Status  Confirmed 
Seminar Series  SEMLPTMC 
Subjects  condmat.meshall 
Date  Monday 19 November 2018 
Time  10:45 
Institute  LPTMC 
Seminar Room  Jussieu, tower 1312, room 523 
Speaker's Last Name  Texier 
Speaker's First Name  Christophe 
Speaker's Email Address  christophe [dot] texier [at] upsud [dot] fr 
Speaker's Institution  LPTMS Orsay 
Title  Counting the equilibria of a directed polymer in a random medium and Anderson localisation 
Abstract  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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