On metric structure of ultrametric spaces

Oleg A. Vasilyev 1, Sergei K. Nechaev 1, 2

Journal of Physics A 37 (2004) 3783-3804

In our work we have reconsidered the old problem of diffusion at the boundary of ultrametric tree from a 'number theoretic' point of view. Namely, we use the modular functions (in particular, the Dedekind eta-function) to construct the 'continuous' analog of the Cayley tree isometrically embedded in the Poincare upper half-plane. Later we work with this continuous Cayley tree as with a standard function of a complex variable. In the frameworks of our approach the results of Ogielsky and Stein on dynamics on ultrametric spaces are reproduced semi-analytically/semi-numerically. The speculation on the new 'geometrical' interpretation of replica n->0 limit is proposed.

  • 1. Landau Institute for Theoretical Physics,
    Landau Institute for Theoretical Physics
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud