.29501
AB - We study a Brownian directed polymer in a centered Gaussian environment that is white in time and colored in space having long-range spatial correlations. For product-type covariances Q(x)asymp∏_j=1^d(1+|x_j|)^−α_j, with α_j∈(0,1) and κ=∑_jα_j, we identify the disorder transition at the marginal value κ=2. For κ>2, weak disorder holds at sufficiently small inverse temperature; for κ<2, the quenched free energy p(β) satisfies −p(β)asympβ^4/(2−κ) as β↓0. For κ=2, strong disorder holds for every β>0, while p(β)=0 for all sufficiently small β, so β_c=0<β̅_c. We also consider the radial covariance cases, where $ Q(x)\asymp(1+|x|)^{-\vartheta}$, when d≥3,ϑ=2 and d=2,ϑ≥2, which was left unanswered in Lacoin~\cite{Lacoin2011}. When d=2, we get ln(−p(β))asymp−β^−2 for ϑ>2 and ln(−p(β))asymp−β^−1 for ϑ=2. The proofs consist of replica coupling, Feynman--Kac variational formula, overlap methods, and continuous-space fractional moments with ordered Wiener-chaos changes of measure.
ER -