Fisher’s geometric model and spin glasses
Joachim Krug (Uni Cologne)
Fisher’s geometric model describes biological fitness landscapes by combining a linear map from the discrete space of genotype sequences of length L to an n-dimensional Euclidean phenotype space with a nonlinear, single-peaked phenotype-fitness map. 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. I will present new results for the distribution of the number of maxima. The typical scale of the number of maxima is derived for general n, and the full scaled probability density and two point correlation function of maxima are determined for the one-dimensional case. The model is closely related to the anti-ferromagnetic Hopfield model with n random continuous pattern vectors, and most results carry over to this setting. The talk is based on joint work with Sungmin Hwang and Su-Chan Park.