The inhomogeneous multispecies PushTASEP: Dynamics and symmetry

The multispecies asymmetric simple exclusion process (ASEP) is an interacting particle system originating in statistical mechanics and probability, with significant connections to different areas of mathematics such as combinatorics and representation theory. It has been the focus of considerable study in the last few decades. The stationary distribution of the multispecies TASEP (i.e. ASEP with asymmetry parameter $t=0$) on a ring of size $n$ was described by Ferrari and Martin [ferrari-martin-2007] in terms of multiline diagrams. The connection between the multispecies ASEP with content $\lambda$ and the Macdonald polynomial $P_{\lambda}(x_{1},\dots,x_{n};q,t)$ was realised by Cantini–de Gier–Wheeler [cantini-degier-wheeler-2015] and by Chen–de Gier–Wheeler [chen-degier-wheeler-2020]. In these works, they define ASEP polynomials (which are called relative Macdonald polynomials in [guo-ram-2021], and which are special cases of the permuted basement Macdonald polynomials studied in [ferreira-2011, alexandersson-2019]). These are polynomials in variables $x_{1},\dots,x_{n}$ whose coefficients are rational functions in $q,t$, which, for $q=1$ and all $x_{i}$ equal, are proportional to the stationary distribution of the multispecies ASEP. Later on, Corteel–Mandelshtam–Williams [corteel-mandelshtam-williams-2022] showed that the ASEP polynomials can be computed using the multiline diagrams stated above by assigning a weight depending on $q,t$ and the variables $x_{1},\dots,x_{n}$.

The ASEP itself is not thought to have nice algebraic properties when the jump rates are allowed to differ between sites – but it is natural to ask if there is some multispecies particle system with site-wise inhomogeneity whose stationary distribution is proportional to the ASEP polynomial at $q=1$ for general values of the parameters $x_{1},\dots,x_{n}$.

In this and a companion paper [ayyer-martin-williams-2024], we show that this is indeed the case, for a multispecies inhomogeneous version of the PushTASEP, with deformation parameter $t\geq 0$. The case of general $t$ is covered in [ayyer-martin-williams-2024]. In this work, we focus on various interesting aspects of the particular case $t=0$.

The arguments in [ayyer-martin-williams-2024] involve developing relationships between the ASEP polynomials and the non-symmetric Macdonald polynomials. Here for the case $t=0$, we explain an alternative proof of a different nature, constructing a Markov chain on multiline diagrams whose bottom row projects to the PushTASEP process, and whose stationary distribution is proportional to the weight defined in [corteel-mandelshtam-williams-2022]. This extends the approach of [ferrari-martin-2006] to the inhomogeneous setting.

A particular focus is on symmetry properties under interchange of the rates between different sites. We show that probabilities of events depending on a given interval of sites remain unchanged when the rates at other sites are permuted. Such results are proved for the general case $t\geq 0$ in [ayyer-martin-williams-2024], for events depending on the configuration at a single time in the stationary distribution. In the special case $t=0$ we can prove significantly stronger properties, applying to systems out of equilibrium, and to events which depend not just on a single time but on the path of an evolving process. These results are obtained by using coupling methods to prove interchangeability results for PushTASEP stations analogous to those previously proved for exponential queueing servers [Weber1979, TsoucasWalrand1987, MartinPrabhakar2010].

Finally we study various observables for the system in stationary, such as the density of particles of a given species at a given site, and the current between sites for particles of a given species. We give explicit formulas for nearest-neighbour two-point correlations in terms of Schur functions.

The PushTASEP, although somewhat less widely known than its cousin the TASEP, has nonetheless been widely studied in recent years. Spitzer [spitzer-1970] introduced it as an example of a long-range exclusion process, but more recently the name PushTASEP has become established. It has often been studied on the infinite lattice $\mathbb{Z}$ – see for example [petrov-2020] for extensive references – but there have been a few studies on finite graphs as well [ayyer_schilling_steinberg_thiery.sandpile.2013, ayyer-2016]. The multispecies case was already considered in [ferrari-martin-2006], under the guise of the discrete-space Hammersley–Aldous–Diaconis process, to which it is equivalent by particle-hole duality. A related multispecies process in discrete time (dubbed the “frog model”) was recently used by Bukh–Cox [bukh-cox-2022] to study problems involving the longest common subsequence between a periodic word and a word with i.i.d. uniform entries. For an inhomogeneous version on the line, limit shape and fluctuation results are obtained by Petrov [petrov-2020]. The multispecies inhomogeneous model with $t>0$ was studied on an open interval by Borodin and Wheeler [borodin-wheeler-2022, Section 12.5].

As this project was being completed, related work by Aggarwal, Nicoletti and Petrov appeared [aggarwal-nicoletti-petrov-2023], which studies an inhomogeneous $t$PushTASEP on the ring (in fact, a more general model in which the sites can have capacities greater than $1$). They obtain a description of the stationary distribution of such systems in terms of “queue vertex models” on the cylinder – these objects are closely related to multiline diagrams (and to matrix product formulae). Their methods are very different to ours, making extensive use of the Yang-Baxter equation.

In this section, we focus on the PushTASEP with a single species of particle, i.e. $s=1$. As before, we take $n$ sites. Therefore, $\lambda=\langle 0^{m_{0}},1^{m_{1}}\rangle$ with $m_{1}<n$ particles (and so $m_{0}=n-m_{1}$ vacancies) and periodic boundary conditions.

Recall that $\Omega_{\lambda}$ is the configuration space. For example, consider the case $m_{0}=m_{1}=2$. With the lexicographic ordering of the configurations, i.e.,

the transition graph of the chain is given in Figure 2.

We are interested in the stationary distribution of this process. It is easy to see that the single species PushTASEP is irreducible. Therefore, the stationary distribution is unique. The the following proposition is straightforward to prove using the master equation.

Recall that the density at site $i$ is the probability of finding a particle at site $i$ in the stationary distribution. We will use angular brackets $\langle\cdot\rangle$ to denote expectations in the stationary distribution. As is usual, we let $\eta_{j}$ be the indicator variable for a particle at site $j$, so that $\langle\eta_{j}\rangle$ is the density at site $j$. Because of the factorized nature of the stationary distribution in 4, we immediately obtain the following.

The current of a particle across a given edge (say $(n,1)$) is the number of particles per unit time that cross that edge in stationarity. Because of particle conservation, the current is the same for all edges. We will denote the current by $J$. In terms of the stationary distribution for the PushTASEP, this is given by

We have the following projection property for our multispecies dynamics. If we view all particles of species $r,\dots,s$ as “particles”, and all particles of lower species $1,\dots,r-1$ as vacancies, then we obtain a single species process with $M_{i}=m_{i}+\dots+m_{s}$ particles and $n-M_{i}$ vacancies.

Considering such projects for all $r=1,2,\dots,s$, we can see the multispecies process as a coupling of $s$ single species processes. This is the basic coupling [liggett-1985, Chapter VIII, Section 2] (under which the bells ring at the same sites at the same types in all the coupled single species systems).

More generally, we can project from a “finer” multispecies system to another “coarser” one, allowing the merging of two or more adjacent classes into one:

To map to a single species system as described just above, we would apply the map $\phi^{r}$ with $\phi^{r}(i)=0$ for $i<r$, and $\phi^{r}(i)=1$ for $i\geq r$.

Weber [Weber1979] proved an interchangeability result for exponential queueing servers in tandem. Consider two independent $./M/1$ queueing servers (i.e. servers who offer service at the times of a Poisson process) in tandem, with service rates $\mu_{1}$ and $\mu_{2}$. The first queue has some arrival process $A$, with an arbitrary distribution (for example, it could be deterministic), which is independent of the service processes. By “tandem” we mean that a customer leaving the first queue immediately joins the second queue; the departure process from the first server is the arrival process of the second.

Then Weber’s result is that the law of the departure process from the system (i.e. of the departure process from the second queue) is the same if the rates $\mu_{1}$ and $\mu_{2}$ are interchanged.

Various different proofs of this result were subsequently given; a significant one from our point of view was a coupling proof by Tsoucas and Walrand [TsoucasWalrand1987]. They constructed a coupling of two pairs of independent Poisson processes, $(S_{1},S_{2})$ with rates $\mu_{1},\mu_{2}$ and $({\widetilde{S}}_{1},{\widetilde{S}}_{2})$ with rates $\mu_{2},\mu_{1}$, such that for any arrival process $A$, the output of the system with arrival process $A$ and service processes $(S_{1},S_{2})$ is the same as that of the system with arrival process $A$ and serice processes $({\widetilde{S}}_{1},{\widetilde{S}}_{2})$.

This stronger result, holding simultaneously for all arrival processes, makes it possible to extend to a multispecies framework (since, along the lines we have already seen above, a multispecies configuration can be seen as a coupling of several single species configurations). See for example [MartinPrabhakar2010] for extensive discussion and applications.

In this section we develop similar ideas in a new context, to obtain interchangeability of rates of PushTASEP stations. We will ultimately be able to apply them to get the result of 2.

We consider the PushTASEP on the ring for this work, but the result below applies equally to any one-dimensional lattice (such as a closed or open interval or all of $\mathbb{Z}$). Each site $j$ has an associated parameter $x_{j}$. Bells ring independently as Poisson processes at each site, with rate $1/x_{j}$ at site $j$. Recall that when a bell rings at site $j$ in the single species PushTASEP:

• •

  if $j$ is empty, nothing happens.

• •

  if $j$ is occupied, then:

  • –

    $j$ becomes empty;

  • –

    the first empty site $k$ to the right of $j$ becomes occupied;

  which we call a transfer of a particle from $j$ to $k$.

For each lattice bond $(j,j+1)$, we have a “flux process” across the bond; a point process $F_{j,j+1}$ which records the times when there is a transfer of a particle from $j$ or a site to its left to $j+1$ or a site to its right.

Fix a site $j$. Consider the system started from, say, time $0$. If we know the flux process $F_{j-1,j}$ for the bond on the left of $j$, the initial occupancy of the site $j$, and the Poisson process of bells at site $j$, then we can obtain the flux process $F_{j,j+1}$ for the bond on the right of $j$.

Iterating, this now works for any finite interval $j,j+1,\dots,j+k$, where $k\geq 0$. Given the flux process $F_{j-1,j}$ at the left end of the interval, the initial configuration inside the interval, and the bell processes inside the interval, one can obtain the flux process $F_{j+k,j+k+1}$ at the right end. For the moment we only need the case $k=1$.

We write $\mathrm{PP}(\rho)$ to denote a Poisson process with rate $\rho$. This is our interchangeability result.

See Figure 3 for an illustration. This theorem says that the bell rates $1/x_{j}$ and $1/x_{j+1}$ at the neighbouring sites $j$ and $j+1$ are interchangeable.

Before we begin the proof, we describe the coupling. First, we will use $\rho=1/x_{j}$ and $\rho^{\prime}=1/x_{j+1}$ in the rest of this section to keep the notation simple. Recall that a basic fact about Poisson process is the thinning theorem [durrett-2016], which in our context reads as follows. We can describe $(A,B)\sim\mathrm{PP}(\rho)\otimes\mathrm{PP}(\rho^{\prime})$ via a single Poisson process of rate $\rho+\rho^{\prime}$ which is the superposition of $A$ and $B$, with a mark $a$ or $b$ attached to each point, according to whether it comes from the process $A$ or $B$ respectively. Given the points of $A+B$, the marks are i.i.d. and each is $a$ with probability $\rho/(\rho+\rho^{\prime})$ and $b$ with probability $\rho^{\prime}/(\rho+\rho^{\prime})$. If we index the points of $A+B$ with $\mathbb{Z}$ in increasing order, say with point $0$ the last to occur before time $0$ and point $1$ the first to occur after time $0$, then we can identify the sequence of marks with an element $(w_{i},i\in\mathbb{Z})\in\{a,b\}^{\mathbb{Z}}$.

Under our coupling, the superposition ${\widetilde{A}}+{\widetilde{B}}$ will be the same as $A+B$; we will just change (some of) the marks. Consider occurrences of the motif $ab$, that is, look for $i$ such that $w_{i}=a$, $w_{i+1}=b$. The set $\mathcal{C}$ of such $i$ has a distribution which is invariant under interchanging $\rho$ and $\rho^{\prime}$. This is because if $i_{1}<i_{2}<\dots<i_{n}$, then $P(i_{1},\dots,i_{n}\in\mathcal{C})$ is equal to $0$ if $i_{k}-i_{k-1}=1$ for some $k$, and otherwise is equal to $\rho^{n}(\rho^{\prime})^{n}/(\rho+\rho^{\prime})^{2n}$. This means there exists a coupling of $(A,B)$ and $({\widetilde{A}},{\widetilde{B}})$ with the required distributions such that:

• •

  the superpositions $A+B$ and ${\widetilde{A}}+{\widetilde{B}}$ are the same;

• •

  the occurrences of the motif $ab$ are the same in both processes.

These are the only properties we will need, but we can make the coupling explicit as follows. The $ab$ motifs are separated by words of the form $b^{*}a^{*}$, consisting of some number (maybe $0$) of $b$’s followed by some number (maybe $0$) of $a$’s. Conditional on the length $n$ of the string, the probability that it consists of $n_{1}$ $b$’s followed by $n_{2}$ $a$’s (where $0\leq n_{1},n_{2}$, with $n_{1}+n_{2}=n$) is proportional to

An explicit way to realise the coupling is to obtain the process $({\widetilde{A}},{\widetilde{B}})$ from $(A,B)$ as follows. Leave the $ab$ motifs unchanged. As for the $b^{n_{b}}a^{n_{a}}$ separating them, replace it instead by $b^{n_{a}}a^{n_{b}}$. This is illustrated in Figure 4. Because of (3.1), this has the effect of interchanging the probability of $a$ and $b$ as desired.

The fact that our coupling of $(A,B)$ and $({\widetilde{A}},{\widetilde{B}})$ works simultaneously over all input processes $L$ means that we can pass from a single species result to a multispecies result.

In this section we discuss how the strategy of Ferrari and Martin [ferrari-martin-2006] adapts to give the result of 1. Theorem 4 of [ferrari-martin-2006] refers to a multispecies version of the Hammersley-Aldous-Diaconis process – see the comment at the end of Section 2 of that paper for the relationship between the HAD and the PushTASEP (or “long-range exclusion process”) via reversing the order of the particles. The main novelty in our setting is the introduction of site-wise inhomogeneity via the parameters $x_{1},\dots,x_{n}$.

Here is the outline of the strategy for studying the stationary distribution of the multispecies process on $\Omega_{\lambda}$, where the partition $\lambda$ gives the contents on the system.

• •

  We consider a set ${\widehat{\Omega}}_{\lambda}$ of multiline diagrams and a map $\Pi$ from ${\widehat{\Omega}}_{\lambda}$ to the set $\Omega_{\lambda}$ of PushTASEP configurations.

• •

  We construct a Markov chain on the set of multiline diagrams (the multiline process) and find its stationary distribution using a time-reversal argument (LABEL:thm:stat_dist_multiline). In the homogeneous case, this stationary distribution was simply the uniform distribution; now that we introduce site-wise inhomogeneity, we get a distribution with weights proportional to monomials in the parameters $x_{1},\dots,x_{n}$.

• •

  We show that the projection of the multiline process under the map $\Pi$ is the multispecies PushTASEP (LABEL:lem:projection). From this we deduce the stationary distribution $\pi$ of the multispecies PushTASEP, with weights proportional to sums of monomials in the parameters $x_{1},\dots,x_{n}$ (LABEL:thm:stat_prob_sum).

To obtain the form of the stationary distribution given in 1, we finally apply the correspondence between ASEP polynomials and weights of multiline diagrams given by [corteel-mandelshtam-williams-2022].

We concentrate particularly on the argument for LABEL:thm:stat_dist_multiline, where the inhomgoneity plays a significant role. In contrast, in the argument for LABEL:lem:projection, the rates are irrelevant, and we outline the idea briefly.

Before we move on to the proof, we state important properties of the multispecies PushTASEP. First, one can see that it is irreducible if all $x_{i}$’s are positive and finite by the following argument. Suppose $\eta,\eta^{\prime}\in\Omega_{\lambda}$. Starting from $\eta$, initiate transitions starting at the locations of species $s$’s until they are in their correct positions in $\eta^{\prime}$. Now, initiate transitions starting at the $(s-1)$’s. These will clearly not affect the $s$’s. Continue this way until all particles are in their correct location in $\eta^{\prime}$.