PreprintCC BY 4.0arXiv
Maximum of the Two-Dimensional Directed Polymer in the Critical Window
Abstract
Let ZNβN(x) denote the partition function of the two-dimensional directed polymer started at x∈ ℤ2, in Gaussian environemnt, in the critical window, with parameter βN=√((1−ϑ/log N)/RN). Here RN∼ (log N)/π is the truncated collision Green function at the origin for the simple random walk on ℤ2. Let MN denote the maximum of log ZN(x) over |x|∞≤ √N. We prove that, for any ϑ∈ ℝ, in probability, MN/√(log N)→ 2√2 as N→∞.
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BibTeX
@misc{cosco2026maximum,
title = {{Maximum of the Two-Dimensional Directed Polymer in the Critical Window}},
author = {Clement Cosco and Shuta Nakajima and Ofer Zeitouni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41989},
eprint = {2610.41989},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41989},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maximum of the Two-Dimensional Directed Polymer in the Critical Window
AU - Clement Cosco
AU - Shuta Nakajima
AU - Ofer Zeitouni
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41989
UR - https://arxiv.org/abs/2610.41989
AB - Let Z_N^β_N(x) denote the partition function of the two-dimensional directed polymer started at x∈ ℤ^2, in Gaussian environemnt, in the critical window, with parameter β_N=√((1−ϑ/log N)/R_N). Here R_N∼ (log N)/π is the truncated collision Green function at the origin for the simple random walk on ℤ^2. Let M_N denote the maximum of log Z_N(x) over |x|_∞≤ √N. We prove that, for any ϑ∈ ℝ, in probability, M_N/√(log N)→ 2√2 as N→∞.
ER -