From Combinatorics to Large Deviations for the Invariant Measures of Some Multiclass Particle Systems

#### D. Gabrielli

2008, v.14, Issue 3, 365-402

ABSTRACT

We prove large deviation principles (LDP) for the invariant measures of the multiclass totally asymmetric simple exclusion process (TASEP) and the multiclass Hammersley - Aldous - Diaconis (HAD) process on a torus. The proof is based on a combinatorial representation of the measures in terms of a collapsing procedure introduced in [O. Angel, The stationary measure of a $2$-type totally asymmetric exclusion process. J. Combin. Theory Ser. A, 2006, v.113, 625-635] for the $2$-class TASEP and then generalized in [P.A. Ferrari and J.B. Martin, Stationary distributions of multi-type totally asymmetric exclusion processes. Ann. Prob., 2007, v. 35, 807-832], [P.A. Ferrari and J.B. Martin, Multiclass processes, dual points and M/M/1 queues. Markov Processes Relat. Fields, 2006, v. 12, 273-299], [P.A. Ferrari and J.B. Martin, Multiclass Hammersley - Aldous - Diaconis process and multiclass-customer queues. Preprint arXiv:0707.4202v1, 2007. To appear in Ann. Inst. H. Poincare] to the multiclass TASEP and the multiclass HAD process. The rate functionals are written in terms of variational problems that we solve in the cases of $2$-class processes.

Keywords: large deviations,interacting particle systems,stationary states