Fisher’s infinitesimal model (or polygenic model) is a widely used statistical model in quantitative genetics. It was proposed by Ronald Fisher to describe the propagation of a quantitative trait along generations of a population which is subjected to sexual reproduction. Recently, this model has pulled the attention of more theoretical communities and some integro-differential equations have been proposed to study the precise dynamics of traits under the coupled effect of sexual reproduction and natural selection. Whilst some partial results have already been obtained, the complete understanding of the long-term behavior is essentially unknown. In this talk, I will present a simpler time-discrete version inspired in the previous time-continuous models. Our novel approach relies on a better understanding of the propagation of the trait value across the ancestors in the family tree. For trait dependent selection with quadratic shape, our technique unravels an ergodicity-type property leading to quantitative convergence rates towards a unique global equilibrium. This is a joint work with Vincent Calvez and Thomas Lepoutre.