Statut  Confirmé 
Série  SEMLPTMC 
Domaines  condmat.meshall 
Date  Lundi 19 Novembre 2018 
Heure  10:45 
Institut  LPTMC 
Salle  Jussieu, tower 1312, room 523 
Nom de l'orateur  Texier 
Prenom de l'orateur  Christophe 
Addresse email de l'orateur  christophe [dot] texier [at] upsud [dot] fr 
Institution de l'orateur  LPTMS Orsay 
Titre  Counting the equilibria of a directed polymer in a random medium and Anderson localisation 
Résumé  I will discuss a new connection between two different problems: the counting of equilibria of a directed polymer in a random medium (DPRM) and the problem of Anderson localisation for the 1D Schrödinger equation. Using the KacRice formula, it is possible to express the mean number of equilibria of a DPRM in terms of functional determinants. In the onedimensional situation, these functional determinants can be calculated thanks to the GelfandYaglom method, showing that the mean number of equilibria of the DPRM growth exponentially with the length of the polymer, with a rate controlled by the generalized Lyapunov exponent (GLE) of the localisation problem (cumulant generating function of the log of the wave function). The GLE is solution of a spectral problem studied by combining numerical approaches and WKBlike approximation. Furthermore, the formalism can be extended in order to obtain the number of equilibria at fixed energy, providing the (annealed) distribution of the energy density of the line over the equilibria. 
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