Ultrametricity in the theory of complex systems

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.