The high-temperature behavior for the directed polymer in dimension 1+2

Quentin Berger (1) , Hubert Lacoin (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 IMPA-Brazil

We investigate the high-temperature behavior of the directed polymer model in dimension 1+2. More precisely we study the difference $\Delta F (\beta)$ between the quenched and annealed free energies for small values of the inverse temperature $\beta$. This quantity is associated to localization properties of the polymer trajectories, and is related to the overlap fraction of two replicas. Adapting recent techniques developed by the authors in the context of the disordered pinning model (Berger and Lacoin, arXiv:1503.07315 [math-ph]), we identify the sharp asymptotic high temperature behavior \[ \lim_{\beta\to0} \beta^2 \log \Delta F (\beta) = -\pi . \]

Mots clés: Disordered Systems; Directed Polymer; Free Energy; Localization

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]

Quentin Berger, Hubert Lacoin. The high-temperature behavior for the directed polymer in dimension 1+2. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2017, 53 (1), pp.430-450. ⟨10.1214/15-AIHP721⟩. ⟨hal-01170204⟩