Ultrametricity in the theory of complex systems

Abstract

We review applications of p-adic and ultrametric methods in the theory of complex systems. We consider the following examples: the p-adic parameterization of the Parisi matrix in the replica method; the method of hierarchical (interbasin) kinetics, which allows describing macromolecular dynamics by models of ultrametric diffusion; the two-dimensional 2-adic parameterization of the genetic code, which demonstrates that degenerations of the genetic code are described by local constancy domains of maps in the 2-adic metric. We discuss clustering methods for a family of metrics and demonstrate that the multiclustering (ensemble clustering) approach is related to the Bruhat–Tits building theory.