  We introduce and analyze a natural class of nonlinear dynamics for spin
systems such as the Ising model. This class of dynamics is based on the
framework of mass action kinetics, which models the evolution of systems of
entities under pairwise interactions, and captures a number of important
nonlinear models from various fields, including chemical reaction networks,
Boltzmann's model of an ideal gas, recombination in population genetics and
genetic algorithms. In the context of spin systems, it is a natural
generalization of linear dynamics based on Markov chains, such as Glauber
dynamics and block dynamics, which are by now well understood. However, the
inherent nonlinearity makes the dynamics much harder to analyze, and rigorous
quantitative results so far are limited to processes which converge to
essentially trivial stationary distributions that are product measures. In this
paper we provide the first quantitative convergence analysis for natural
nonlinear dynamics in a combinatorial setting where the stationary distribution
contains non-trivial correlations, namely the Ising model at high temperatures.
We prove that nonlinear versions of both the Glauber dynamics and the block
dynamics converge to the Gibbs distribution of the Ising model (with given
external fields) in times $O(n\log n)$ and $O(\log n)$ respectively, where $n$
is the size of the underlying graph (number of spins). Given the lack of
general analytical methods for such nonlinear systems, our analysis is
unconventional, and combines tools such as information percolation (due in the
linear setting to Lubetzky and Sly), a novel coupling of the Ising model with
Erd\H{o}s-R\'enyi random graphs, and non-traditional branching processes
augmented by a fragmentation process.
