Ultrametricity for physicists

Complexity of hierarchical ensembles

“…. , n according to the power law [15] M l = (l + 1) a (8) with an exponent a > 1. In the limit l 1, the condition (3) takes the form…”

“…The formal basis of this theory is that states of hierarchically constrained objects are related to an ultrametric space whose geometrical image is the Cayley tree with nodes corresponding to statistical (sub)ensembles [8]. The relaxation of hierarchical structures had been considered first [9] as a diffusion process on either uniformly or randomly multifurcating trees characterized by a diversity being a measure of the tree's complexity [10].…”

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“…Применение ультраметричности в статистической физике не-упорядоченных систем (в первую очередь, спиновых стёкол) об-суждалось Мезардом, Паризи, Вирасоро [211] и другими автора-ми, см., например, обзор [232]. Обзор метода реплик можно най-ти в [43].…”

How the quasispecies evolution depends on the topology of the genome space

“…This analogy was brought to biology within the spin-glass model of evolution [29]. The main result of the spin-glass theory, which will be relevant for us here, is the tree-like (or ultrametric) structure of minima in the free-energy landscape [28,32]. We therefore assume that the same ultrametric structure of peaks will be characteristic for the rugged fitness landscape.…”

“…We will present the basic facts on ultrametric spaces, p-adic numbers and p-adic Fourier transform here [32,39,41,42].…”