Next seminar : Séminaire du LPTMS: Christophe Texier

Tuesday, December 18 2018 at 11:00:00

Counting the equilibria of a directed polymer in a random medium and Anderson localisation

Christophe Texier (LPTMS, Université Paris-Sud)

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. Reference:
  • Yan V. Fyodorov, Pierre Le Doussal, Alberto Rosso and Christophe Texier, Exponential number of equilibria and depinning threshold for a directed polymer in a random potential, Annals of Physics 397, 1-64 (2018)

Last Highlight : C’est chaud une ville la nuit

Il tend à faire plus chaud en centre-ville qu’en banlieue, mais certains centres chauffent davantage que d’autres. Une équipe de recherche internationale vient d’établir que la disposition des immeubles conditionne la température nocturne des villes – une découverte clé pour l’urbanisme de demain.

Voir la suite de l'article sur  CNRS - le journal

Ref. : J. M. Sobstyl, T. Emig, M. J. Abdolhosseini Qomi, F.-J. Ulm et R. J.-M. Pellenq, Role of City Texture in Urban Heat Islands at Nighttime, Phys. Rev. Lett. 120, 108701 (2018)

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