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

Clément Cosco; Shuta Nakajima; Ofer Zeitouni

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arXiv · 2503.17236

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

Clément Cosco·Shuta Nakajima·Ofer Zeitouni

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Abstract

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

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Clément Cosco, Shuta Nakajima, Ofer Zeitouni. 2025-03-21. The maximum of the two dimensional Gaussian directed polymer in the subcritical regime. https://arxiv.org/abs/2503.17236

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