The maximum of the two dimensional Gaussian directed polymer in the subcritical regime

We study the maximum ${\phi }_N^{\ast }$ of the partition function of the two dimensional (subcritical) Gaussian directed polymer over an $\sqrt{N}\times \sqrt{N}$ box. We show that ${\phi }_N^{\ast }\mathrm{/}\log N$ converges towards a constant ${\sigma }^{\ast }$, 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.