Sampling the Fermi statistics and other conditional product measures

Through a Metropolis-like algorithm with single step computational cost of order one, we build a Markov chain that relaxes to the canonical Fermi statistics for k non-interacting particles among m energy levels. Uniformly over the temperature as well as the energy values and degeneracies of the energy levels we give an explicit upper bound with leading term kmln k for the mixing time of the dynamics. We obtain such construction and upper bound as a special case of a general result on (nonhomogeneous) products of ultra log-concave measures (like binomial or Poisson laws) with a global constraint. As a consequence of this general result we also obtain a disorder-independent upper bound on the mixing time of a simple exclusion process on the complete graph with site disorder. This general result is based on an elementary coupling argument, illustrated in a simulation appendix and extended to (non-homogeneous) products of log-concave measures.