aa r X i v : . [ m a t h . P R ] N ov THE TAZRP SPEED PROCESS
GIDEON AMIR, OFER BUSANI, PATRÍCIA GONÇALVES, AND JAMES B. MARTINA
BSTRACT . In [1] Amir, Angel and Valkó studied a multi-type version of the totally asymmetric simple ex-clusion process (TASEP) and introduced the TASEP speed process, which allowed them to answer delicatequestions about the joint distribution of the speed of several second-class particles in the TASEP rarefactionfan. In this paper we introduce the analogue of the TASEP speed process for the totally asymmetric zero-rangeprocess (TAZRP), and use it to obtain new results on the joint distribution of the speed of several second-classparticles in the TAZRP with a reservoir. These is a close link from the speed process to questions about station-ary distributions of multi-type versions of the TAZRP; for example we are able to give a precise description ofthe contents of a single site in equilibrium for a multi-type TAZRP with continuous labels. INTRODUCTION
In the totally asymmetric simple exclusion process (TASEP), each site of Z contains either a particleor a hole. If a particle has a hole to its right, they exchange places at rate 1. In [9], Ferrari and Kipnisconsidered the TASEP with Riemannian initial data – that is, where there exists an asymptotic density ofparticles to the left of the origin, and also a (perhaps different) asymptotic density of particles to the rightof the origin – and with a second-class particle placed at the origin. The second-class particle interacts withholes as if it were a particle, and with particles as if it were a hole.As the configuration evolves, the position of the second-class particle, X ( t ) , changes, and a naturalquestion is whether the limit U = lim t → ∞ t − X ( t ) , (1)exists, and if so, in what sense. Consider for example the case where the initial condition has particles atall negative sites and holes at all positive sites. It was shown in [9] that the limit in (1) exists in distribution,and that U ∼ U [ − , ] , (2)The hydrodynamics of the TASEP are described by the inviscid Burgers equation; for these initial condi-tions, the equation displays an entire interval of characteristics emanating from the origin (the so-called“rarefaction fan"), and one has the interpretation that the speed of the second-class particle is distributeduniformly across the set of characteristics.The natural question of whether the convergence in (1) can be strengthened to almost sure convergencewas resolved in [18] by Mountford and Guiol, for the rarefaction-fan initial condition, using large deviationsfor last-passage percolation and microscopic approximation of the Hamilton-Jacobi equation related to the G. Amir was supported by the Israel Science Foundation through grant 575/16 and by the German Israeli Foundation throughgrant I-1363-304.6/2016.O. Busani was supported by EPSRC’s EP/R021449/1 Standard Grant.P. Gonçalves thanks FCT/Portugal for support through the project UID/MAT/04459/2013. This project has received fundingfrom the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grantagreement No 715734).
TASEP hydrodynamics. A different proof was given by Ferrari and Pimentel [12] using a direct couplingbetween the path of the second-class particle and an interface in a two-type last-passage percolation model.In [8] Ferrari, Gonçalves and Martin considered the TASEP process (and partially-asymmetric versionsof it) starting from a configuration with two second-class particles P and Q at positions 0 and 1 respectively,with only first class particles to their left and only holes to their right. They showed, for example, that forthe TASEP, the probability that P attempts a jump over Q at some time t > .In order to answer further questions about the joint distribution of the speed of several second-classparticles at the rarefaction fan – such as, what is the probability that the two second-class particles developthe same speed? – Amir, Angel and Valkó [1] introduced the TASEP speed process. In this model one startsfrom an initial condition in which every site of Z contains a particle of a different type, with a hierarchydetermined by their initial position. Each particle sees itself as a second-class particle viewing all particlesto its left as first-class particles, and all particles to its right as holes. In this way, the particle positioned atany site i ∈ Z develops a speed almost surely, and one obtains the so-called TASEP speed process { U i } i ∈ Z , (3)a process indexed by Z which encodes the joint speed of all particles. This process proved to be a rich modelencoding much information about the joint behaviour of second-class particles around the rarefaction fan.In the case of two second-class particles in the rarefaction fan, an explicit joint distribution of the speedwas obtained, in particular, it was shown that with positive probability ( ) the two particles develop thesame speed. In fact, it was shown that with probability 1, the set of speeds attained is dense in [ − , ] , andthat for any speed v which is attained, there are in fact infinitely many particles, called a convoy, with speed v . The TASEP speed process was also used in [6] and [5] by Coupier and Heinrich to show that in the last-passage percolation model, there are no three geodesics with the same direction. Results from [1] aboutthe speed process of the TASEP, and about related questions concerning speeds of particles in partiallyasymmetric systems, were recently extended to models with inhomogeneity in space and time by Borodinand Bufetov [3].A closely related and also widely-studied interacting particle system is the (constant-rate) Totally Asym-metric Zero-Range Process (TAZRP). In this process each site of Z can contain any finite number n ofparticles. Each site is equipped with a Poisson clock with rate 1, upon ringing, if there is a particles at site x it jumps to site x +
1. Note that for the TASEP the full rarefaction fan is obtained by taking the maxi-mum density (1) to the left of the origin and minimum density (0) to the right of it. As for the TAZRP thenumber of particles at each site is unbounded, it seems that the analogue to the full rarefaction fan initialcondition for the TASEP is the initial condition where to the left of the origin, the density is infinite and tothe right it is zero. This initial condition can be modelled by setting a reservoir for the TAZRP at the leftof the origin. The TAZRP with a reservoir is simply the TAZRP on {− , , , ... } where at site − { , , ... } whilethe reservoir itself is equipped with a Poisson clock of rate one, which whenever rings, a particle jumpsfrom the reservoir to site 0. In [13] it was shown that the TAZRP with a reservoir has a hydrodynamic limitgiven by the function h ( x , t ) = − √ xt √ xt xt ∈ ( , ) xt > . (4)In [13] Gonçalves considered second-class particles positioned at time t = g and independent Riemannian initial data. In particular, this includes the case with HE TAZRP SPEED PROCESS 3 a reservoir at site − V almost surely, andthat V = ( + U ) where U ∼ U [ − , ] . In [2], Balázs and Nagy obtained the distribution of the speed of asecond-class particle at the rarefaction fan for a large set of models including the TASEP and ZRP using asigned measure on the configurations.In this paper we continue the study of speeds of particles in the TAZRP, and of related questions con-cerning stationary distributions of multi-type versions of the process.We consider an initial condition η ∗ in which each site of Z has an infinite column of particles (with abottom particle but no top particle). Every particle has higher priority than all the particles above it, andalso than all the particles at sites to its right. In this way, every particle sees itself as a second-class particlesitting on top of a finite stack of first-class particles at its own site, with an infinite reservoir of first-classparticles to its left, and empty space to its right.We show that every particle develops a speed with probability 1, leading to an array U = { U z , i } z ∈ Z , i ∈ N where U z , i is the speed of the particle positioned at column z on top of i particles in η ∗ . Furthermore, thedistribution of this “speed process" U is shown to be a stationary distribution for a multi-type version ofthe TAZRP, whose particles have types in R . Indeed, all translation-invariant stationary distributions canbe obtained via appropriate rescalings of the speed process. Although any individual speed is a continuousrandom variable, any pair of speeds have positive probability to be equal.The properties above are analogous to ones known for the TASEP from [1]. However, in the case of theTAZRP we can go much further than has been possible for the TASEP in describing the joint distribution ofseveral speeds. In particular, we give an explicit description of the joint distribution of the speeds of all theparticles in a given column, and hence of the contents of a typical site in a stationary multi-type TAZRP.Our approach begins with the coupling between configurations with second-class particles in TASEPand configurations with second-class particles in TAZRP, in particular the connection between the speed ofa second-class particle in the TAZRP with the flux of holes seen by a second-class particle in the TASEP.This is combined with the results in [11] showing that the second-class particle in the TASEP startingfrom Riemann initial data has a speed with probability 1, and an expression of the flux of holes seen by asecond-class particle as a function of this particle’s speed,To get more precise information about the joint distribution of speeds, we then develop a new approachinvolving fixed points of multi-type queues. We can think of a site z of the multi-type TAZRP as a priorityqueue whose service process is a Poisson process of rate 1. When a service occurs, the highest-priorityparticle present leaves the queue, moving from z to z +
1. In a translation-invariant equilibrium, the dis-tribution of the queue’s arrival process (the process of particles moving from z − z ) is the same as thedistribution of the queue’s departure process. Taking as a starting point results of Martin and Prabhakar[17], we are able to build up a detailed description of the possible distributions of the contents of the queuefor systems with some finite number n of types; by taking appropriate limits, we can then pass to the fullpicture of multi-type equilibria.The rest of the paper is organized as follows. In the next section we define the models and give themain results. In Section 3 we describe the coupling between the TASEP and the TAZRP, with and withoutsecond-class particles. In Section 4 we prove that distribution of the TAZRP speed process is stationarywith respect to the TAZRP dynamics (Theorem 1) and start to obtain results on the distributions of thespeeds. In Section 5 we study the fixed points of multi-type priority queues, and prove Theorem 2 describ-ing the equilibrium distributions of a single column in the multi-type TAZRP with a finite number of types.In Section 6 we use the results of Section 5 to prove results about the TAZRP speed process (Theorem 3 HE TAZRP SPEED PROCESS 4 and Theorem 4). In Section 7 we prove a result concerning overtaking between particles which have thesame speed (Theorem 5). 2.
MAIN RESULTS
The totaly asymmetric simple exclusion process (TASEP) on Z is a Markov process on Y = { , } Z whose generator is defined for cylinder functions by f : Y → R (5) L EP f ( ξ ) = ∑ x ∈ Z ξ ( x ) ( − ξ ( x + )) (cid:0) f (cid:0) ξ x , x + (cid:1) − f ( ξ ) (cid:1) , where ξ x , x + ( z ) = ξ ( x + ) z = x ξ ( x ) z = x + ξ ( z ) otherwise . Define the measures { ν α : 0 ≤ α ≤ } as the i.i.d. product measures on Y s.t ν α ( ξ ( ) = ) = α . It iswell known that any stationary measure with respect to (5) that is also translation invariant is a convexcombination of { ν α : 0 ≤ α ≤ } (see [15]). Another way to describe the TASEP is through the so-calledHarris construction. In the Harris construction, we attach to each bond connecting two adjacent sites x and x + T ( x , x + ) of rate one. The dynamics of the process is as follows. At the ring of theclock T ( x , x + ) at time t , if there is a particle at site x and no particle at x + t − then at time t theparticle at site x jumps to site x +
1; otherwise, there is no change in the configuration. This construction iswell defined since on any finite time interval, a.s. the graph can be broken into finite subgraphs on whichthe dynamics depends only on its clocks (and not those of other subgraphs).The totally asymmetric zero range process (TAZRP) on Z is a Markov process on X = N Z whose generatoris given by L ZR f ( η ) = ∑ x ∈ Z g ( η ( x )) (cid:0) f (cid:0) η x , x + (cid:1) − f ( η ) (cid:1) , where g : N → R + satisfies a Lipschitz condition and vanishes at 0, and where η x , x + ( z ) = η ( x ) − z = x η ( x + ) + z = x + η ( z ) otherwise.We shall be interested in the case where g ≡ g ( x ) = x ≥ g ≡ z ∈ Z a finite number of particles are stacked one on top of the other. We attach a Poisson clock T ( x , x + ) to eachpair of adjacent sites; upon ringing, if there is at least one particle at site x then the bottom particle at x makes a jump to the top of the stack at x +
1, otherwise there is no change in the configuration. Alternativelythe constant-rate TAZRP can be thought of as a system of M/M/1 queues in tandem, one at each site of Z .Using the same arguments as before, one can show that the dynamics is well defined. The stationary andtranslation invariant distributions are well known for the TAZRP and in the case where g ≡ (cid:8) µ ρ : 0 ≤ ρ < ∞ (cid:9) where µ ρ is product measure whose marginals are geometric with mean ρ , i.e. µ ρ ( η ( x ) = k ) = (cid:18) ρ + ρ (cid:19) k + ρ k ∈ N . (6)More generally, we can study the multi-type TAZRP on Z = n η ∈ R Z × N : η ( z , i ) ≥ η ( z , i + ) o . (7)To each particle we assign a “class” in R and now the queues become priority queues with infinitely manycustomers. At each service the highest priority (greatest value) particle jumps to the next queue. We will HE TAZRP SPEED PROCESS 5 think of the particles at each queue as sorted according to their class, with the strongest at the bottom.The value η ( z , i ) represents the class of the i th strongest particle at site z . One can get a similar Harrisconstruction using the same clocks T ( x , x + ) as before: when an adjacent pair rings the bottom particle at x jumps to x + Z can be definedthrough the generator(8) L f ( η ) = ∑ x ∈ Z ( f ( σ x η ) − f ( η )) , where the operator σ x is defined in the following way: let i sort z ( α ) = min { i : η ( z , i ) < α } . In other words, i sort z is the lowest index for which η ( z , · ) is smaller than α . The operator σ x is defined through σ x η ( z , i ) = η ( x , i + ) z = x η ( x , ) z = x + , i = i sort x + ( η ( x , )) η ( x + , i − ) z = x + , i > i sort x + ( η ( x , )) η ( z , i ) otherwise . (9)In words, σ x takes the lowest-positioned (and hence of highest value) particle in column x , η ( x , ) , andputs it at position i sort x + ( η ( x , )) in column x + η ( x , ) upward by one (see Figure 1). Remark . At this point it is not clear why the dynamics in (8) is well defined on the set (7) as it could bethe case that for some x η ( x , ) < inf i η ( x + , i ) . (10)Nevertheless, we shall point out where needed, why on the set of configurations in Z the dynamics is welldefined.We shall also need the operator σ ∗ x on Z , which takes η ( x + , ) and puts it in the correct position incolumn x . More precisely, we define σ ∗ x η ( z , i ) = η ( x + , i + ) z = x + η ( x + , ) z = x , i = i sort x ( η ( x + , )) η ( x , i − ) z = x , i > i sort x ( η ( x + , )) η ( z , i ) otherwise . (11)We would like to consider a process analogous to the TASEP speed process introduced in [1]. In [1], an . . σ ( A ) One application of the operator σ . . . σ ∗ ( B ) One application of the operator σ ∗ . F IGURE The two operators σ and σ ∗ acting on columns 0 and 1. ergodic process { U i } i ∈ Z was constructed where U ∼ U [ − , ] . The marginal U i , represents the speed of a HE TAZRP SPEED PROCESS 6 second-class particle positioned between infinite first class particles to its left and infinite holes to its right,under the TASEP dynamics. The coupling between the different marginals is obtained by starting from aninitial condition where there is a hierarchy between the particles. In this initial condition, each particle isstronger than its neighbour to its right. On the ring of the bell at the edge connecting sites x and x +
1, theparticle at site x jumps to site x + x + x .In the ZRP the number of particles at each site is not bounded. We consider starting the dynamics froma configuration where each site has an infinite number of particles.We denote by p z , n the particle sitting at site z with n other particles below it. Here too, we impose a fullhierarchy (order relation) on the initial particles according to the lexicographical order: p i , j is stronger than p k , l (denoted p i , j > p k , l ) if i < k , or if i = k and j < l . (That is, each particle is stronger than those at sitesto its right, or at the same site and directly above it).Consider a specific particle in our initial configuration. If we only care about the dynamics of thatparticle, then we do not care about the hierarchy between the other particles. Thinking of that particle assecond-class, we may consider all particles underneath it or at sites to its left as first-class particles, and allthe particles above it or at sites to its right as holes. We will show in Section 3 that, under the multi-typeTAZRP dynamics, this particle will develop a speed with probability 1. We record this speed in U i , j , the ( i , j ) ’th element of the array { U i , j } i , j ∈ Z × N , the TAZRP speed process.Another way to visualize the configuration of particles denoted above by p z , n is by considering an arrayof numbers η ∗ ∈ Z with(12) η ∗ ( z , i ) < η ∗ ( w , j ) if and only if ( w = z and i > j ) or ( w < z ) where Z is the set defined in (7) (see Figure 2). Here, particle p z , i is identified with the number η ∗ ( z , i ) and the index ( z , i ) throughout the dynamics. The number η ∗ ( z , i ) plays the role of the class, or type of theparticle which determines its interaction with other particles in the configuration in time. Note that strongerparticles correspond to higher values, as opposed to the set-up in [1]. Between each pair of neighbouringcolumns in the array we assign a Poisson clock, where upon ringing the largest number in the left column(sitting at the bottom of the column) makes a jump to the right column and positions itself on top of all thenumbers that are strictly larger than itself. We shall go from the picture of the array of numbers to the arrayof particles often. We also use the words class and type interchangeably, so that p z , i has higher class than p w , j ⇔ η ∗ ( z , i ) > η ∗ ( w , j ) . (13)Let X z , i ( t ) denote the position of the particle p z , i at time t , that is, the site (column) X z , i ( t ) ∈ Z where the − . . . p − , p − , ... p − , i ... p , p , ... p , i ... p , p , ... p , i ... p , p , ... p , i ... . . . F IGURE The initial configuration η ∗ HE TAZRP SPEED PROCESS 7 particle p z , i can be found at time t under the dynamics of (8) and the initial condition η ∗ , i.e. the multi-typeTAZRP. Let p z , i and p ′ w , i be two particles in the configurations η and η ′ respectively. We say the particle p z , i sees the same environment as particle p ′ w , i if for every l ∈ N and k ∈ Z p z + k , l ≤ p z , i ⇐⇒ p ′ w + k , l ≤ p ′ w , i , p z + k , l ≥ p z , i ⇐⇒ p ′ w + k , l ≥ p ′ w , i . In other words, two particles see the same environment if the relative order between them and other particlesin the configuration is preserved relative to their position.We say that p z , i has a speed if lim t → ∞ t − X z , i ( t ) exists and call the limit the speed of p z , i . We have thefollowing result, a proof of which we give in Section 3. Lemma 1.
For every z ∈ Z and i ∈ N the particle p z , i has a strictly positive speed with probability one.That is, the following limit exists and is strictly positive a.s. (14) U z , i = lim t → ∞ t − X z , i ( t ) > . We are now in a position to define the TAZRP speed process.
Definition 1.
The
TAZRP speed process U = { U z , i } Z × N is given by U z , i = lim t → ∞ t − X z , i ( t ) for z ∈ Z , i ∈ N , where the limit is with probability one. We define µ to be the distribution of the process U on Z .We also show the following property of the TAZRP speed process, which together with the previousLemma show that U ∈ Z and that on the support of µ in the set Z the dynamics in (8) is well defined. Lemma 2. ∑ ∞ i = E [ U z , i ] = and P ( inf ≤ i U z , i = ) = . Note that although, clearly, the measure µ is translation invariant, it may not be reflection invariant. Let π denote the reflection operator on Z defined by πη ( x ) = η ( − x ) . Then π operates on measures on Z inthe usual way, and we define µ π = πµ . We also denote by µ π the distribution of the 0’th column of µ π (and by stationarity the distribution of any column).Let G : R → R be a non-decreasing function. For η ∈ Z , we write G ( η ) for the configuration G ( η ) z , i = G ( η z , i ) . Note that G ( η ) ∈ Z . An easy yet important observation is that the dynamics of the multi-typeTAZRP (and likewise the TASEP) are conserved under a monotone relabelling of the types. (See Lemma4 and Corollary 1 in Section 4 below.)We can now state our first main result. Theorem 1.
The distribution µ π is an ergodic stationary distribution of the multi-type TAZRP. Any othertranslation-invariant ergodic stationary distribution is the distribution of G ( η ) where η ∼ µ π , for somenon-decreasing function G from R to R . Our next result is about the stationary measures for the n -type TAZRP. In the n -type TAZRP there are n different classes of particle which may be present at any site. The first-class particles have the highestpriority, followed by the second-class, and so on. We may imagine the particles at a site (or column)ordered according to their type, with the highest-priority particles at the bottom. When the clock rings atsite x , the particle of the highest priority jumps to site x + x +
1. The n -type TAZRP can be obtained by restricting the multi-type TAZRP to a subset of theset Z in (7). Let Z n = Z ∩ R Z × N n , where R n = {− , .., − n , − n − } . Let η ( t ) be the multi-type TAZRP HE TAZRP SPEED PROCESS 8 on Z . Then the set Z n is closed under the dynamics of η ( t ) , that is, if η ( ) ∈ Z n then η ( t ) ∈ Z n for all t > n -type TAZRP to be the multi-type TAZRP restricted to Z n .The interpretation is that for i = , , . . . , n , a particle of type i has a label of type − i ∈ R n . If the totalnumber of particles of types 1 up to n at a site x in a configuration η ∈ Z n is k , then η x , i ≥ − n for i ≤ k − η x , i = − n − i ≥ k . We interpret the label − n − n -type TAZRP. The choice of R n is not crucial – one could take any ordered set of size n + R n , one can read off the class of the particles by removing the minus sign from the particlelabel. The TAZRP is the n -type TAZRP for n = α , p ∈ ( , ) . We say a random variable X has geometric distribution with parameter α , denoted X ∼ Geom ( α ) , if P ( X = k ) = ( − α ) α k − for k ≥
1, and that X has a Bernoulli-geometric distribution,denoted X ∼ Ber ( p ) Geom ( α ) , if P ( X = k ) = ( − p ) k = p ( − α ) α k − k ≥ . For 1 ≤ i ≤ n let us denote by Q i the number of particles of class i in the column 0 of configurations in Z n . Theorem 2.
For any translation-invariant ergodic stationary distribution of the n-type TAZRP, with non-zero and finite density of particles of types , , . . . , n, there are λ , . . . , λ n > with ∑ ni = λ i < such thatthe random variables Q i are independent, andQ i ∼ Ber (cid:18) λ i − ( λ + ... + λ i − ) (cid:19) Geom ( λ + ... + λ i ) . (15) Remark . Note that as one should expect from the well-known geometric i.i.d. distribution of the TAZRP,or basic results on stationary distributions on stationary distributions of M / M / Q is geometric, with parameter λ . In general, the sum a geometric and an independentBernoulli-geometric is not geometric, but for particular values of the parameters such relations do hold,and in this case one obtains, also as expected, that Q + · · · + Q i is geometric for each 1 ≤ i ≤ n , withparameter λ + · · · + λ i . We may also intepret λ i as the intensity at which particles of type i move from site0 to site 1. Also λ i is the probability that the highest-priority particle at site 0 is of type i .Now that we have stated the result that the distribution of the speed process U is a stationary measurefor the multi-type TAZRP dynamics (Theorem 1), we turn to investigating this measure. As each of thesecond-class particles has speed, the column of speeds { U , i } i ∈ N can be thought of as a marked pointprocess where the points are the set of speeds in [ , ] attained by the particles at column zero and the markassociated with the point v ∈ [ , ] is the number of particles attaining a specific speed. The following resultcharacterizes the distribution of a column of the speed process. Theorem 3.
Let U be the speed process. The distribution of { U , i } i ∈ N is a marked Poisson process on [ , ] , with intensity x and mark distribution Geom ( − √ x ) . In particular, for a fixed j > , the sequenceof speeds { U , i } ∞ i = j + conditioned on U , j is independent of { U , i } j − i = . Theorem 3 shows that the set of values of U , · accumulates at 0. We also see that conditioning on someparticle attaining the speed v , the probability of finding another particle with speed v is positive. Moreover,it gives a Markovian property for the column of speeds.Note that a related marked-Poisson-process structure was recently found by Fan and Seppalainen [7] inthe description of joint distributions of Busemann functions for the last-passage percolation model (see forexample their Theorem 3.4).Our next result states the joint distribution of two second-class particles starting at column 0. HE TAZRP SPEED PROCESS 9
Theorem 4.
Let U be the TAZRP speed process and let f ( x ) = − √ x. Then for i < j and x > x P (cid:0) x ≥ U , i , x ≥ U , j (cid:1) = − f ( x ) i + − f ( x ) j + − (cid:18) f ( x ) f ( x ) (cid:19) i + ! , and P (cid:0) U , i = U , j ∈ dx (cid:1) = ( i + ) f ( x ) j √ x dx . Theorem 4 again says that there is a positive probability for two second-class particles at a column tohave the same speed; we also see that conditional on having the same speed, the distribution of the speedhas a density.In general, obtaining results on the joint distribution of two columns is hard. We have the followingresult in this direction.
Proposition 1.
Let U be the speed process of the TAZRP, and let f ( x ) = − √ x and j , k ∈ N . Then, P (cid:0) U , > x , U − , j − > x > U − , j > ... > U − , j + k − > x (cid:1) (16) = ( f ( x ) − f ( x )) f ( x ) j f ( x ) k . Take two particles, p , j and p i , k where 0 < i . Both particles develop speed v and u respectively. Onthe event that v = u , what is the probability that p , j overtakes p i , k ? that is, what is the probability that X , j ( t ) > X i , k ( t ) for some t >
0? Our next result shows that overtaking occurs with probability 1.
Theorem 5.
Let U be the TAZRP speed process. Suppose i > and condition on U , j ≥ U i , k . Then withprobability 1, p , j overtakes p i , k . THE COUPLING BETWEEN
TASEP
AND
TAZRP3.1.
The basic coupling.
We begin by describing a coupling between the exclusion and zero-range pro-cess on Z . There are, in fact, two natural ways to define such a coupling, particle-hole and particle-particle,and both work for ASEP-AZRP as well. In the particle-hole coupling, each particle in the TASEP config-uration will correspond to a column in the TAZRP, and consecutive holes between particles in the TASEPcorrespond to particles sitting in the same column (the column that corresponds to the first particle to theirleft). The advantage of this coupling is that clocks on the particles in the TASEP correspond naturally toclocks on the sites (columns) in the TAZRP, though the direction of movement is reversed. This couplingwas originally introduced by Kipnis in [14], where he used it to relate several observables between TASEPand TAZRP, e.g. the position of a tagged particle at time t in the TASEP with the current through a bondup to time t in the TAZRP.We will be more interested in the particle-particle coupling as it can be generalized to deal with second-class particles. In the particle-particle coupling, each hole in the TASEP configuration corresponds to acolumn in the TAZRP, and the particles between consecutive holes become particles sitting in the columncorresponding to the first hole to their right. As this is the coupling we plan to use, we describe it morerigorously. Let ξ t be a TASEP configuration. Denote by { y i ( ) } i ∈ Z the positions of all the holes at time0, ordered so that ... < y − ( ) < y ( ) < y ( ) < .. , with y ( ) denoting the first hole in position > y i ( t ) denote the position of the i ’th hole at time t . We construct a configuration η t from ξ t bysetting η t ( i ) = y i + ( t ) − y i ( t ) −
1. It is not hard to check that under this coupling η t follows TAZRP dy-namics, and that, in fact, the clocks on the columns correspond to the clocks of the particles in the TASEPthat may indeed jump. We denote by Φ : Y → X (Figure 3) the mapping between TASEP and TAZRPconfigurations described above. HE TAZRP SPEED PROCESS 10 Φ F IGURE The mapping Φ between the TASEP and the TAZRP. ⊛ ⊛ F IGURE The four steps in mapping a TASEP configuration with a second-class particle to aTAZRP configuration with a second-class particle.
The coupling with second-class particle.
In [13], Gonçalves generalized the coupling discussedin the preceding subsection and introduced a coupling between the TASEP and the TAZRP where theconfigurations in both dynamics have one second-class particle. Let ξ ∈ Y be a TASEP configurationwith a second-class particle q , and i ∈ Z be the index of the first hole to the right of q . Replace thesecond-class particle q with a hole to obtain the configuration ξ ′ ∈ Y , so that Φ ( ξ ′ ) = η ′ ∈ X . Finally,put a second-class particle p on top of the i ’th column in η ′ to obtain η . The mapping (Figure 4)just defined is a bijection between TASEP and TAZRP configurations with second-class particles so thatthroughout the dynamics of ξ t (TASEP starting from ξ ) and η t (TAZRP starting from η ) the position ofthe second-class particle in the TAZRP can tell the flux of holes seen by the second-class particle in theTASEP. The position, and hence the speed of the second-class particle p , can be found by considering theflux of holes passing across the second-class particle q in the TASEP configuration. Let H tasep ( t , ξ ) = inf { i : the hole y i is to the right of q } (17)be the number of holes that have crossed the second-class particle q under TASEP dynamics starting from ξ (here we assume that the q was positioned at time t = y − and y ). It is not hard to verifythat X p ( t ) = H tasep ( t , ξ ) , i.e. the position of the particle p equals the flux of holes crossing q . Let ξ bea Riemann initial data; that is, the limits ρ : = lim m → ∞ m − ∑ k = − m ξ ( k ) and λ : = lim m → ∞ m m ∑ k = ξ ( k ) (18)exist. It was shown in [11][Proposition 2.2 and Theorem 3] thatlim t → + ∞ H tasep ( t , ξ ) t = (cid:16) + U (cid:17) almost surely , (19)where U is the speed of the second-class particle q . It now follows that the speed of the second-classparticle p starting from a Riemann initial condition equals (cid:16) + U (cid:17) . In [11], it was also shown that whenthe initial configuration is i.i.d. on either side of the origin and λ < ρ then U has uniform distribution onthe interval [ − ρ , − λ ] . In this paper, we are mostly interested in the case where λ = ρ = − U HE TAZRP SPEED PROCESS 11 of p is given by P ( U ≤ v ) = √ v v ∈ [ , ] . (20)The coupling described above between configurations with second-class particles can be extended to con-figurations with finitely many second-class particles up to the point in time where one second-class particleattempts a jump to a site where there is another second-class particle. Let ξ be a TASEP configuration with m second-class particles such that between the positions of any two second-class particles there is at leastone hole. This corresponds in the TAZRP to the case where each column has at most one second-classparticle. First we register in { i j } mj = the indices of the holes to the right of each second-class particle.Then replace the second-class particles with holes, apply the mapping Φ on the new configuration andfinally place second-class particles at the top of columns i , ..., i m . We will use this coupling in the proof ofTheorem 5. Remark . In [4], in the context of last-passage percolation, Cator and Pimentel obtained the distributionof the speed of a second-class particle in any Riemann initial condition. Using the coupling in Subsection3.1 one can translate the results in [4] to results for the distribution of the speed of a second-class particlein the TAZRP starting from a larger set of initial conditions.Before we turn to the proof of Lemma 1, we note that it is a straightforward consequence of (19).Nevertheless, for the sake of self-containment we give here a proof that uses only the results in [11], thatthe second-class particle positioned at the origin between all particles to the left and holes to its right has aspeed with probability 1, and that this speed is > − Proof of Lemma 1.
Step 1:
We first show that the particle p , (the particle located at the bottom of the 0’th column) has aspeed. By calling p , a 2nd class particle, all particles to its left are 1st class compared to it as they areof higher class. Similarly all particles to its right or above it are seen as 3rd class (holes), and using thecoupling of the TAZRP with the TASEP we get to the TASEP configuration . . . . . .. It was shown in [18, 11] that particle 2 has speed U a.s., and as explained in Subsection 3.2, the speed U , equals U , = (cid:16) + U (cid:17) . (21)And in particular it is strictly positive with probability 1. Step 2:
We now claim that particle p , i also develops a strictly positive speed for all i >
0. Consider theevent that i − p , p − , − ( i − ) ... p − , . Under this event p , sees the same environment as p , i sees at η ∗ (recall (12)), that is, all the particlesbelow it or to its left are first class particles while all particles to its right or above it are holes. Thus if p , i HE TAZRP SPEED PROCESS 12 has positive probability not to have a strictly positive speed under η ∗ , then so does p , , contradicting thefact that with probability 1 p , has a strictly positive speed. (cid:3) We end this section with a proof of Lemma 2.
Proof of Lemma 2.
We will use the mass transport principle (See e.g. [16] Chapter 8). For each k ≥ t define a mass-transport function f kt : Z × Z → [ , ∞ ] by f kt ( z , y ) = ∑ ki = X z , i ( t ) > y y ≥ z y < z . One may think of this as each of the first k particles in every column sending a mass of 1 to each po-sition they jump from up to time t . Define Z kt = t ∑ ∞ y = − ∞ f kt ( , y ) and Y kt = t ∑ ∞ z = − ∞ f kt ( z , ) . Y kt is justthe averaged rate at which particles of (initial) height ≤ k jumped from 0, and therefore E [ Y kt ] ≤ k → ∞ E [ Y kt ] = t . On the other hand, lim t → ∞ Z kt = ∑ ki = U , i . Since the distribution of f kt is trans-lation invariant, the mass-transport principle gives that E [ Z kt ] = E [ Y kt ] , and therefore ∑ ∞ i = E [ U i , ] = i ≥ U z , i . Fix any ε >
0. Since U z , i ≥
0, by Markov’s inequality P ( U z , i ) > ε ) < E [ U z , i ] ε ,and therefore ∑ i ≥ P ( U z , i > ε ) < ∞ . By the Borel-Cantelli Lemma this a.s. happens only for finitely many i -s, and therefore P ( inf i ≥ U z , i ≤ ε ) =
1. Since ε was arbitrary, we are done. (cid:3) STATIONARITY OF THE
TAZRP
SPEED PROCESS
The key to understanding why the distribution of U (or more precisely, its reflection) gives a stationarydistribution for the multi-type TAZRP is to understand the effect of a small change to the initial condition η ∗ on the speed process.Specifically, in the next lemma we consider how the speed process starting from σ η ∗ is different fromthat starting from η ∗ , where σ x is the operator given in (9) and η ∗ is the initial condition given in (12).More precisely, consider the two initial conditions η ( ) = η ∗ and η ′ ( ) = σ η ∗ . Let T be a Poissonprocess on Z × R + , representing the different clocks on the sites of Z . Apply now the Harris constructionwith T to the two initial conditions η ( ) and η ′ ( ) . Let U be the speed process associated with the process η as defined in Definition 1. We define the speed process U ′ to be the speed process associated with theprocess η ′ ( t ) , that is U ′ z , i = lim t → ∞ t − X z , i ( t ) , (22)where we assume the dynamics starts from η ′ . It is important to note that particles in η ( ) and η ′ ( ) are thesame particles indexed by Z × N according the their position in η ( ) . More precisely, the particle η ∗ ( , ) is identified with ( , ) and its class is the number η ∗ ( , ) . The operation σ will move the particle ( , ) and place it at the bottom of column 1 (this is due to the initial order imposed on η ∗ where every particlein the 0’th column is stronger than any particle in column 1). However it is important to note that we stillidentify this particle by its initial position in η ∗ , i.e. ( , ) . This means that the position of the particle ( , ) in η ∗ is X , ( ) = σ η ∗ is X ′ , ( ) =
1. Hence, the arrays { U z , i } ( z , i ) ∈ Z × N and { U ′ z , i } ( z , i ) ∈ Z × N register the speed of particle ( , ) at position ( , ) of the array, despite the fact that X ′ , ( ) = Lemma 3.
Let η and η ′ be two TAZRPs defined by the Harris construction with initial condition η ∗ and σ η ∗ respectively, and a Poisson process T on Z × R + . Let U and U ′ be the TAZRP speed processesassociated with η and η ′ respectively, then (23) σ ∗ U = U ′ . HE TAZRP SPEED PROCESS 13 ( , )( , )( , ) ... ( , )( , )( , ) ... ( , )( , ) ... ( , )( , )( , )( , ) ... F IGURE Illustration of the sorting process dynamics. The particle ( , ) interacts with the onlyparticle that is not ordered with respect to, ( , ) , to create two new particles - ( , ) and ( , ) . In order to prove the lemma we shall make use of a process we call the sorting process. The configuration η of the sorting process compares two initial configurations η and ξ of the multi-type TAZRP. Applyingthe dynamics of the sorting process on the initial configuration keeps track of the development of the twoinitial configurations η and ξ when one applies on them the same Poisson clocks in the Harris construction.For ( x , y ) , ( x , y ) ∈ R we write ( x , y ) ≤ ( x , y ) whenever x ≥ x and y ≥ y . Let W = n η ∈ (cid:0) R (cid:1) Z × N : η z , i ≤ η z , j , for all j ≤ i and z o . (24)Note that W is simply an array indexed by Z × N that contains pairs of real numbers. We say that ( x , y ) , ( x , y ) are ordered if either ( x , y ) ≤ ( x , y ) or ( x , y ) ≥ ( x , y ) , otherwise we say that they areunordered. Let η ∈ W , and for k ∈ { , } let η kz , i denote the k ’th component of the pair η z , i . We attachindependent Poisson clocks of rate 1 to each site (column) x ∈ Z , at the ring of the clock of the column x , the largest (with respect to the order on pairs) pair sitting at the bottom of the column, jumps to thecolumn to its right where the pairs rearrange into elementwise order. More precisely, if the pair η z , jumpsto column z +
1, then we arrange the sets of numbers A = η z , ∪ n η z + , i o i ∈ N (25) B = η z , ∪ n η z + , i o i ∈ N , according to their order to obtain the decreasing sequences { a i } ∞ i = and { b i } ∞ i = . Then replace the column η z + , · by a new column whose i ’th element is ( a i , b i ) . We call this process on W the sorting process. Wesay the pairs ( x , y ) = η z , ( t − ) and ( x , y ) ∈ η z + , · ( t − ) interact if the jump of the pair ( x , y ) at time t to column z + ( x , y ) / ∈ η z + , · ( t ) . Note that ( x , y ) interacts only with pairs in η z + , · ( t − ) thatare unordered with respect to itself (see Figure 5). We make the following observations:(1) If ( x , y ) is a pair in η that is ordered with respect to all other pairs in η , then ( x , y ) will not interactthroughout the dynamics.(2) If η , ξ ∈ Z , then η z , i = ( η ( z , i ) , ξ ( z , i )) ∈ W . Proof of Lemma 3.
Define η by η z , i = ( η ( z , i ) ∗ , σ η ( z , i ) ∗ ) , where η ∗ is as in (12), and let η z , i ( t ) be thesorting process starting from the initial condition η . The idea of the proof is that the sorting processmarginals η and η are the multi-type TAZRP with initial conditions η ∗ and σ η ∗ respectively, thisallows us to compare the position of the same particle in the two processes. First note that all pairs in η areordered with respect to any other pair except the pairs in η , · ( ) and the pair η , ( ) . It follows that pairsthat are not in η , · ( ) ∪ η , ( ) do not interact throughout the dynamics. Let A = (cid:8) i : η , i ( ) interacts with a pair ( x , y ) s.t. x = η ∗ ( , ) (cid:9) , (26) HE TAZRP SPEED PROCESS 14 and let i f ast = − , A = /0sup A , A = /0 . Note that if U , i < U , then i f ast < i as particle p , i cannot overtake particle p , and so η , i ( ) cannotinteract with any pair of the form ( η ∗ , , y ) for some y ∈ η ∗ , · . As lim i → ∞ U , i = i f ast = ∞ .On the event that i f ast ≥
0, the pairs (cid:8) η , i (cid:9) i fast i = will interact with particles whose first coordinate is p , according to their order. Once the pair η , has interacted with η , at some time t >
0, then the twopairs are ordered into two new pairs ( p , , p , ) and ( p , , p , ) . The pair ( p , , p , ) is ordered w.r.t. allpairs in η ( t ) and therefore will not interact at later times t > t . The next interaction (if i f ast >
0) will bebetween the pairs η , = ( p , , p , ) and ( p , , p , ) at some time t > t . The interaction will lead to theformation of the pairs ( p , , p , ) and ( p , , p , ) at time t and we see that the pair ( p , , p , ) is orderedw.r.t. all other pairs and so will not interact again (see Figure 6). We continue in the same way until allpairs { ( p , i , p , i ) } i fast i = have formed by time t i fast as well as the pair (cid:16) p , , p , i fast + (cid:17) . By the definition of i f ast , no interactions will occur at time t > t i fast . Now, let X t and X ′ t be the processes that keep track of thehorizontal position of the different particles in η and η ′ respectively i.e. X z , i ( t ) = n ⇐⇒ p z , i ∈ η ( n , · )( t ) (27) X ′ z , i ( t ) = n ⇐⇒ p ′ z , i ∈ η ′ ( n , · )( t ) . This implies that for t > t i fast X z , i ( t ) = X ′ z , i ( t ) if z / ∈ { , } ∨ ( z = , ≤ i ≤ i f ast ) (28) X z , i + ( t ) = X ′ z , i ( t ) if z = , i > X , i = X ′ , i + if i > i f ast + X , i fast + ( t ) = X ′ , ( t ) . Multiplying by t − and letting t go to infinity we obtain U z , i = U ′ z , i if z / ∈ { , } ∨ ( z = , ≤ i ≤ i f ast ) (29) U z , i + = U ′ z , i if z = , i > U , i = U ′ , i + if i > i f ast + U , i fast + = U ′ , . One can now verify that the relations in (29) between U and U ′ as arrays indexed by Z × N are equivalentto (23) as configurations in W , and the result is proved (That i f ast + = i sort is a consequence of Theorem5, but we do not need it here). (cid:3) We are now ready for the proof of Theorem 1. We defer the proof of the uniqueness of µ π to Section 6. Proof of Theorem 1 without uniqueness.
Let T be a Poisson process on Z × R with rate 1, and let T s = T + × ( , s ) be the translation of T by s units of time. Also define T + s = T s ∩ Z × R + , the restriction of T s to Z × R + . Define U ( s ) to be the speed process constructed through the Harris construction with initialcondition η ∗ and the Poisson process T + s . For each s > U ( s ) has distribution µ , and it is enough to showthat U ( s ) satisfies the TAZRP dynamics. Starting from U , adding an infinitesimal time s adds, at each site i , the operator σ i at rate 1. According to Lemma 3 this should result in applying σ ∗ i to U ( ) to obtain U ( s ) at rate one. It is straightforward to see that πσ ∗ i η = σ − i − πη which implies that the process π U ( s ) is HE TAZRP SPEED PROCESS 15 − ... ( p − , , p − , )( p − , , p − , ) ... ( p − , i , p − , i ) ... ( p , , p , )( p , , p , ) ... ( p , i , p , i + ) ... ( p , , p , )( p , , p , ) ... ( p , i , p , i − ) ... ( p , , p , )( p , , p , ) ... ( p , i , p , i ) ... ... ( A ) The initial configuration η . Only the pairs in red are not ordered. Any other couple in the configurationis ordered with respect to all other pairs. ( p , , p , )( p , , p , ) ... ( p , i , p , i + ) ... ( p , , p , )( p , , p , ) ... ( p , i , p , i − ) ... ( p , , p , )( p , , p , ) ... ( p , i , p , i ) ...0 1 2 0 1 2 ( p , , p , )( p , , p , ) ... ( p , i , p , i + ) ... ( p , , p , )( p , , p , ) ... ( p , i , p , i − ) ... ( p , , p , )( p , , p , ) ... ( p , i , p , i ) ... ( B ) One step in the sorting process starting from η . The pairs ( p , , p , ) and ( p , , p , ) interact and giverise to two new pairs in column 1 - ( p , , p , ) , which is ordered with respect to any other particle in theconfiguration, and ( p , , p , ) which is unordered with respect to any pair in column 0. F IGURE The sorting process. defined through the generator (8) and the initial condition π U = µ π which is exactly what we need. To seethat µ π is ergodic, it is enough to note that µ π is generated by applying some deterministic mapping G onthe Poisson process T , which is ergodic w.r.t. the translation operator τ , and that τ G ( T ) = G ( τ T ) . (cid:3) Let G : R → R be a non-decreasing function. Let η ∈ Z , we write G ( η ) for the configuration G ( η ) z , i = G ( η z , i ) . Note that G ( η ) ∈ Z . An easy yet important observation is that the dynamics of the multi-typeTAZRP (and likewise the TASEP) are conserved under a monotone relabelling of the types. Lemma 4.
Let G : R → R be a non-decreasing function. Let η ∈ Z , and let T be a Poisson process on Z × R + and consider η ( t ) and η G ( t ) , the multi-type TAZRP defined through the Harris construction with T and the initial conditions η and G ( η ) respectively. ThenG ( η ( t )) = η G ( t ) ∀ t ≥ . (30) Proof.
By the definition of η G , (30) holds for t =
0. Now, following the Harris construction, it is enoughto show, that σ i G ( η ) = G ( σ i η ) for every i ∈ Z , (31)which is not hard to verify. (cid:3) Corollary 1.
Let G : R → R n be an increasing function. Then the distribution of the process G ( π U ( · )) isa stationary and ergodic distribution for the n- type TAZRP. HE TAZRP SPEED PROCESS 16
Proof.
Since G is increasing, G ( π U ( t )) ∈ Z n for every t >
0. By Lemma 4, the stationarity of π U ( · ) implies the stationarity of G ( π U ( · )) . Moreover, since τ G ( π U ( · )) = G ( τπ U ( · )) we see that the ergodicityof U ( · ) implies that of G ( π U ( · )) . (cid:3) One can use the one-point marginals of the 1- type TAZRP along with Corollary 1 to obtain the one-pointmarginal of U . Lemma 5.
Let U be the TAZRP speed process. Then, for every j ∈ N (32) P (cid:0) U , j ≤ v (cid:1) = − (cid:0) − √ v (cid:1) j + v ∈ [ , ] . Proof.
Let(33) G v ( x ) = ( − x > v − x ≤ v . Note that by Corollary 1, G v ( U ) is a stationary and ergodic measure of the 1-type TAZRP and that P ( U , ≤ v ) = P (cid:16) n i : G ( U ) , i = − o = (cid:17) = P µ α ( η = )= + α , where in the second equality we used the well-known unique stationary ergodic measures for the TAZRPmentioned in (6). By (20) we see that P ( U , ≤ v ) = √ v , and therefore that(34) α = − √ v √ v . Similarly, we see that P (cid:0) U , j ≤ v (cid:1) = P µ α ( η ≤ j ) (35) = − (cid:18) α + α (cid:19) j + . Plugging (34) in equation (35) we obtain the result. (cid:3)
Remark . Lemma 5 implies the result in [13][Theorem 2.1, case ρ = ∞ ]. Indeed, the equality there canbe written with our notation and by using the monotonicity of the speeds of particles in one column, aslim t → ∞ ∞ ∑ i = P ( X , i ( t ) ≥ ut ) = − √ u √ u , which follows easily by using (32).5. STATIONARY MEASURES FOR THE n - TYPE
TAZRP5.1.
One-column distribution in stationarity.
Our approach to investigating the n -type TAZRP is throughthinking of each column of the n -type TAZRP as a queue. Such a queue has services at times of a Poissonprocess of rate 1, and its arrival process contains particles of types from 1 to n . The server attends to par-ticles according to their class; when a service occurs, the particle with the highest priority is served (if anyparticle is present), and departs from the queue. HE TAZRP SPEED PROCESS 17
Let λ i be the intensity of arrivals of type i . We are interested in the case where the behaviour of thequeue is stationary in time and ergodic, with a finite average number of particles of each type present in thequeue, and so we need n ∑ i = λ i < . (36)We wish to consider stationary distributions of the n -type TAZRP which are translation-invariant. In thiscase the departure process from the queue (say, the process of particles moving from site x to site x +
1) hasthe same distribution as the arrival process to the queue (say, the process of particles moving from site x − x ). In this sense we say that the distribution of the arrival process is a fixed point for the queueingserver. Using a coupling approach analogous to that used by Mountford and Prabhakar [19], one can showthat for any λ , . . . , λ n satisfying (36), there is a unique ergodic fixed point with intensity λ i of arrivals oftype i (see [17] for discussion). We denote this process by F ( n ) , or F ( n ) λ ,..., λ n when we need to emphasise thedependence on the arrival intensities.Let us mention a few immediate properties of the processes F ( n ) : • By Burke’s Theorem, the process F ( ) λ is a Poisson process of rate λ . • More generally, again by Burke’s Theorem, for each i , the combined process of all points in F ( n ) λ ,..., λ n of types 1 , . . . , i is a Poisson process with rate ∑ ij = λ i . • The process F ( n ) λ ,..., λ n restricted to types 1 , . . . , n −
1, i.e. removing the type- n points, gives theprocess F ( n − ) λ ,..., λ n − .The following proposition, which is the starting-point of our analysis of n -type equilibrium distributions,shows that F ( n ) can be obtained by feeding F ( n − ) into a queue with service rate ∑ nj = λ i . It was shown asa by-product of the construction of the multi-type Hammersley process by Ferrari and Martin in [10], andmore directly using interchangeability properties of queues by Martin and Prabhakar in [17]. Proposition 2.
Consider an exponential server with rate ∑ ni = λ i , and an arrival process with distributionF ( n − ) . Take the departure process and add to it a point of type n whenever the queue has an unusedservice. The resulting output process has distribution F ( n ) . For 0 < s ≤ P ( s ) λ ,..., λ n for the distribution of the vector ( Q , ..., Q n ) , where Q i is the numberof particles of type i at some fixed time in the queue with arrival process F ( n ) λ ,..., λ n and with an exponentialserver of rate s . Where there is no room for confusion we abbreviate by P ( s ) . Remark . For a fixed 1 ≤ i ≤ n , let c = ∑ ij = λ j . Note that the distribution of P ( c ) λ ,..., λ i is equal to that of P ( ) λ / c ,..., λ n / c restricted on ( Q , ..., Q i − ) . Proof of Theorem 2.
To prove Theorem 2, we need to show that under P ( ) , the distribution of Q , . . . , Q n is that given by ( ) . The proof of (15) is by induction on n and as it is a bit technical, we first prove thetheorem for the case where n =
2. We then continue to prove the induction for general n .As observed at Remark 2, the result for n = M / M / a ≥ b >
0. Define an event A ε as follows: the process F ( ) contains a b ( , ε ) . As ε gets small this event becomes unlikely; we will look at the dominantcontribution to the probability computed in two different ways.Firstly, by definition of F ( ) as a fixed point, F ( ) is the output process of a rate-1 server with arrivalprocess also distributed as F ( ) , and hence with queue distributed as P ( ) λ , λ . If ε is very small, the dominantway to get the event A ε is not to rely on any arrivals to the queue, but to suppose that the queue already HE TAZRP SPEED PROCESS 18 contains precisely a b a + b services before time ε . Theprobability of this event will decay as ε a + b and any other way of achieving it decays quicker. Since the rateof service is 1, we get(37) P ( A ε ) ∼ P ( ) λ , λ ( Q = a , Q ≥ b ) ε ( a + b ) ( a + b ) ! , where by f ( ε ) ∼ g ( ε ) we mean that f ( ε ) / g ( ε ) → ε → F ( ) is the output process of rate- ( λ + λ ) server fed by F ( ) (which isjust a Poisson process of rate λ ), with unused services designated as type-2 departures. In terms of sucha queue, the dominant way to get the event A ε as ε → a a + b services before time ε . Again this is better than relying on any new arrivals to thequeue. In this case we get(38) P ( A ε ) ∼ P ( λ + λ ) λ ( Q = a ) ( ε ( λ + λ )) ( a + b ) ( a + b ) ! . Comparing (37) and (38) we get P ( ) ( Q = a , Q ≥ b ) = ( λ + λ ) ( a + b ) P ( λ + λ ) λ ( Q = a )= ( λ + λ ) ( a + b ) P ( ) λ / λ + λ , λ / λ + λ ( Q = a )= ( λ + λ ) ( a + b ) (cid:18) − λ λ + λ (cid:19) (cid:18) λ λ + λ (cid:19) a = ( − λ ) λ a λ − λ ( λ + λ ) b − = P ( ) ( Q = a ) λ − λ ( λ + λ ) b − , where in the second equality we used Remark 5. From this it follows quickly that under P ( ) λ , λ , Q and Q are independent, and Q has Bernoulli-geometric distribution with parameters λ / ( λ ) and λ + λ asclaimed.We now turn to the proof for general n ∈ Z . As an initial part of the induction step for general n , it isuseful to state a lemma relating P ( λ + ... + λ n ) λ ,..., λ n − and P ( ) λ ,..., λ n . Lemma 6.
Assume the induction hypothesis (15) for n − . Then for all a , ..., a n − ∈ Z + , (39) ( λ + ... + λ n ) ∑ n − j = a j P ( λ + ... + λ n ) ( Q = a , ..., Q n − = a n − )= λ n λ + ... + λ n − ( λ + ... + λ n − ) P ( ) ( Q = a , ..., Q n − = a n − ) . Proof.
Recall that we can move from P ( ) to P ( λ + ··· + λ n ) by replacing λ i by λ i / ( λ + ··· + λ n ) for each i . Bythe induction hypothesis, under P ( ) , the Q i , 1 ≤ i ≤ n − P ( λ + ··· + λ n ) . It will be enough to show that for each i , for any a i ∈ Z ,(40) ( λ + ... + λ n ) a i P ( λ + ... + λ n ) ( Q i = a i ) = λ i + + ... + λ n λ i + ... + λ n − ( λ + ... + λ i − ) − ( λ + ... + λ i ) P ( ) ( Q i = a i ) . Then the claim in (39) will follow by multiplying (40) over i = , , ..., n − Q i under P ( ) is given by (15), so to obtain the distribution of Q i under P ( λ + ··· + λ n ) , werescale the parameters as above to get Q i ∼ Ber (cid:18) λ i λ i + ... + λ n (cid:19) Geom (cid:18) λ + ... + λ i λ + ... + λ n (cid:19) HE TAZRP SPEED PROCESS 19
Now one can check (40) directly for each value of a i . There are essentially two cases, a i = a i > (cid:3) Now we can carry out the induction step. Following what we did in the case n =
2, fix a , ..., a n − ≥ b ≥
1. Let A ε be the event that, during the time interval ( , ε ) , the process F ( n ) contains a a a n − points of type ( n − ) , and then finally b points of type n . Again we let ε → A ε . First we look at F ( n ) as the outputof a queue of rate 1 fed by an arrival process whose distribution is F ( n ) . As ε becomes small, the dominantway for the event A ε to occur is that at time 0 the queue already contains precisely a i customers of type i for 1 ≤ i ≤ n −
1, and at least b customers of type n , and that then a + · · · + a n − + b services occur duringthe interval ( , ε ) . This gives(41) P ( A ε ) ∼ P ( ) ( Q = a , ..., Q n − = a n − , Q n ≥ b ) ε (cid:16) b + ∑ n − j = a i (cid:17) (cid:16) b + ∑ n − j = a i (cid:17) ! . On the other hand, look at F ( n ) as the output of a queue of rate λ + · · · + λ n , fed by an arrival process whosedistribution is F ( n − ) , and with points of type n added at times of unused service. Then the dominant wayfor A ε to occur for small ε is that at time 0 the queue already contains precisely a i customers of type i for1 ≤ i ≤ n −
1, and then a + ... + a n − + b services occur during ( , ε ) . This leads to(42) P ( A ε ) ∼ P ( λ + ... + λ n ) ( Q = a , ..., Q n − = a n − ) ( ε ( λ + ... + λ n )) b + ∑ n − j = a i (cid:16) b + ∑ n − j = a i (cid:17) ! . Comparing (41) and (42) and continuing using equation (39), we get that for b ≥ P ( ) ( Q = a , ..., Q n − = a n − , Q n ≥ b )= ( λ + ... + λ n ) (cid:16) b + ∑ n − j = a i (cid:17) P ( λ + ... + λ n ) ( Q = a , ..., Q n − = a n − )= λ n λ + ... + λ n − ( λ + ... + λ n − ) P ( ) ( Q = a , ..., Q n − = a n − ) ( λ + ... + λ n ) b = P ( ) ( Q = a , ..., Q n − = a n − ) λ n − ( λ + ... + λ n − ) ( λ + ... + λ n ) ( b − ) . (43)From equation (43) we see that Q n is independent of Q , ..., Q n − , and has the Bernoulli- geometric distri-bution of (15) as claimed. This completes the induction step and the proof . (cid:3) Two-column distribution in stationarity.
Using the same ideas as in the preceding proof we canalso say something about two neighbouring queues. Denote by Q zi the number of particles of type i in thecolumn z in equilibrium and define P , ( ) as the joint probability of two queues Q , Q (the joint distributionof (cid:0) Q , ..., Q n , Q , ..., Q n (cid:1) ) , where the departure of Q is the arrival process of Q and where the serverprocess of both queues is of rate 1. Lemma 7.
Let P , ( ) be the distribution of two queues in tandem at stationarity, then P , ( ) (cid:0) Q ≥ , Q = a , Q ≥ b (cid:1) = λ λ a + ( λ + λ ) b . Proof.
We think of the process F ( ) in two ways. First we think of it as the departure process of the queue Q fed by F ( ) and served at rate 1. Let N and N two independent Poisson processes of rate 1 that areindependent of Q and Q . Let A be the event where one sees in the departure process the sequence that HE TAZRP SPEED PROCESS 20 begins with a b [ , ε ) . The probability of A is dominated by P , ( ) (cid:0) Q ≥ , Q = a , Q ≥ b (cid:1) P ( In the interval [ , ε ] , N has exactly a + b epochs(44) before N has its first epoch after which there is another epoch of N ) . To see that, note that we need to have at least one first class particle in Q , a first class particles in Q andat least b second-class particles in Q , then, in the time interval [ , ε ) the following must happen in order:(1) ( a + b ) customers must be served in Q before any customer is served in Q ;(2) one service in Q ;(3) one service in Q .Recall that if X is the sum of n i.i.d. exponential r.v’s of rate λ then X ∼ Γ ( n , λ ) , i.e. P ( X ∈ dx ) = f n , λ ( x ) dx = λ n x n − ( n − ) ! e − λ x dx . (45)We have P ( A ) ∼ P , ( ) (cid:0) Q ≥ , Q = a , Q ≥ b (cid:1) Z ε f a + b , ( r ) Z ε r f , ( r )( − e − ( ε − r ) ) dr dr (46) ∼ P , ( ) (cid:0) Q ≥ , Q = a , Q ≥ b (cid:1) Z ε h r a + b − ( a + b − ) ! e − r i Z ε r e − ( r − r ) ( − e − ( ε − r ) ) dr dr ∼ P , ( ) (cid:0) Q ≥ , Q = a , Q ≥ b (cid:1) Z ε h r a + b − ( a + b − ) ! i Z ε r e − r ( − e − ( ε − r ) ) dr dr ∼ P , ( ) (cid:0) Q ≥ , Q = a , Q ≥ b (cid:1) Z ε r a + b − ( a + b − ) ! ( ε − r ) dr . Using integration by parts twice Z ε r a + b − ( a + b − ) ! ( ε − r ) dr = Z ε r a + b + ( a + b + ) ! dr (47) = ε a + b + ( a + b + ) ! . Plugging (47) into (46) P ( A ) ∼ P , ( ) (cid:0) Q ≥ , Q = a , Q ≥ b (cid:1) ε a + b + ( a + b + ) ! . (48)On the other hand, we can think of the two queues under P , ( λ + λ ) , that is, having F ( ) as their arrivalprocess and served at rate λ + λ . We can obtain F ( ) by interpreting an unserved epoch in Q as asecond-class particle. We thus have P ( A ) ∼ P , ( λ + λ ) (cid:0) Q ≥ , Q = a (cid:1) P (cid:16) N has exactly a + b epochs(49) in the interval [ , ε ] before N has its first epoch in the interval [ , ε ] after which N has an epoch (cid:17) ∼ P , ( λ + λ ) (cid:0) Q ≥ , Q = a (cid:1) Z ε f a + b , λ + λ ( r ) Z ε r f , λ + λ ( r ) (cid:16) − e − ( λ + λ )( ε − r ) (cid:17) dr dr ∼ P , ( λ + λ ) (cid:0) Q ≥ , Q = a (cid:1) Z ε h ( λ + λ ) a + b r a + b − ( a + b − ) ! i Z ε r e − ( λ + λ ) r ( − e − ( λ + λ )( ε − r ) ) dr dr ∼ P , ( λ + λ ) (cid:0) Q ≥ , Q = a (cid:1) ( λ + λ ) a + b + Z ε r a + b − ( a + b − ) ! ( ε − r ) dr ∼ P , ( λ + λ ) (cid:0) Q ≥ , Q = a (cid:1) ( λ + λ ) a + b + ε a + b + ( a + b + ) ! . HE TAZRP SPEED PROCESS 21
Comparing (46) and (49) and letting ε go to zero, we conclude that P , ( ) (cid:0) Q ≥ , Q = a , Q ≥ b (cid:1) = P , ( λ + λ ) (cid:0) Q ≥ , Q = a (cid:1) ( λ + λ ) a + b + = P , ( λ + λ ) (cid:0) Q ≥ (cid:1) P , ( λ + λ ) (cid:0) Q = a (cid:1) ( λ + λ ) a + b + = λ λ + λ (cid:18) − λ λ + λ (cid:19) (cid:18) λ λ + λ (cid:19) a ( λ + λ ) a + b + = λ λ a + ( λ + λ ) b − , where in the second equality we used the independence of the number of first class particles across differentcolumns and in the third equality the fact that the distribution of Q i is geometric and Remark 5. (cid:3) MARGINALS OF THE
TAZRP
SPEED PROCESS
In this section we apply the results in Section 5 to obtain more refined results on the speed process. Wedivide the results into two subsections. The first deals with the distribution of one column of the speedprocess, whereas the second deals with the distribution of two columns.6.1.
Distribution of the speeds at a single column.
One may think of a column in the multi-type TAZRPin stationarity as a queue with a countable number of classes. For example, the column of the speed process, U , can be thought of as a marked point process P on [ , ] . Each realization of U is a countable set ofnumbers in [ , ] , where we label each number in that set (the speed U z , i for some i which is to be thoughtof as the class of a particle) with a number in N that denotes the number of particles in that class. Forexample, if U , i − > U , i = U , i + = / > U , i + , then (cid:0) , (cid:1) ∈ P , that is, there are two particles of class / in the column. In what follows we would like to show that the TAZRP speed process can be viewedas the continuum version of the stationary measure discussed in Subsection 5.2. In fact, we prove this byapproximating U by n -type TAZRP for large n .For each n ∈ N , fix 1 > x > ... > x n + =
0, and define the function G n x : [ , ] → R n by(50) G n x ( x ) = − min { i : x ≥ x i } , where x = ( x , ..., x n ) . By Corollary 1 applying the map G n x on each element η ( z , i ) of η ∈ Z gives anelement of Z n so that G n x ( π U ) is a stationary and ergodic distribution for the n -type TAZRP. Proof of Theorem 3.
Let G n x : [ , ] → R n be the function defined in (50) associated with x i = − in − , for1 ≤ i ≤ n +
1. Applying G n x on π U we obtain an ergodic and stationary measure for the n -type TAZRP.By the uniqueness of the stationary and ergodic measures of the n -type TAZRP, we see that the queue Q n = ( Q n , ..., Q nn ) has a Bernoulli-geometric product distribution as in Theorem 2. The arrival rates to thequeue Q n are given by λ i = q − ( i − ) n − − p − in − = √ x i − − √ x i , for 1 ≤ i ≤ n . (51)To see that, note that by stationarity of G ( n ) , the arrival rate of customers of type i to the queue equals therate of departure of customers of type i under P ( ) . By the stationarity of P ( ) , the rate of departure ofparticles of type i equals the probability that the i ’th customer is the first in the queue, that is, particle oftype i is next to be served in the queue. λ i = P ( ) ( Q = , ..., Q i − = , Q i > ) (52) = P ( U , ∈ ( x i − , x i ]) = − √ x i − ( − √ x i − ) , HE TAZRP SPEED PROCESS 22 where in the third equality we used the marginal distribution of U , in (20). By Theorem 2 we see that Q n , ..., Q nn are independent and that Q ni ∼ Ber (cid:18) √ x i − − √ x i √ x i − (cid:19) Geom ( − √ x i ) (53) = Ber p − ( i − ) n − − √ − in − p − ( i − ) n − ! Geom (cid:16) − p − in − (cid:17) . Fix ε > i ∈ [ , n − ⌊ ε n ⌋ ] . In what follows we use C to denote a constant that may depend on somevariables and that changes form line to line. Note that by Taylor expansion around 1 − ( i − ) n − thereexists C ( i , ε ) > p − ( i − ) n − − √ − in − p − ( i − ) n − = ( p − ( i − ) n − ) − n − p − ( i − ) n − + Cn − (54) = n − ( − ( i − ) n − ) + Cn − = n − x i − + Cn − , where for a fixed ε and every n , C ( · , ε ) is bounded uniformly on i ∈ [ , n − ⌊ ε n ⌋ ] . Fix y ∈ [ ε , ) , and let i n = ⌈ ny ⌉ . Then, y ∈ [ x n − i n , x n − i n − ) , and if y ∈ U then Q nn − i n =
0. Plugging (54) in (53), there exists a C ( i , ε ) such that Q nn − i n ∼ Ber (cid:18) n − x n − i n − + Cn − (cid:19) Geom (cid:0) − √ x n − i n − − Cn − (cid:1) . (55)As | y − x n − i n − | ≤ n − , plugging y into (55), there exists C ( y , ε ) > C ( · , ε ) is bounded on [ ε , ) ,such that Q nn − i n ∼ Ber (cid:18) n − y + Cn − (cid:19) Geom (cid:0) − √ y + Cn − (cid:1) . (56)Now let P n = { ( p i , l i ) } ni = be the marked point process associated with the queue Q n by p i = x n − i ≤ i ≤ n (points)(57) l i = Q nn − i ≤ i ≤ n (marks) . Let P n , = S l i = p i . If t , t , t ∈ ( ε , ] such that t < t < t , then by Theorem 2 { P n , ∩ [ t , t ) } , { P n , ∩ ( t , t ] } (58)are independent. By (56) we see that δ − lim δ → lim n → ∞ P ( P n , ∩ [ y , y + δ ) = /0 ) = y , (59)which implies that P n , converges to an inhomogeneous Poisson process with intensity x on [ ε , ) . Next,again by (56), we see that conditioned on the event P n , ∩ [ y , y + δ ] = /0 , (60)if p i ∈ P n , ∩ [ y , y + δ ) , l i ∼ Geom ( r ( δ , n )) andlim δ → lim n → ∞ r ( δ , n ) = − √ y , (61)which implies the result on [ ε , ) . Taking ε → (cid:3) HE TAZRP SPEED PROCESS 23
Proof of Theorem 4.
Fix 0 = x < x < x < x , x .First note that U and π U have the same distribution and therefore, so do G ( π U ) and G ( U ) a fact we usein the computations below. Since G ( π U ) is stationary w.r.t. the 2- type TAZRP with some λ and λ , wecan relate x , x to λ , λ . By the definition of the projection G ( π U ) P (cid:16) ( π U ) , < x (cid:17) = P ( U , < x ) = P ( ) ( Q = , Q = ) P (cid:16) ( π U ) , < x (cid:17) = P ( U , < x ) = P ( ) ( Q = ) . Using Lemma 5 and Theorem 2 we see that λ = − √ x λ = √ x − √ x , (62)and P ( ) ( Q = k ) = ( − λ ) λ k (63) P ( ) ( Q = k ) = − λ − λ k = λ − λ ( − ( λ + λ ))( λ + λ ) k − k > . For i < j P (cid:0) x > U , i , x > U , j (cid:1) = P ( ) ( Q ≤ i , Q + Q ≤ j ) (64) = i ∑ l = P ( ) ( Q = l ) P ( ) ( Q ≤ j − l ) . Using (63) P ( ) ( Q ≤ m ) = − ( λ + λ ) − λ + λ ( − ( λ + λ ) m ) − λ (65) = − λ ( λ + λ ) m − λ . Plugging (65) into (64) and using (62) P (cid:0) x ≥ U , i , x ≥ U , j (cid:1) (66) = i ∑ l = ( − λ ) λ l (cid:18) − λ ( λ + λ ) j − l − λ (cid:19) = − λ i + − ( λ + λ ) j + − (cid:18) λ λ + λ (cid:19) i + ! = − ( − √ x ) i + − ( − √ x ) j + − (cid:18) − √ x − √ x (cid:19) i + ! , which is what we wanted.Next we compute the joint distribution on the diagonal, that is, the probability that the i ’th and j ’th particleshave the same speed. HE TAZRP SPEED PROCESS 24 P (cid:0) x > U , i ≥ U , j > x (cid:1) = P ( ) ( Q ≤ i , Q + Q ≥ j + ) (67) = i ∑ l = P ( ) ( Q = l ) P ( ) ( Q ≥ j + − l | Q = l )= i ∑ l = h ( − λ ) λ l i " λ ( λ + λ ) j − l − λ = λ ( λ + λ ) j i ∑ l = (cid:18) λ λ + λ (cid:19) l = ( λ + λ ) j + − (cid:18) λ λ + λ (cid:19) i + ! , where we used the independence of Q and Q . Plugging (62) into (67) we obtain P (cid:0) x > U , i ≥ U , j > x (cid:1) = ( − √ x ) j + − (cid:18) − √ x − √ x (cid:19) i + ! . Dividing by x − x and letting x → x we conclude that P (cid:0) U , i = U , j ∈ dx (cid:1) = ( i + ) ( − √ x ) j √ x dx . (cid:3) At this point we can also give the proof of the uniqueness statement in Theorem 1. We wish to showthat if µ is the distribution of the speed process, so that µ π is a stationary distribution of the multi-typeTAZRP, then every translation-invariant ergodic stationary distribution of the multi-type TAZRP is of theform G ( µ π ) for some non-decreasing function G . Proof of the uniqueness statement in Theorem 1.
The coupling approach of Mountford and Prabhakar [19]shows that for given λ , . . . , λ n , there is a unique translation-invariant ergodic stationary distribution of the n -type TAZRP µ λ ,..., λ n such that the rate of jumps of particles of type i from site 0 to site 1 is λ i . Infact, since in stationarity the rate of such jumps is just the probability that the highest-priority particle atsite 0 has type i , these distributions are characterised by the distribution of the type of that particle; under µ λ ,..., λ n , the probability that the highest-priority particle at site 0 has type i is λ i .Any distribution ν on Z is characterised by the probabilities of cylinder events of the form { η ( z , ) ≤ a , . . . , η ( z , k ) ≤ a k } . Hence in fact ν is characterised by its projections G n x ( ν ) where G n x is a function of the form defined at (50).If ν is a stationary distribution for the multi-type TAZRP, then we know that any such G n x ( ν ) is stationaryfor the n -type TAZRP. Suppose that ν and ˜ ν are two translation-invariant ergodic stationary distributionsfor the multi-type TAZRP, such that the distribution of η ( , ) is the same under ν and ˜ ν . Then, by thecharacterisation of the distributions µ λ ,..., λ n above, the n -type stationary distributions G n x ( ν ) and G n x ( ˜ ν ) arein fact the same for any such x . Hence ν and ˜ ν are the same. So for any given distribution of η ( , ) , thereis at most one translation-invariant ergodic stationary distribution.But under µ π , the distribution of η ( , ) is non-atomic. So for any desired target distribution, we canfind a non-decreasing function G with the desired distribution of η ( , ) under G ( µ π ) . Hence indeed alltranslation-invariant ergodic stationary distributions are of the form G ( µ π ) , as desired. (cid:3) HE TAZRP SPEED PROCESS 25
Joint distribution of multiple columns.
In this section we apply the results in Section 5.2 to Propo-sition 1.
Proof of Proposition 1.
Let Q and Q be two queues with arrival process F ( ) in stationarity, s.t the de-parture process of Q is the arrival process of Q . It is not hard to see that P (cid:0) U , > x , U − , j − > x > U − , j > ... > U − , j + k − > x (cid:1) (68) = P (cid:16) ( π U ) , , ( π U ) , j − > x > ( π U ) , j > ... > ( π U ) , j + k − > x (cid:17) = P ( ) (cid:0) Q ≥ , Q = j , Q ≥ k (cid:1) . By Lemma 7 we see that P ( ) (cid:0) Q ≥ , Q = j , Q ≥ k (cid:1) = λ λ j ( λ + λ ) k − . Using (62) we obtain the result. (cid:3)
Remark . Fix v ∈ ( , ) . Apply the function G v on the reflected speed process π U and define the events A i = (cid:8) the number of particles in the column G v ( π U ) i , · whose speed exceeds v (cid:9) i ∈ Z . (69)As the distribution of G v ( π U ) is stationary with respect to the 1-type TAZRP we see that the events { A i } i ∈ Z are independent. 7. OVERTAKING
Consider the initial condition η ∗ . Let i ≤ i and j , j be such that p i , j > p i , j . We define theirmeeting time T ∈ R + ∪ ∞ as the first time that the particle p i , j is at the same column as p i j . We say theparticle p i , j overtakes the particle p i , j if T < ∞ . Note that X p i , j ( t ) < X p i , j ( t ) t < TX p i , j ( t ) ≥ X p i , j ( t ) t ≥ T . Proof of Theorem 5.
The case where U , j > U i , k is clear and so we assume U , j = U i , k . The proof will relyon the following observations:(1) As we are concerned only with the positions of the particles p , j and p i , k , we may change the typesof the particles in the configuration (even using a non-monotone relabelling) as long as the relativepriority is preserved with respect to the two particles .(2) We may also ignore any part of the dynamics that does not affect the positions of the two particles.(3) We are only interested in the dynamics until the overtaking time T .We will use these guidelines to simplify the TAZRP configuration. We will then use the coupling with theTASEP and results on the TASEP speed process to conclude that overtaking occurs.We divide the proof into several cases according to the values of i , j , k . Case 1:
First assume that i = j = k =
0, in other words, we assume the particle p , is at the bottomof the column 0 has the same speed as that of the particle at the bottom of column 1. As the particlesabove p , are weaker then p , we may consider them as holes with respect to both p , and p , , as thiswould not affect the dynamics of the two particles until overtaking occurs. We now re-label the rest of theparticles as follows (Figure 7): • p , = − p , = − • p i , j = − i < j ∈ N . • p i , j = − i > i = , j > HE TAZRP SPEED PROCESS 26
It is straightforward to check that this keeps the order of priority with respect to the two particles. Usingthe coupling of the TAZRP with the TASEP we see that this configuration translates to (for the TASEP weuse the convention in [1] that stronger particles are those with smaller value)(70) . . . . . .
Recall from Subsection 3.2 that the coupling of the TAZRP with several second-class particles holds untilthe first time that two second-class particles are in the same column, which in this case is up to time T .Since particle p , and p , have the same speed, so do the particles 2 and 3 in (70). In [1, Theorem 1.14] itwas shown that with probability 1 particle 2 overtakes particle 3. By the coupling with the TAZRP we seethat p , overtakes p , . Case 2:
Next assume i = , j ≥ k ≥
0. (if j = k = p m , l where m = ≤ l < j or m = ≤ l < k which are labeled as first class particles, i.e. p m , l = − . Although the latter ( m = ≤ l < k ) are of smaller value than the particle p , j , until time T there is no interaction between them and p , j (asthey are always strictly to the right of p , j ) so the labelling is consistent with the dynamics up to the pointof overtaking. This translates to the following multi-type TASEP configuration(71) . . . ... |{z} j ... |{z} k . . . We now claim that particle 2 overtakes particle 3 in (71). Assume it does not, then there is some positiveprobability p > Case 3
Finally we prove the theorem for i >
1. We use induction on i . Suppose U , j = U i + , k for some j , k ∈ N , and that our hypothesis holds for 1 ≤ i ′ ≤ i . There are two possibilities:(1) There exists 1 ≤ m ≤ i and l ∈ N s.t U , j = U m , l = U i + , k .(2) For every 1 ≤ m ≤ i and l ∈ N U m , l = U , j = U i + , k .For case 1 we use the induction hypothesis twice to conclude that particle p m . l overtakes particle p i + , k and that particle p , j overtakes particle p m , l which together implies that p , j overtakes p i + , k . It remainstherefore to deal with case 2. Note that by (32) we see that for every m ∈ Z w.p 1 we have(72) lim l → ∞ U m , l = . Equation (72) implies that for 1 ≤ m ≤ i , column m has only a finite number of particles whose speedexceeds U , j and all the speeds of all other particles in the column are strictly smaller than U , j . Particleslocated at column m for 1 ≤ m ≤ i and whose speed is smaller than U i + , k cannot overtake particle p i + , k and are of value smaller than that of p , j and therefore will not change the dynamics of p , j and p i + , k and can be considered as holes by both particles. Particles located at column m for 1 ≤ m ≤ i and whosespeed is larger than U i + , k cannot be overtaken by particle p , j (whose speed is smaller then theirs butwhose value is greater) and we can therefore label them as first class particles. Since we are in case 2 allparticles in columns 1 ≤ m ≤ i fall into one of the above two options. The particles at columns 0 and i + . . . ... |{z} j ... |{z} N ... |{z} N i ... |{z} k . . ...., HE TAZRP SPEED PROCESS 27 where for 1 ≤ m ≤ i N m is the number of particles whose speed is greater than U , j . We now argue asbefore: assume particle 2 does not overtake particle 3, then there is some positive probability p ′ > i = (cid:3) − ( A ) Case 1. The particles above particles 2 and3 are of lower class and do not affect the dy-namics up to the meeting time. − ( B ) Case 2. Although the particles below particles 2 and3 are of different class in the speed process, treating themas first class particles does not affect the dynamics betweenparticles 2 and 3. − ( C ) Case 3 (2). Any particle in columns 0 − F IGURE Illustration of the three configurations described in cases 1, 2 and 3 of the proof ofTheorem 5, with the minus signs omitted for neater presentation. The blue particle will ultimatelymeet the red particle.
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IDEON A MIR , B AR -I LAN U NIVERSITY , 5290002, R
AMAT G AN , I SRAEL
E-mail address : [email protected] O FER B USANI , U
NIVERSITY OF B RISTOL , S
CHOOL OF M ATHEMATICS , F RY B UILDING , W
OODLAND R D ., B RISTOL
BS81UG, UK.
E-mail address : [email protected] URL : https://people.maths.bris.ac.uk/~di18476/ P ATRÍCIA G ONÇALVES , C
ENTER FOR M ATHEMATICAL A NALYSIS , G
EOMETRY AND D YNAMICAL S YSTEMS , I
NSTITUTO S UPERIOR T ÉCNICO , U
NIVERSIDADE DE L ISBOA , 1049-001 L
ISBOA , P
ORTUGAL
E-mail address : [email protected] URL : https://patriciamath.wixsite.com/patricia J AMES
B. M
ARTIN , D
EPARTMENT OF S TATISTICS , U
NIVERSITY OF O XFORD , UK
E-mail address : [email protected] URL ::