Determinantal expressions in multi-species TASEP

Consider an inhomogeneous multi-species TASEP with drift to the left, and define a height function which equals the maximum species number to the left of a lattice site. For each fixed time, the multi-point distributions of these height functions have a determinantal structure. In the homogeneous case and for certain initial conditions, the fluctuations of the height function converge to Gaussian random variables in the large-time limit. The proof utilizes a coupling between the multi-species TASEP and a coalescing random walk, and previously known results for coalescing random walks.


Introduction
Consider a multi-species TASEP (totally asymmetric simple exclusion process) on the infinite lattice Z. The species numbers are indexed by the integers, so any particle configuration can be expressed as η(t) = (ηx(t)) x∈Z , where ηx(t) denotes the species number of the particle at site x. By convention, let ηx(t) = −∞ denote a hole (no particle) at lattice site x .The particle at site x jumps one step to the left with exponential jump rates of rate λx, and particles with higher species have priority to jump. In other words, if a particle of species j attempts to jump to a site of species i, then the particles switch places if i < j, and the jump is blocked if i > j. All jumps occur independently of each other. For µ ∈ Z ∪ {±∞}, let ξµ(t) denote the location of a single random walker which starts at µ, and jumps one step to the left with exponential rate λx when at site x. If µ = {±∞} then the particle remains at {±∞} for all times t ≥ 0. In other words, ξµ(t) describes the evolution of the single-species inhomogeneous TASEP with only a single particle starting at lattice site µ.
In words, this is the maximum species number among all particles to the left of y. The main result of this paper is the following theorem: Theorem 1.1. Fix k lattice points x1 < . . . < x k and k species numbers m1 < . . . < m k . Let where inf ∅ = +∞ by convention. Then where G is the k × k matrix with entries In the homogeneous case when all λx ≡ 1, it is readily seen (by the central limit theorem) that the fluctuations are Gaussian: Corollary 1.2. Suppose that all λx are equal to 1. Assume that x1, . . . , x k and m1, . . . , m k depend on t in such a way that for some νi, wi, ν ′ j , w ′ j ∈ R. Then where M has entries Here, Φ is the cumulative distribution of a standard Gaussian: Remark 1. If mi ≥ mj for some i < j in Theorem 1.1, then so there is no loss of generality in assuming that m1 < . . . < m k . Remark 2. Since m1 < . . . < m k , then Remark 3. Two initial conditions which satisfy the assumptions of Corollary 1.2 are the step initial conditions ηx(t) = x for x ∈ Z and the flat initial conditions In these two cases, respectively, µi = mi and µi = 2mi. Then Remark 4. For the (single-species) TASEP with various initial conditions (step, flat, stationary, and their mixtures), it has been shown in [Joh03], [BFPS07], [BFP07], [BFS08a], [BFP10], [BFS08b] that the fluctuations in the usual height function converges to various Airy processes; see also [QR19]. These fluctuations are asymptotically independent of the fluctuations of hx i in Corollary 1.2. For example, letting hx(t) denote the number of particles to the left of x at time t, then with step initial conditions.

Proof of Theorem
The theorem is proved by coupling the multi-species TASEP to a coalescing random walk, and using a known determinantal expression for the distributions of coalescing random walk from Proposition 2.5 of [Ass18] (see also Proposition 9 of [War05] for a similar expression in the context of coalescing Brownian motions). This coupling does not seem to have appeared in the literature so far; but note the upcoming preprint [BB], in which a similar coupling is being used.
In the coalescing random walk, if a particle tries to jump to a site that is already occupied, then the two particles coalesce into a single particle. The particle at site x has an exponential clock of rate λx, and jumps one step to the left when the clock rings, just as in the multi-species TASEP. The coupling is constructed as follows. The coalescing random walk and the multi-species TASEP begin with the same initial conditions. We couple the exponential clocks of each particle in the multi-species TASEP with the exponential clocks of the corresponding particle in the coalescing random walk. In this way, each particle in the multi-species TASEP is coupled with a particle in the coalescing random walk at time t = 0. The evolution occurs with the following rules: 1. If a particle in the multi-species TASEP attempts to jump but the jump is blocked, then no change occurs in either system.
2. If a particle in the multi-species TASEP jumps one step to the left and that site is unoccupied, then the corresponding coupled particle (if there is one) in the coalescing random walk makes the same jump.
3. If a particle in the multi-species TASEP jumps one step to the left, and that site is occupied by a lower species particle, then the corresponding coupled particle (if there is one) makes the same jump in the coalescing random walk. Additionally, the lower species particle is then absorbed into the higher species particle in the coalescing random walk (if it had not already been absorbed previously), and is no longer coupled with the corresponding particle in the multi-species TASEP. 4. If a particle in the multi-species TASEP attempts a jump but there is no coupled particle in the coalescing random walk, then no jump occurs in the coalescing random walk.
One can see that two particles are coupled if and only if they occupy the same location. Letηx(t) denote the species of the particle located at site x at time t in the coalescing random walk. As before, letηx(t) = −∞ by convention if there is no particle at site x and time t. Now let gy(t) = max{ηx(t) : x < y}.
Lemma 2.1. For x1 < . . . < x k and any fixed time t, the joint distribution of (gx 1 (t), . . . , gx k (t)) equals the joint distribution of (gx 1 (t), . . . ,gx k (t)). In other words, for any m1 < . . . < m k , Proof. At time t = 0, the multi-species TASEP and the coalescing random walk are in the same configuration, so (1) holds. The only potential way that the two sides of (1) could be different is when a particle jump causes the two processes to be updated in a different manner -that is, update rules 3 or 4. Suppose that an update under rule 4 occurs. This means that a particle as jumped in multi-species TASEP when there is no coupled particle in the coalescing random walk. This means that the particle in the coalescing random walk has already been absorbed into a higher species particle. Correspondingly, this means that in the multi-species TASEP, there is a higher species particle to the left of that particle. This means that the evolution of the uncoupled / absorbed particle no longer affects the values of gx i (t) orgx i (t), since the particle cannot cross xi unless the higher species particle has also crossed xi. Similarly, if an update under rule 3 occurs, then the multi-species TASEP and the coalescing random walk differ in the uncoupled / absorbed particle. For identical reasons, the evolution of this particles does not affect the values of gx i (t) orgx i (t).
By Lemma 2.1, it suffices to find the right-hand-side of (1). As mentioned in Remark 2, µ1 ≤ . . . ≤ µ k , where µi is defined as in Theorem 1.1. Letxµ(t) denote the position of the particle in the coalescing random walk that started at lattice site µ at time 0. Then it is straightforward to see that there is the inclusion of events {gx i (t) ≥ mi} ⊇ {xµ i (t) ≤ xi}.
To see the reverse inclusion, note thatxµ i (t) is the left-most particle which has species higher than mi, and remains the left-most such particle after coalescence. Therefore, P (gx 1 (t) ≥ m1, . . . ,gx k (t) ≥ m k ) = P(xµ 1 (t) ≤ x1, . . . ,xµ k (t) ≤ x k ) The probability on the right-hand-side can be found from Proposition 2.5 of [Ass18]. The cited proposition applies more generally (it allows for right jumps as well), and includes a reflecting boundary. We can take the limit of this boundary point to −∞ and apply the proposition. The result is that the probability equals det[Gij ], as needed.