Exclusion processes with multiple interactions

We introduce the mathematical theory of the particle systems that interact via permutations, where transition rates are assigned not to the jumps from a site to a site, but to the permutations themselves. These permutation processes can be viewed as the natural generalization of symmetric exclusion processes, where particles interact via transpositions. We develop a number of innovative coupling techniques for the permutation processes and establish the needed conditions for them to apply. We use duality, couplings and other tools to explore the stationary distributions of the permutation processes with translation invariant rates.