The TASEP speed process\thanksrefT1

# The TASEP speed process\thanksrefT1

[ [[    [ [[    [ [[ Bar Ilan University, University of British Columbia and University of Wisconsin G. Amir
Department of Mathematics
Bar Ilan University
52900 Ramat Gan
Israel
O. Angel
Department of Mathematics
University of British Columbia
Vancouver BC, V6T 1Z2
B. Valkó
Department of Mathematics
480 Lincoln Dr.
USA
\smonth8 \syear2009\smonth4 \syear2010
\smonth8 \syear2009\smonth4 \syear2010
\smonth8 \syear2009\smonth4 \syear2010
###### Abstract

In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order. Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on . We study the joint distribution of these speeds: the TASEP speed process.

We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes.

By combining this result with known stationary measures forTASEPs with finitely many types, we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed.

We also study the partially asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations of , are symmetric in distribution. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.

[
\kwd
\doi

10.1214/10-AOP561 \volume39 \issue4 2011 \firstpage1205 \lastpage1242 \newproclaimexample[theorem]Example \newproclaimfact[theorem]Fact \newproclaimdefn[theorem]Definition

\runtitle

The TASEP speed process

\thankstext

T1Research was carried out while all authors were at the University of Toronto. Supported in part by NSERC.

{aug}

A]\fnmsGideon \snmAmir\correflabel=e1]amirgi@math.biu.ac.illabel=u1,url]http://u.math.biu.ac.il/~amirgi/, B]\fnmsOmer \snmAngellabel=e2]angel@math.ubc.calabel=u2,url]http://www.math.ubc.ca/~angel and C]\fnmsBenedek \snmValkólabel=e3]valko@math.wisc.edulabel=u3,url]http://www.math.wisc.edu/~valko

class=AMS] \kwd[Primary ]82C22 \kwd[; secondary ]60K35 \kwd60K25. Exclusion process \kwdTASEP \kwdASEP \kwdmulti-type \kwdsecond class \kwdstationary measure.

## 1 Introduction

The exclusion process on a graph describes a system of particles performing continuous time random walks, interacting with other particles via exclusion: attempted jumps to occupied sites are suppressed. When the graph is and particles jump only to the right at rate one the process is called the totally asymmetric simple exclusion process (TASEP). We denote configurations with where particles are denoted by and empty sites by .222The common practice is to denote empty sites by . However, under various common extensions of the TASEP including those used here, it is convenient to denote empty sites by a label larger than the labels of all particles. The TASEP is a Markov process with generator

 Lf(η)=∑nf(σnη)−f(η), (1)

where is the operation that sorts the coordinates at in decreasing order

 (σnη)n = max(ηn,ηn+1),(σnη)n+1=min(ηn,ηn+1), (2) (σnη)k = ηkif k∉n,n+1.

A second class particle is an extra particle in the system trying to perform the same random walk while being treated by the normal (first class) particles as an empty site. It is an intermediate state between a particle and an empty site, and is denoted by a .333th class particles will be denoted by , even for . That is why it is convenient to use for holes rather than . This means that the second class particle will jump to the left if there is a first class particle there who decides to jump onto the second class particle. This is still a Markov process, with the same generator (1) and state space . Note that empty sites can just be considered as particles with the highest possible class. Thus we can equally well consider state space with holes represented by ’s.

More generally, we shall consider the multi-type TASEP which has the same generator with state space . Thus we allow particle classes to be nonintegers or negative numbers. If there are particles with maximal class they can be considered to be holes. A special case is the -type TASEP (without holes) where all particles have classes in . If particles of class are interpreted as holes instead of maximally classed particles, this process becomes the traditional -type TASEP (with holes). To avoid confusion, from here on all multi-type configurations shall be without holes. (Holes will appear only in individual lines in the multi-line configurations defined below.)

The following result is this paper’s foundation. We let denote the TASEP configuration at time , with the value at position . This strengthens results of Ferrari and Kipnis FerrariKipnis () that get the same limit in distribution.

###### Theorem 1.1 ((Mountford and Guiol MountfordGuiol ()))

Consider the TASEP with initial condition

Let denote the position of the second class particle at time , defined by . Then , where is a uniform random variable on .

Thus a second class particle with first class particles to its left and third class particles to its right “chooses” a speed , uniform in and follows that speed: . (See FP (), FMP () for alternative proofs of Theorem 1.1.)

Now, consider any other starting configuration such that for all and for all . The particle starting at 0 does not distinguish between higher classes, or between lower classes, so its trajectory has the same law. This applies in particular to every particle in a multi-type TASEP with starting configuration . Let be the location of particle at time , so that [ is the inverse permutation of ]. An immediate consequence is the following:

###### Corollary 1.2 ((The speed process))

In the TASEP with starting configuration , a.s. every particle has a speed: for every

 Xn(t)−nta.s.⟶t→∞Un,

where is a family of random variables, each uniform on .

{defn}

The process is called the TASEP speed process. Its distribution is denoted by .

Thus is a measure supported on . It is clear from simulations (and our results below) that is not a product measure, that is, that the speeds are not independent. Figure 1 shows

a portion of the process. Some aspects of this process were studied in FGM ().

### 1.1 Main results

In order to study the TASEP speed process we prove two results, which are our main tools in understanding the joint distributions of speeds. These results are of significant interest in and of themselves. The following is a new and surprising symmetry of the TASEP. A version of this theorem was proved in DPRW (), in the context of the TASEP on finite intervals. We extend it here also to the ASEP444Some sources use PASEP/ASEP, respectively, for what other sources call ASEP/TASEP (PASEP stands for partially). We adopt the latter convention. (defined in Section 1.3).

###### Theorem 1.3

Consider the starting configuration and as above. For any fixed the process has the same distribution as . This holds also for the ASEP.

At any time we have that and are permutations of , one the inverse of the other. Thus this theorem implies that as a permutation has the same law as its inverse. It is not hard to see that this holds only for a fixed , and not as processes in [e.g., changes by at most 1 at each jump].

The next result gives additional motivation for considering the speed process, as it relates its law to stationary measures of the multi-type TASEP (and ASEP).

###### Theorem 1.4

is itself a stationary measure for the TASEP: the unique ergodic stationary measure which has marginals uniform on .

This means that if we consider a TASEP in where the initial configuration has distribution then at any time the distribution of is also given by .

It is known that the -type process has ergodic stationary measures, and that the distribution of among the classes determines this distribution uniquely. Standard techniques (see below) can be used to show that the same holds also with infinitely many classes. Specifically, for any distribution on there is a unique ergodic stationary measure for the TASEP with (and any ) having that distribution. For any two nonatomic distributions on , these measures are related by applying pointwise a nondecreasing function to the particle classes (see Lemma 5.3), so every such measure can be deduced from the measure with marginals uniform on . If a distribution has atoms, then the corresponding stationary measure can still be deduced from the speed process’ law in the same way, but the operation is nonreversible. Thus we have the following characterization:

###### Corollary 1.5

Every ergodic stationary measure for the TASEP can be deduced from by taking the law of for some nondecreasing function .

### 1.2 Results: Joint distribution

Computer simulations suggested early on that are not independent (see Figure 2). Recent results of Ferrari, Goncalves and Martin FGM () confirm this prediction. They proved (among other things)

that the probability that particle eventually overtakes particle (we identify a particle with its class) is . It follows that (not necessarily equal since does not a priori imply overtaking). Our first theorem describing the joint distribution of speeds is the following:

###### Theorem 1.6

The joint distribution of , supported on , is

 f(x,y)dxdy+g(x)\mathbh1(x=y)dx

with

In particular, , and .

#### Remarks

Note that the density in (linear in , so that there is repulsion between the speeds) can be deduced using only Theorem 1.3 (we do not include this argument here). However, proving the—seemingly simpler—constant density on and deriving the singular component on the diagonal requires the power of Theorem 1.4. It is interesting to compare the power of Theorem 1.3 with that of the methods of FGM (). It appears that both methods run into similar difficulties and have similar consequences, suggesting a fundamental connection (there are also some parallels in the proofs). Specifically, can the density in the region be derived using the techniques of FMP ()? Finally, it is interesting that our proof relies nontrivially on the extension of the TASEP to infinitely many different classes of particles, though the question and answer can both be posed using only classes (including holes). A similar remark holds about some other results below as well.

Additional information about the joint distribution of speeds is derived in Section 7. We derive certain properties of the -dimensional marginals of , and in Theorem 7.7 we compute the joint distribution of three consecutive speeds.

A surprising aspect of Theorem 1.6 is that there is a positive probability () that , even though each is uniform on . Indeed, for any two particles there is a positive probability that their speeds are equal. This phenomenon can be thought of as a spontaneous formation of “convoys,” sets of particles that have the same asymptotic speed, so their trajectories remain close. Our next result gives a full description of such a convoy.

###### Theorem 1.7

Let the convoy of be , that is, the set of all with the same speed as . Then is -a.s. infinite with density. Moreover, conditioned on , is a renewal process, and the nonnegative elements of have the same law as the times of last increase of a random walk conditioned to remain positive, with step distribution

 P(X=1)=P(X=−1)=1−U204,P(X=0)=1+U202.

The “times of last increase” of a walk are those indices for which implies . In particular the convoys are infinite and they provide a translation invariant partition of the integers into infinitely many infinite sets. The convoys are essentially the process with density for second class particles, seen from a second class particle, as studied by Ferrari, Fontes and Kohayakawa in FFK ().

### 1.3 The ASEP

As the name suggests, the totally asymmetric simple exclusion process is an extremal case of the asymmetric simple exclusion process: the ASEP. The ASEP is defined in terms of a parameter , with being the TASEP. While most quantities involved depend on , the dependence will usually be implicit.

In the ASEP particles jump one site to the right at rate and to the left at rate (we use the convention ). The generator of this Markov process is

 Lf(η)=∑np(f(σnη)−f(η))+¯¯¯p(f(σ∗nη)−f(η)), (3)

where and sort the values in in decreasing and increasing order, respectively.

While some of the questions above make sense also in this setting, there is a key difficulty in that the analogue of Theorem 1.1 for the ASEP (conjectured below) is still unproved. Using the methods of Ferrari and Kipnis FerrariKipnis () it can be proved that converges in distribution to a random variable uniform in , where hereafter we denote . Note that the particles in the exclusion process try to perform a random walk with drift (and they cannot go faster than that), that explains why the support of the limiting random variable is changed. In fact, in many ways the ASEP behaves similarly to the TASEP slowed down by a factor of .

###### Conjecture 1.8

In the ASEP, exists a.s. (and the limit is uniform on ).

By the discussion preceding Corollary 1.2 this is equivalent to the following:

###### Conjecture 1.9

The ASEP speed process measure is well defined and translation invariant with each uniform on .

In order for statements about the ASEP speed process to make sense we must assume this conjecture, and therefore some of our theorems are conditional on Conjecture 1.8. It should be noted that with minor modifications our results also hold assuming a weaker assumption, namely a joint limit in distribution of the speeds . In that case, the speed process measure is still defined, even though the particles may not actually have an asymptotic speed.

As noted there, Theorem 1.3 holds also for the ASEP, with no additional condition. Theorem 1.4 becomes conditional:

###### Theorem 1.10

Assume Conjecture 1.8 holds. Then is a stationary measure for the ASEP: the unique ergodic stationary measure which has marginals uniform on .

As in the case of the TASEP, this can be interpreted as follows: if an ASEP is started with initial configuration in with distribution , then at any time the distribution of the process is also given by . Note that both the dynamics and the measure depend implicitly on the asymmetry parameter .

A useful tool in studying the speed process is the understanding of the stationary measures of the type TASEP in terms of a multi-line process described below, developed by Angel Angel () and Ferrari and Martin FerrariMartin (). There is no known analogue for these results that describes the stationary measure of the multi-type ASEP. Thus we need to use other (and weaker) techniques to extract information about the marginals of the ASEP speed process. This explains the contrast in the level of detail between the following results and the corresponding theorems above about the TASEP.

###### Theorem 1.11

We have the following limit:

 limt→∞P(X0(t)

Theorem 2.3 of FGM () proves that the probability that particles and interact at least once (i.e., one of them tries to jump onto the other) is . In the next section we will show that this is equivalent to the just stated theorem.

Our next theorem provides information about the joint distribution of , assuming Conjecture 1.8 holds.

###### Theorem 1.12

Assume Conjecture 1.8 holds. Let the measure on be the marginal of under . Denote by the reflection of about the line . Then on we have

 p⋅μ(2)−¯¯¯p⋅~μ(2)=y−x4ρ2dxdy.

We finish this section with a statement concerning the case . Consider the total amount of time that particles and spend next to each other, that is, .

###### Theorem 1.13

In the TASEP, if and only if . If Conjecture 1.8 holds, then the same holds for the ASEP.

In the TASEP implies that there is at least one interaction between 0 and 1 which means that they are a.s. swapped. (See the next section for a more detailed discussion.) Thus if , then eventually . In fact, this holds for any two particles in the same convoy: in Lemma 9.9 we will prove that in the TASEP, particle 0 will eventually overtake all the particles in its convoy with positive index.

### 1.4 Overview of the paper

The rest of the paper is organized as follows. Section 2 provides some of the background: constructions of the processes and the multi-line description of the stationary measure for the multi-type TASEP. Section 3 includes the proof of the symmetry property (Theorem 1.3) and Section 4 proves the stationarity of the speed process (Theorems 1.4 and 1.10). Sections 6 and 7 include the results about various finite-dimensional marginals of the TASEP speed process. Section 8 deals with the proof of Theorem 1.7. Finally, in Section 9 we prove our results on the ASEP speed process.

## 2 Preliminaries

### 2.1 Construction of the process

There are several formal constructions of the TASEP and ASEP. The one that best suits our needs seems to be Harris’s approach Liggett (). We include the construction since there are several variations and the exact details are used in some of our proofs. The process is a function defined on . will denote the class of the particle at position at time . The configuration at time is . The classes of particles will be real numbers, hence the configuration at any given time is in . Setting gives the initial configuration .

We define the transposition operator , acting on by exchanging and , while keeping all other classes equal. Using this we can alternately describe the sorting operator by

Thus has the effect of sorting in decreasing order, keeping other classes the same.

The TASEP is defined using the initial configuration and the location of “jump” points. The probability space contains a standard Poisson process on , that is, a collection of independent standard Poisson processes on , denoted . If is a point of , then at time the values of and may be switched. In the TASEP they are sorted, that is, . This can be described as applying each of the operators at rate 1 independently. A simple percolation argument shows that this dynamic is a.s. well defined. (For any fixed there are a.s. infinitely many integers so that there are no Poisson points on which means that to define the process up to time it suffices to consider finite lattices.)

#### The ASEP

Defining the partially asymmetric exclusion process requires additional randomness. Given the parameter , we attach to each point in the Poisson process an independent Bernoulli random variable with . We can now define the probabilistic sorting operator as follows:

Thus with probability the smaller classed particle is moved to the right position and with probability it is moved to the left position. When such an event happens we say that and have an interaction (regardless of whether they were actually swapped). Note that if particles interact in this way, then their order after the swap is independent of the order before the swap. The key observation is that after interact in this way at least once, has probability of being to the right of , and this is unchanged by further interactions. Moreover, if we condition on (the total time spend next to each other until time ), then

 P(Xi(t)

where the expression on the right is just the probability that there were no interaction between and until time plus the probability that there was some interaction, and at time particle is to the left of . One of the consequences of (4) is that

 limt→∞P(Xi(t)

Thus Theorem 1.11 implies which gives . But is exactly the probability that there is at least one interaction between and 1 which shows why Theorem 2.3 of FGM () and our Theorem 1.11 are equivalent.

In the TASEP case if there is an interaction between , then after that. Thus in that case from (5) we get

 P(eventually Xi(t)>Xj(t)|Ji,j=∞)=1,

which explains the remark after Theorem 1.13.

There is an alternate construction for the ASEP, which will be used in Section 3. Consider a Poisson process with lower intensity on , but whenever it has a point we apply at time the operator rather then , where is defined by

 πnY=⎧⎨⎩τnY,\quadYnYn+1 with prob. q=(1−p)/p,Y,\quadYn>Yn+1 with prob. ¯q=(2p−1)/p.

Thus if the pair is in increasing order it is always swapped, while if it is in decreasing order it is swapped only with probability . It is easy to see that every possible swap occurs at the same rate in the two constructions; hence the resulting processes have the same generator.

### 2.2 Stationary measures for the multi-type TASEP

The following theorem can be proved by standard coupling methods (see, e.g., Liggettcouple () where the same theorem is proved for the 2-type TASEP).

###### Theorem 2.1

Fix every with . There is a unique ergodic stationary distribution for the -type TASEP with . The measures are the extremal stationary translation invariant measures. They are the only stationary translation invariant measures with the property that for each , the distribution of is product Bernoulli measure with density .

For the ordinary TASEP (with particles and holes) this stationary distribution is just the product Bernoulli with a fixed density. If we have an -type TASEP then the structure of the stationary distribution is more complicated. The first description of for was given by the matrix method DJLS (). Reference FFK () gave probabilistic interpretations and proofs of the measure and its properties. Recently combinatorial descriptions of have appeared as well. The -type TASEP was treated by Angel Angel () (see also Duchi–Schaeffer DuchiSchaeffer ()). These results were extended for all by Ferrari and Martin FerrariMartin (). They give an elegant construction of using systems of queues.

We will now briefly describe the -line description of for the -type TASEP. The two-line case suffices for most of our results, with the exception of the results of Section 7. For a more detailed description and proofs see FerrariMartin ().

From here on we shall fix the parameters . Consider independent Bernoulli processes on denoted where has parameter (these are the lines). From these lines we construct a system of coupled queues. The lines give the service time of the queues, and the departures from each queue are the arrivals to the next queue.

It is important to observe that the time for the queues goes from right to left, that is, is followed by and so on. The resulting system of queues is positively recurrent, so it can be defined starting at and going over the lines toward .

The th queue will consist of the particles that departed from the th line and are waiting for a service in . This queue will consist of particles of classes . When a service is available in the lowest classed particle in the th queue is served and departs (to the next queue). If the queue is empty then a particle of class is said to depart the queue. The departure process of each queue (i.e., the times and sequence of classes of departing customers) is the arrival process for the next queue.

It is convenient to think of an additional queue with as its service times. This queue has no arrivals (so it is always empty). The unused services introduce first class particles, which join the second queue whenever there is a service in . These operations are evaluated for each from line to line in order. Let be the number of particles of type in the th queue after column of the multi-line process has been used.

Note that each queue has a higher rate of service than of arrivals, so the queues sizes are tight, and the state with all queues empty is positively recurrent. In practice, the th queue has types of particles in it, so the whole system of queues is described by nonnegative integers.

###### Theorem 2.2 ((Ferrari–Martin))

is the distribution of the departure process of , with class (or empty sites) at those when there is no service.

As an example, and to clarify the graphic representation we use later, consider the following segment of a configuration of the three-line process for . Suppose both queues are empty at time 5. (This is denoted by the exponent.) Here, \CIW denotes a in the corresponding line, and \CIB a 1. Later, in cases where we do not care about a specific value we may use to denote that

At time , reading the rightmost column from top to bottom, there is no service in , so no first class particle joins the second queue, which therefore remains empty. There is a service in , and no particles in the first queue, so a second class particle joins the second queue. There is service in , so the second class particle departs immediately. Thus at time the queue states are .

At time a first class particle arrives to the first queue, and stays there since there is no service in the second queue. There is no further service in column , so the state at time is . There is no departure, which is denoted by a (or hole). At time another first class particle arrives, and there is no particle in the second queue so the service in gives rise to a third class particle departing. The states are now . Finally, at time a first class particle is served at both and , departing and leaving queue states . The resulting segment of is .

## 3 Symmetry

Recall the operators defined above. These act randomly on configurations, and the ASEP can be defined by applying each of the Markov operators at rate .

Formally, is defined as acting on : probability measures on . Given a measure on , we let be the distribution of applied to a sample from . Since and also act naturally on the measures (in the same way), one finds the operator relation

 πi=qτi+¯¯¯qσi.

Note that gives so in that case . In the case we get and , so the process reduces to a symmetric random walk on .

The crucial observation leading to Theorem 1.3 is the following lemma.

###### Lemma 3.1

Fix any , and sequence . Then

 πin⋯πi1⋅id\lx@stackreld=(πi1⋯πin⋅id)−1. (6)

That is, applying a sequence of ’s in the reverse order to the identity leads to the inverse permutation. This is trivially true when and , but requires proof for other . When the operator is deterministic and this distributional identity is an equality of permutations. {pf*}Proof of Theorem 1.3 The theorem follows from Lemma 3.1 since at any finite time at each there is positive probability () that no swap has occurred. Each such separates into two parts with independent behavior, so the state of the process is a product of finite, mutually commuting permutations. The distribution of the sequence of applied operators between such inactive locations is symmetric in time.

We now prove Lemma 3.1. In the case of the TASEP, Lemma 3.1 and Theorem 1.3 were first proved in DPRW (). To prove the lemma in the general case, we start with the following facts about the transposition operators. The identities are readily verified, and the last claim is known as Matsumoto’s lemma (see, e.g., coxetergroups (), Theorem 3.3.1). {fact} The operators satisfy the relations

 τ2i = I, (7) τiτj = τjτifor |i−j|>1, (8) τiτi+1τi = τi+1τiτi+1, (9)

where denotes the identity operator. With these relations the operators generate the symmetric group. Furthermore, it is possible to pass between any two minimal words of the same permutation (i.e., words of minimal length representing that permutation) using only relations (8), (9).

The ’s satisfy similar relations:

###### Lemma 3.2

The operators satisfy the relations

 π2i = qI+¯¯¯qπi, (10) πiπj = πjπifor |i−j|>1, (11) πiπi+1πi = πi+1πiπi+1. (12)

Note that only the first relation differs from the corresponding relation for . When these reduce to the relations for . In the case the first relation becomes . In that case, the only nontrivial relation is (12) which is true since both sides have the effect of sorting the three terms involved in decreasing order. {pf*}Proof of Lemma 3.2 Equation (10) is easy to check, and (11) is trivial. For (12), using and expanding, we need to show that

 q3(τiτi+1τi)+q2¯¯¯q(τiτi+1σi+τiσi+1τi+σiτi+1τi) +q¯¯¯q2(τiσi+1σi+σiτi+1σi+σiσi+1τi)+¯¯¯q3(σiσi+1σi)

is unchanged by exchanging and . It is easy to verify that

 τiτi+1τi = τi+1τiτi+1,τiτi+1σi=σi+1τiτi+1,τiσi+1τi=τi+1σiτi+1, σiσi+1σi = σi+1σiσi+1,σiτi+1τi=τi+1τiσi+1,

so it remains to show

 τiσi+1σi+σiτi+1σi+σiσi+1τi=τi+1σiσi+1+σi+1τiσi+1+σi+1σiτi+1.

We may assume . Since only the relative order of matters, we may assume these are in some order. Applying these operators to the possible orders gives Table 1.

In each column, the entries in the top half are a permutation of the entries in the bottom half, so adding the first three operators gives the same result as adding the last three. {pf*}Proof of Lemma 3.1 Given , let . If this is a minimal (w.r.t. length) word for in , then with probability 1. In this case, the reverse word is minimal for , so the claim holds.

The proof proceeds by induction on . Take some sequence . If the representation is minimal, then the claimed identity holds. Otherwise, let be maximal such that is a minimal representation. By maximality of we see that has a shorter representation, so there is a representation . (The length is and not since its parity is opposite that of .) Thus is another minimal representation of .

Starting with , we can repeatedly apply relations (11) and (12) to the first terms in the product, to get

 πi1⋯πin=πj1⋯πjk−1π2ik+1⋯πin.

Here appears twice since it is both the last term in the alternate representation of and the first in the remainder of the sequence. Relation (10) now gives

 πi1⋯πin=q(πj1⋯πjk−1πik+2⋯πin)+¯¯¯q(πj1⋯πjk−1πik+1⋯πin). (13)

Similarly, working with the reverse sequence,

 πin⋯πi1=q(πin⋯πik+2πjk−1⋯πj1)+¯¯¯q(πin⋯πik+1πjk−1⋯πj1). (14)

Applying (14) and (13) to to , and using the induction hypothesis for the shorter sequences and completes the proof.

Note: the proof actually shows that any word in the ’s can be reduced (as an operator) to some convex combination of words corresponding to minimal words.

###### Corollary 3.3

Consider the infinite type TASEP with initial condition . Then converges weakly to as .

{pf}

For any this process has the same law as , which converges a.s. to a process with law .

## 4 Stationarity

We will give two different proofs of the stationarity of the distribution of the speed process. The first is specific to the TASEP, and is reminiscent of coupling from the past. It uses the Harris construction directly. The second proof is based on the symmetry between and (or more specifically Corollary 3.3). The second proof holds also for the ASEP, word by word, under the assumption that Corollary 3.3 is true for the ASEP (which is weaker then Conjecture 1.8).

### 4.1 Coupling proof

###### Lemma 4.1

Consider two TASEPs defined via the Harris construction as the function of the same Poisson process on . We set the initial conditions as and (i.e., particles 0 and 1 are switched initially in ). Let denote the speed process of , and denote the speed process of . Then .

{pf}

All particles other than are either larger or smaller than both and , so any swaps involving a particle other than will occur or not occur equally in and . It follows that for any we have and hence . Similarly, since 0 and 1 must fill the only vacant trajectories, as an unordered pair.

In particle is always to the right of particle , so and , completing the proof. {pf*}Proof of Theorem 1.4 using coupling Consider a Poisson process on . Half of the process, namely the restriction to is used in the Harris construction of the TASEP. Similarly, for any we can translate the Poisson process by [i.e., take all points of the form where is in the original process], and take the restriction to , which can be used in the Harris construction to get a different (though highly dependent) instance of the TASEP.

Let be the speed process resulting from the Harris construction using the translated Poisson process. Clearly for every , has the same law , so we are done if we show that evolves as a TASEP (with time parameter ). Consider the effect of an infinitesimal positive shift . The shift adds new operations, to be applied before the original sequence of operations. These are added at rate 1 at each location. By the previous lemma, the effect on the resulting speeds of applying before using the same Poisson process is to apply to the speeds, which is exactly what we need.

It is interesting to note that in the Poisson process , the part on is used to determine the “initial” speed process , and the restriction to is used exactly as in the Harris construction to generate the TASEP dynamics of .

### 4.2 Symmetry based proof

{pf*}

Proof of Theorems 1.4 and 1.10 using symmetry We write the proof for , but it holds verbatim for under Conjecture 1.8.

Informally, we argue as follows. Fix and let . Both and converge a.s. to a sample of . By Theorem 1.3 these have the same law as and , so for large both of these have law close to . However, the result of letting evolve for an additional time is , which is close to .

Let be the evolution operator for the Markov process corresponding to the generator on [see (1)]. To prove stationarity it is enough to show that for every and every bounded continuous local function we have

 ∫Psf(η)dμ(η)=∫f(η)dμ(η). (15)

Consider the process started from and denote the distribution of by . By Corollary 3.3 the weak limit of is which means that for every local bounded continuous function we have

 ∫f(η)dνt(η)⟶t→∞∫f(η)dμ(η).

For any fixed

 ∫Psf(η)dνt(η)⟶t→∞∫Psf(η)dμ(η).

But which (for any fixed , as ) converges to . Now (15) and the theorem follow.

## 5 Basic properties of stationary distributions

In this section we present a medley of simple results concerning the (T)ASEP and its stationary distributions. These are only weakly related to each other, and are collected here for convenience.

###### Proposition 5.1

is ergodic for the shift. Under Conjecture 1.8, so is .

{pf}

Consider the setup of Corollary 1.2 and use the Harris construction with independent standard Poisson processes on the interval to define and the variables . Then the limit process is measurable with respect to the -algebra generated by the i.i.d. processes (). Since is generated by i.i.d. processes any translation invariant event in has to be trivial. But then the same thing must be true for any translation invariant event in the -algebra generated by as this is a sub--algebra of .

There are three possible “reflections” for the ASEP. One may reverse the direction of space, so that (low classed) particles flow to the left and not right; one can consider the time reversal of the dynamics, and one can reverse the order of classes (or keep the same generator but replace class with , or , etc.). It is easy to see that reversal of both space and class order preserves the original dynamics. This is called the space-class symmetry of the TASEP/ASEP.

The following proposition is the space-class symmetry of the speed process, and follows directly from the corresponding symmetry of the ASEP process.

###### Proposition 5.2

For the TASEP . This also holds for the ASEP, assuming Conjecture 1.8 holds.

The following observation and its corollary provide an important connection between the distribution of the speed process and the stationary measures of multi-type ASEP. These connections will be used to extract information on the joint distribution of the speeds of several particles in Sections 6 and 7.

###### Lemma 5.3

Let be an ASEP, and let be a nondecreasing function. Then is also an ASEP (with the same asymmetry parameter).

{pf}

The ASEP is defined as applying to each of the operators independently at rate . Applying a nondecreasing function to each coordinate commutes with every , hence is just the ASEP with initial configuration .

###### Corollary 5.4

If is nondecreasing, then for the TASEP the distribution of is the unique ergodic stationary measure of the multi-type TASEP with types and densities .

This also holds for the ASEP (and its corresponding multi-type stationary measure) under Conjecture 1.8.

{pf}

Let denote the distribution of . Since is ergodic, so is . The marginals are as claimed since each is uniform on .

To prove that is stationary, start a TASEP with initial configuration . By Lemma 5.3 is a -type TASEP. Since is stationary, also has law , and so , hence is also stationary.

The result for the ASEP follows the same way.

The next proposition shows that a TASEP started with uniform i.i.d. classes must converge to the speed process. In particular, even though classes in the i.i.d. initial distribution are a.s. all different, the process converges to the speed process which has infinite convoys of particles with the same class (see Section 8). Thus the TASEP dynamics has the effect of aggregating particles with increasingly closer speeds next to each other.

###### Proposition 5.5

Consider a TASEP where are i.i.d. uniform on . Then converges weakly to . The same holds for the ASEP under Conjecture 1.8

{pf}

Let be the distribution of for the process of the lemma. We need to show that for any fixed bounded and continuous function .

If we start the -type TASEP with an i.i.d. product measure initial distribution then its distribution converges to an ergodic stationary measure with the same one-dimensional marginal. (This can be shown by standard coupling arguments introduced by Liggett; see, e.g., Liggettcouple () or Liggett (), Chapter 8.)

Using Lemma 5.3 and Corollary 1.5 it follows that for any nondecreasing step function on the process converges in distribution to .

For an integer let , which maps to , . Define the operator on configurations, as the operator that applies to each coordinate: . Since is continuous we can select such that . By the triangle inequality we have

 ∣∣∣∫gdνt−∫gdμ∣∣∣≤2ε+∣∣∣∫g∘F⊗Mdνt−∫g∘F⊗Mdμ∣∣∣,

and is applied to a TASEP with finitely many types, so it can be made smaller than by taking large enough.

## 6 Two-dimensional marginals of the TASEP speed process

The key tool for analyzing the joint densities of the speed process is Corollary 5.4. This states that if the speed process is monotonously projected into , then the result is the stationary measure of the multi-type TASEP with suitable densities. In the TASEP, the latter is given in terms of the multi-line process (see Section 2.2). More explicitly, we will use the following projections, to which we refer as canonical projections. Let be an increasing sequence taking values in , with the conventions that and . Define by

 F(u)=Fx(u)=min{i\dvtx^u

Note that if is uniform on