4 - Ultrametricity in the Theory of Complex Systems

Summary

Introduction

In the present chapter we give a short review of some of the applications of p-adic and more general ultrametricmethods in the statistical physics of disordered systems, dynamics of macromolecules, and genetics. The application of p-adic analysis to mathematical physics was initiated in [436]. These methods were used in string theory, high-energy physics, cosmology, and other fields [434], [135], [275], [35], [90], [322], [323], [324], [42], [411], [412], [349], [325], [326], [152], [191], [198], [190], [470].

Applications to the physics of complex systems and other related areas were discussed in [336], [403], [162], [83], [376]. The starting point of applications of ultrametric methods to complex systems was the replica symmetry-breaking approach [336], [372], [335], where ultrametric spaces were used to describe spaces of states of spin glasses. In the replica symmetry-breaking approach the ultrametric was a result of the branching process in a space of high dimension, see for mathematical discussion [359], [61], [461].

The relation between replica symmetry-breaking and p-adic analysis (p-adic parametrization of the Parisi matrix) was found in [53], [373]. In particular, in [373] correlation functions of the replica approach were expressed in the form of p-adic integrals. Generalizations of these results to more general ultrametric spaces were considered in [234, 235, 239]. In [97, 118] the application of the Fourier transform on some Abelian groups to the diagonalization of Parisi matrices was discussed (the p-adic case is treated in a similar way).

The p-adic Potts model was considered in [352, 246, 351, 350]. Hierarchical models of quantum statistical mechanics related to trees were considered in [186], [329], [327], [328], [311], [347], [346]. In [10, 11], [3] the p-adic random walk was considered. p-Adic Brownian motion was considered in [146], [148]. In [359] a random walk on the border of the tree in the complex plane was considered (the tree was related to the energy landscape of the Dedekind function). In [170] the quantum dynamics on a complex energy landscape was discussed in relation to Anderson localization. The application of p-adic methods to fractals was discussed in particular in [211].

The statistical mechanics of complex systems is also related to the dynamics on complex energy landscapes, in particular in applications to the dynamics of macromolecules [41], [366], [86], [201], [166], [399], [70], [165].