Distribution of the number of fitness maxima in Fisher's Geometric Model

Abstract

Fisher's geometric model describes biological fitness landscapes by combining a linear map from the discrete space of genotypes to an $n$-dimensional Euclidean phenotype space with a nonlinear, single-peaked phenotype-fitness map. Genotypes are represented by binary sequences of length $L$, and the phenotypic effects of mutations at different sites are represented by $L$ random vectors drawn from an isotropic Gaussian distribution. Recent work has shown that the interplay between the genotypic and phenotypic levels gives rise to a range of different landscape topographies that can be characterised by the number of local fitness maxima. Extending our previous study of the mean number of local maxima, here we focus on the distribution of the number of maxima when the limit $L \to \infty$ is taken at finite $n$. We identify the typical scale of the number of maxima for general $n$, and determine the full scaled probability density and two point correlation function of maxima for the one-dimensional case. We also elaborate on the close relation of the model to the anti-ferromagnetic Hopfield model with $n$ random continuous pattern vectors, and show that many of our results carry over to this setting. More generally, we expect that our analysis can help to elucidate the fluctuation structure of metastable states in various spin glass problems.