Random evolutionary dynamics driven by fitness and house-of-cards mutations. Sampling formulae

We first revisit the multi-allelic mutation-fitness balance problem, especially when mutations obey a house of cards condition, where the discrete-time deterministic evolutionary dynamics of the allelic frequencies derives from a Shahshahani potential. We then consider multi-allelic Wright-Fisher stochastic models whose deviation to neutrality is from the Shahsha-hani mutation/selection potential. We next focus on the weak selection, weak mutation cases and, making use of a Gamma calculus, we compute the normalizing partition functions of the invariant probability densities appearing in their Wright-Fisher diffusive approximations. Using these results, Generalized Ewens sampling formulae (ESF) from the equilibrium distributions are derived. We start treating the ESF in the mixed mutation/selection potential case and then we restrict ourselves to the ESF in the simpler house-of-cards mutations only situation. We also address some issues concerning sampling problems from infinitely-many alleles weak limits.Comment: \`a paraitre: Journal of Statistical Physic

Karlin-McGregor mutational occupancy problem revisited

Some population is made of n individuals that can be of p possible species (or types). The update of the species abundance occupancies is from a Moran mutational model designed by Karlin and McGregor in 1967. We first study the equilibrium species counts as a function of n, p and the total mutation probability $\nu$ before considering various asymptotic regimes on n, p and $\nu$. Running title: KMG Model with Mutations.Comment: to appear: Journal of Statistical Physic

Limiting search cost distribution for the move-to-front rule with random request probabilities

Consider a list of $n$ files whose popularities are random. These files are updated according to the move-to-front rule and we consider the induced Markov chain at equilibrium. We give the exact limiting distribution of the search-cost per item as $n$ tends to infinity. Some examples are supplied.Comment: move-to-front, search cost, random discrete distribution, limiting distribution, size biased permutatio