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On the metric structure of ultrametric spaces. (English) Zbl 1098.11064

Proc. Steklov Inst. Math. 245, 169-188 (2004) and Tr. Mat. Inst. Steklova 245, 182-201 (2004).
Summary: In our work, we reconsider the old problem of diffusion at the boundary of an ultrametric tree from a ‘number-theoretic’ point of view. Namely, we use modular functions (in particular, the Dedekind \(\eta\)-function) to construct a ‘continuous’ analogue of the Cayley tree isometrically embedded into the Poincaré upper half-plane. Later, we work with this continuous Cayley tree as with a standard function of a complex variable. In the framework of our approach, the results of Ogielski and Stein on the dynamics on ultrametric spaces are reproduced semi-analytically/semi-numerically. Speculation on the new ‘geometrical’ interpretation of the replica \(n\to 0\) limit is proposed.
For the entire collection see [Zbl 1087.46002].

MSC:

11Z05 Miscellaneous applications of number theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics