On the Metric Structure of Ultrametric Spaces

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 frameworks of our approach, the results of Ogielsky 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.