Christophe Texier (LPTMS Orsay)
Counting the equilibria of a directed polymer in a random medium and Anderson localisation
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 Kac-Rice formula, it is possible to express the mean number of equilibria of a DPRM in terms of functional determinants. In the one dimensional situation, these functional determinants can be calculated thanks to the Gelfand-Yaglom 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 WKB-like 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.