We study an $n$-species $t$-PushTASEP, an integrable long-range stochastic process, on a one-dimensional periodic lattice with inhomogeneities $x_1,\ldots,x_L$ and arbitrary capacity $l$ at each lattice site. The Markov matrix is identified with an alternating sum of commuting transfer matrices over all fundamental representations of $U_t(\widehat{sl}_{n+1})$. Stationary probabilities are expressed in a matrix product form involving a fusion of quantized corner transfer matrices for the strange five-vertex model introduced by Okado, Scrimshaw, and the second author. The resulting partition function, which serves as the normalization factor of the stationary probabilities, is obtained from the $l=1$ case by a finite plethystic substitution of length $l$.
PushTASEP is a class of totally asymmetric simple exclusion processes (TASEPs) of interacting particles on one-dimensional lattice. Its characteristic feature is a stochastic dynamics in which multiple particles may move simultaneously over long distances by pushing one another according to prescribed rules. By now, various versions of PushTASEPs have been introduced and studied extensively; see, for example, [ANP23, AK25, AM23, AMW24, BW22, CP13, P19] and the references therein.
In this paper we study a PushTASEP formulated as a continuous-time Markov process on a periodic lattice of length L, in which each local state is given by a length-l row-shaped semistandard tableau with entries in {0, 1, . . . , n}, namely, {(i 1 , . . . , i l ) ∈ Z l | 0 ≤ i 1 ≤ • • • ≤ i l ≤ n}. Such a local state may be interpreted as an assembly of particles of species 1, 2, . . . , n, together with δ i1,0 + • • • + δ i l ,0 vacant slots, within the maximal capacity l at each site. (In the main text, however, we treat 0 as a particle species as well.) The transition rates depend on the parameter t and on inhomogeneity parameters x 1 , . . . , x L attached to each site. The model is Markovian, and hence physically meaningful, in the parameter region t ≥ 0 and x 1 , . . . , x L > 0. We refer to this model, which has n+l l local states at each site, as the n-species capacity-l t-PushTASEP.
A key ingredient underlying our approach is the quantum R-matrix S k l (z) acting on V k ⊗ V l in an appropriate gauge, where V k and V l denote the degree-k antisymmetric tensor representation and the degree-l symmetric tensor representation of the quantum affine algebra U t ( sl n+1 ), respectively. This coupling of antisymmetric and symmetric tensor representations appears to be new in the context of integrable probability. In principle, S k l (z) can be obtained [KRS81] either by l-fold symmetric fusion of S k 1 (z) or by k-fold antisymmetric fusion of S 1 l (z). In this paper, we adopt a more efficient construction based on three-dimensional integrability explored in [K22,Chap. 11.3]. It directly yields an explicit formula for S k l (z) given in (52a)-(53b) in terms of the 3D L-operator (50).
Based on the R-matrix, we construct the commuting family of transfer matrices T k (z|x 1 , . . . , x L ) of the solvable vertex model (cf. [Bax83]) with spectral parameter z and inhomogeneities x 1 , . . . , x L for 0 ≤ k ≤ n+1. In the terminoogy of the quantum inverse scattering method [STF80], it has the auxiliary space V k and acts on the quantum space V ⊗L l . Let H n,l (x 1 , . . . , x L ) be the Markov matrix of the n-species capacity-l t-PushTASEP in (17a) whose transition rates are given by (29) and also described combinatorially in Section 2.3. Our first main result is the following formula (Theorem 11):
Here, D m is a scalar factor defined in (15), determined by the particle multiplicity m. The RHS is an alternating sum of the derivative of the transfer matrices whose auxiliary spaces range over all fundamental representations V 0 , . . . , V n+1 . The identity (1) extends a Baxter-type formula for quantum Hamiltonians (cf. [Bax83, eq. (10.14.20)]) to an inhomogeneous stochastic setting, generalizing our earlier result for the l = 1 case [AK25] to arbitrary capacity. A notable aspect of (1) is that neither the individual transfer matrix T k (z|x 1 , . . . , x L ) nor its derivative at z = 0 is stochastic in general: their matrix elements need not be positive and do not satisfy probability conservation. Nevertheless, the alternating sum in (1) acts as an inclusion-exclusion mechanism, retaining admissible particle motions with correct rates while cancelling forbidden channels.
The result (1) reduces the problem of finding stationary states of the model to that of constructing a joint eigenstate of the commuting transfer matrices. Our second main result provides such a construction explicitly: the stationary probability of a configuration (σ 1 , . . . , σ L ) is expressed in a matrix product form
up to normalization. Here the operator A i (z) associated with a local state i = (i 1 , . . . , i l ) is defined as (cf. ( 120), (123))
where the sum runs over distinct permutations (i ′ 1 , . . . , i ′ l ) of (i 1 , . . . , i l ). For the basic case l = 1, the operators A 0 (z), . . . , A n (z) in (113) coincide, up to a minor conventional change, with those introduced in [KOS24]. They are corner transfer matrices (CTMs [Bax83, Chap.13]) of the strange five-vertex model, which are quantized in the sense that the “Boltzmann weights” take values in the t-oscillator algebra (109). The construction of the CTMs for higher l in (3) may be viewed as a symmetric fusion at the level of matrix product operators.
Let Z l,m (x 1 , . . . , x L ; t) = (σ1,…,σ L )∈S(m) P(σ 1 , . . . , σ L ) denote the normalization factor of the stationary probabilities for capacity l, where S(m) is defined in (13). This quantity defines a symmetric polynomial in x 1 , . .
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