Spectral Gap for Multi-species Exclusion Processes

We consider a multi-species generalization of the symmetric simple exclusion process in homogeneous and non-homogeneous hypercubes of Z^d. In this model, the hyperplanes of configurations with given numbers of particles of each species are not necessarily irreducible. We give a sufficient condition of the dynamics to make them irreducible. In addition, assuming the irreducibility of them, we show some estimates of the spectral gap (the absolute value of the second largest eigenvalue of the generator), which plays an important role in the study of hydrodynamic limit.