The maximum of the two dimensional Gaussian directed polymer in the subcritical regime
Clément Cosco, Shuta Nakajima, Ofer Zeitouni
TL;DR
This paper analyzes the leading-order growth of the maximum of the Gaussian directed polymer's partition function in two dimensions within the subcritical regime, showing φ_N^*/log N converges to a universal constant σ^*. The constant σ^* is expressed via a slowly varying variance profile and coincides with the maximum of a branching random walk with such a variance structure, linking polymer extremes to BRW and log-correlated field phenomena. The authors develop a multiscale barrier approach, coupled with decoupling and diffusive-scale separation, to derive matching upper and lower bounds through variational analysis and moment methods. This work connects 2D polymer extremes to Gaussian multiplicative chaos and broader KPZ/SHE universality classes, and clarifies how scale-inhomogeneous contributions govern extreme value behavior in subcritical regimes.
Abstract
We study the maximum $φ_N^*$ of the partition function of the two dimensional (subcritical) Gaussian directed polymer over an $\sqrt N \times \sqrt N$ box. We show that $φ_N^*/\log N$ converges towards a constant $σ^*$, which we identify to be the same as for the maximum of a branching random walk with a slowly varying variance profile as studied in Fang-Zeitouni, J. Stat. Phys. 2012 and (in the context of the generalized random energy model) in Bovier-Kurkova, Ann. Inst. H. Poincare 2004.