Nonlinear Exchange Dynamics for Spin Systems

This thesis studies a class of nonlinear Markov dynamics for spin systems on finite graphs, in which a transition is the outcome of a collision between two configurations exchanging a single spin. The resulting evolution is a discrete analogue of the Boltzmann equation: it is quadratic in the law of the process, and it conserves the density profile rather than the configuration itself, so that each initial density selects a distinct stationary Gibbs measure. We establish exponential decay of the relative entropy along such dynamics for two models. For the Ising model with positive semi-definite interaction matrix of small spectral radius, and for the hardcore model on graphs of bounded degree below the critical density, the relative entropy contracts at an explicit rate, uniform in the initial datum. The proofs complete Kac's program in this setting: the nonlinear evolution is realised as the mean-field limit of a conservative particle system, for which we prove a modified log-Sobolev inequality with constant independent of the number of particles, and the passage back to the nonlinear equation is carried out through entropic and Fisher chaos. The main ingredients are stochastic localization and comparison with the Down-Up walk. As an algorithmic consequence, we obtain the first sampler that draws from the hardcore model at a prescribed density without computing the corresponding fugacity, at near-linear cost in the size of the graph.