  We study the convergence towards a unique equilibrium distribution of the
solutions to a time-discrete model with non-overlapping generations arising in
quantitative genetics. The model describes the dynamics of a phenotypic
distribution with respect to a multi-dimensional trait, which is shaped by
selection and Fisher's infinitesimal model of sexual reproduction. We extend
some previous works devoted to the time-continuous analogues, that followed a
perturbative approach in the regime of weak selection, by exploiting the
contractivity of the infinitesimal model operator in the Wasserstein metric.
Here, we tackle the case of quadratic selection by a global approach. We
establish uniqueness of the equilibrium distribution and exponential
convergence of the renormalized profile. Our technique relies on an accurate
control of the propagation of information across the large binary trees of
ancestors (the pedigree chart), and reveals an ergodicity property, meaning
that the shape of the initial datum is quickly forgotten across generations. We
combine this information with appropriate estimates for the emergence of
Gaussian tails and propagation of quadratic and exponential moments to derive
quantitative convergence rates. Our result can be interpreted as a
generalization of the Krein-Rutman theorem in a genuinely non-linear, and
non-monotone setting.
