Ultrametric identities in glassy models of Natural Evolution

Abstract: Spin-glasses constitute a well-grounded framework for evolutionary models. Of particular interest for (some of) these models is the lack of self-averaging of their order parameters (e.g. the Hamming distance between the genomes of two individuals), even in asymptotic limits, much as like the behavior of the overlap between the configurations of two replica in mean-field spin-glasses. In the latter, this lack of self-averaging is related to peculiar fluctuations of the overlap, known as Ghirlanda-Guerra identities and Aizenman-Contucci polynomials, that cover a pivotal role in describing the ultrametric structure of the spin-glass landscape. As for evolutionary models, such identities may therefore be related to a taxonomic classification of individuals, yet a full investigation on their validity is missing. In this paper, we study ultrametric identities in simple cases where solely random mutations take place, while selective pressure is absent, namely in {\em flat landscape} models. In particular, we study three paradigmatic models in this setting: the {\em one parent model} (which, by construction, is ultrametric at the level of single individuals), the {\em homogeneous population model} (which is replica symmetric), and the {\em species formation model} (where a broken-replica scenario emerges at the level of species). We find analytical and numerical evidence that in the first and in the third model nor the Ghirlanda-Guerra neither the Aizenman-Contucci constraints hold, rather a new class of ultrametric identities is satisfied; in the second model all these constraints hold trivially. Very preliminary results on a real biological human genome derived by {\em The 1000 Genome Project Consortium} and on two artificial human genomes (generated by two different types neural networks) seem in better agreement with these new identities rather than the classic ones.