Schubert Polynomials, the Inhomogeneous TASEP, and Evil-Avoiding Permutations

Consider a lattice of n sites arranged around a ring, with the n sites occupied by particles of weights {1, 2,..., n}; the possible arrangements of particles in sites thus correspond to the n! permutations in S-n. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on S-n, in which two adjacent particles of weights i < j swap places at rate x(i) - y(n+ 1-j) if the particle of weight j is to the right of the particle of weight i. (Otherwise, nothing happens.) When y(i) = 0 for all i, the stationary distribution was conjecturally linked to Schubert polynomials [18], and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues [4, 5]. In the case of general y(i), Cantini [7] showed that n of the n! states have probabilities proportional to double Schubert polynomials. In this paper, we introduce the class of evil-avoiding permutations, which are the permutations avoiding the patterns 2413, 4132, 4213, and 3214. We show that there are (2+root 2)(n-1)+(2-root 2)(n-1) / 2 evilavoiding permutations in S-n, and for each evil-avoiding permutation w, we give an explicit formula for the steady state probability psi(w) as a product of double Schubert polynomials. (Conjecturally, all other probabilities are proportional to a positive sum of at least two Schubert polynomials.) When y(i) = 0 for all i, we give multiline queue formulas for the z-deformed steady state probabilities and use this to prove the monomial factor conjecture from [18]. Finally, we show that the Schubert polynomials arising in our formulas are f lagged Schur functions, and we give a bijection in this case between multiline queues and semistandard Young tableaux.