On metric structure of ultrametric spaces

TL;DR

This paper explores the metric structure of ultrametric spaces by connecting them with complex analysis and number theory, providing new insights into diffusion processes and potential geometric interpretations of replica limits.

Contribution

It introduces a novel approach using modular functions to model ultrametric spaces as continuous structures embedded in the Poincare upper half-plane.

Findings

Reproduces Ogielsky and Stein's results semi-analytically.

Provides a new geometric interpretation of the replica n->0 limit.

Establishes a connection between ultrametric diffusion and complex analysis.

Abstract

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.