Limits of multilevel TASEP and similar processes

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www.imstat.org/aihp 2015, Vol. 51, No. 1, 18–27

DOI:10.1214/13-AIHP555

Limits of multilevel TASEP and similar processes

Vadim Gorin

^a

and Mykhaylo Shkolnikov

^b

aDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA, and Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow 127994, Russia. E-mail:vadicgor@gmail.com

bDepartment of Statistics, University of California, Berkeley, CA 94720, USA. E-mail:mshkolni@gmail.com Received 11 September 2012; revised 24 February 2013; accepted 27 February 2013

Abstract. We study the asymptotic behavior of a class of stochastic dynamics on interlacing particle configurations (also known as Gelfand–Tsetlin patterns). Examples of such dynamics include, in particular, a multi-layer extension of TASEP and particle dynamics related to the shuffling algorithm for domino tilings of the Aztec diamond. We prove that the process of reflected interlacing Brownian motions introduced by Warren in (Electron. J. Probab.12(2007) 573–590) serves as a universal scaling limit for such dynamics.

Résumé. Nous étudions le comportement asymptotique d’une classe de dynamiques aléatoires sur des configurations entrelacées de particules (dites aussi motifs de Gelfand–Tsetlin). Des exemples de telles dynamiques incluent, en particulier, une extension à plusieurs niveaux du TASEP et des dynamiques de particules reliées à l’algorithme de mélange pour les pavages par dominos du diamant aztèque. Nous montrons que le processus des mouvements browniens réfléchis entrelacés introduit par Warren dans (Electron. J. Probab.12(2007) 573–590) est une limite d’échelle universelle pour ces dynamiques.

MSC:60J27; 60K35; 60F17

Keywords:Interacting particle system; Exclusion process; Reflected Brownian motion

1. Introduction

Consider N (N+1)/2 interlacing particles with integer coordinates x_i^j, j =1, . . . , N, i=1, . . . , j satisfying the inequalities

x_i^j₋₁< x_i^j₋⁻₁¹≤x_i^j. (1)

and denote byGT^{(N )}the set of all vectors inZ^{N (N}⁺^1)/2, which satisfy (1). An element ofGT⁽⁵⁾ is shown in Fig.1.

Elements of GT^{(N )} are often called Gelfand–Tsetlin patterns of size N and under this name are widely used in representation-theoretic context. Whenever it does not lead to confusion, we use the notationx_i^jboth for the location of a particle and the particle itself.

In this article we study a class of Markov chains onGT^{(N )}with simple block/push interaction between particles.

Let us start with an example. LetY (t ),t≥0 be a continuous-time Markov chain taking values inGT^{(N )}and defined as follows. Each of theN (N+1)/2 particles has an independent exponential clock of rate 1 (in other words, the times when clocks of particles ring are independent standard Poisson processes). If the clock of the particleY_i^j rings at time t, we check whetherY_i^j⁻¹(t−)=Y_i^j(t−)+1. If so, then nothing happens; otherwise we letY_i^j(t)=Y_i^j(t−)+1. If Y_i^j(t−)=Y_i^j₊⁺₁¹(t−)= · · · =Y_i^j₊⁺_k^k(t−), then we also setY_i^j₊⁺₁¹(t)=Y_i^j₊⁺₁¹(t−)+1, . . . , Y_i^j₊⁺_k^k(t)=Y_i^j₊⁺_k^k(t−)+1.

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Fig. 1. Interlacing particles.

Informally, one can think of each particle having a weight depending on its vertical coordinate, with higher particles being lighter. The dynamics is defined in such a way that heavier particles push lighter ones and lighter ones are blocked by heavier ones in order to preserve the interlacement conditions.

The Markov chainY was introduced in [4] as an example of a 2d growth model relating classical interacting particle systems and random surfaces arising from dimers. The restriction ofY to theN leftmost particlesx₁¹, . . . , x₁^N is the familiar totally asymmetric exclusion process (TASEP), while the particle configurationY (t )at a fixed timet can be identified with a lozenge tiling of a sector in the plane and with a stepped surface (see the introduction in [4] for the details).

More generally, we can start from anyN (N +1)/2 integer-valued random processes (in discrete or continuous time) with unit jumps of particles (but possibly differing jump rates) and construct a process similar toY onGT^{(N )} through the same block/push interactions between particles (see Section 2 for a formal definition). We will refer to this process as the interlacing dynamics driven by the originalN (N +1)/2 random processes. Some of such processes were studied in [4,23] and [17]. In particular, in the last article it is shown that interlacing dynamics driven byN (N+1)/2 independent simple random walks is closely related to the shuffling algorithm for the domino tilings of the Aztec diamond studied in [11].

In the present article we investigate the convergence to the scaling limits of the process Y and more general interlacing dynamics. The asymptotics of the fixed timedistributions of such processes ast → ∞is well studied.

Most results are easier to prove if the process is started from thedensely packedinitial conditions:Y_i^j(0)=1−i, j =1, . . . , N,i=1, . . . , j. The reason is that such processes often havedeterminantalcorrelation functions (see [4]).

However, many of the results hold in much greater generality. If one takes the limitt→ ∞for a fixedN∈N, then the distribution ofY (t )converges to the so-called “GUE-minors” process, namely the joint distribution of the eigenvalues of anN×NGaussian Hermitian matrix and the eigenvalues of its top-left corners (see [1,16,17,20]).

IfNgoes to infinity together witht, then locally one unveils the Tracy–Widom distribution, the Airy point process and discrete point processes governed by the sine kernel in the limit (see [8,14,15]), whereas the global fluctuations in this model are described by the Gaussian Free Field (see [4]). In the physics terminology the model fits into the anisotropic KPZ universality class (see the introduction in [4] and the references therein for more details).

We perform the next step and study the joint distributions ofY and similar processes atvarioustimes. Currently we restrict ourselves to the case of fixedN. The limit processW_i^j,j=1, . . . , N,i=1, . . . , j was introduced in [22]

and can be constructed inductively as follows.W₁¹is standard Brownian motion; givenW_i^j withj < kthe coordinate W_i^k is defined as a Brownian motionreflectedby the trajectoriesW_i^k₋⁻₁¹ (ifi >1) andW_i^k⁻¹(ifi < k) (see [22] or Section2for a more detailed definition). The processW has many interesting properties: for instance, its projection W_i^N,i=1, . . . , N is anN-dimensional Dyson’s Brownian motion, namely the process ofN independent Brownian motions conditioned not to collide with each other by means of Doob’sh-transform.

Our main result is the following theorem. LetD([0,∞),R^{N (N}⁺^1)/2)stand for the space of right-continuous functions from[0,∞)toR^{N (N}⁺^1)/2having left limits, endowed with the topology of uniform convergence on compact sets.

Theorem 1. LetX(·;n),n=1,2, . . .be a sequence ofZ^{N (N}⁺^1)/2-valued random processes with unit steps in each coordinate and letX(·;n),n=1,2, . . .be the interlacing dynamics driven byX(·;n),n∈N,respectively,with some

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initial conditionsX(0;n)∈GT^{(N )},n∈N,which are independent of the increments ofX(·;n),n∈N.Suppose that there exists a sequence{a_n(·)}n∈Nof continuous real-valued functions on[0,∞)and a sequence{b_n}n∈N of positive reals such that,asn→ ∞,one hasb_n→ ∞,the law of(X(·;n)−a_n(·))/b_nonD([0,∞),R^{N (N}⁺^1)/2)converges to the law of a standardN (N+1)/2dimensional Brownian motion,and the law of(X(0;n)−a_n(0))/bnconverges to some lawW (0).

Then the law of(X(·;n)−a_n(·))/b_nonD([0,∞),R^{N (N}⁺^1)/2)converges to the law of the processW with initial conditionW (0)asn→ ∞.

Remark 2. If one wants to speak about discrete time processes,then it suffices to extrapolate them to continuous time processes with piecewise constant trajectories,to which one can apply Theorem1.

Remark 3. The proof of Theorem1 shows that the result of the theorem is universal in the following sense.Sup- pose that X(·;n), n∈N are such that the processes ^X(^·;ⁿ⁾_b⁻^aⁿ⁽^·⁾

n , n∈N converge in distribution to an arbitrary C([0,∞),R^{N (N}⁺^1)/2)-valued processB asn→ ∞,where{an(·)}n∈Nis a sequence of continuous real-valued func- tions on [0,∞)and {bn}n∈N is a sequence of positive reals such that bn→ ∞as n→ ∞.Then,the laws of the processes ^X⁽^·;ⁿ⁾_b⁻^aⁿ⁽^·⁾

n ,n∈Nconverge to the law of the image ofB under the continuum Skorokhod mapΨ^SK(see Section3.2below for the definition of the latter)inD([0,∞),R^{N (N}⁺^1)/2)asn→ ∞.

Remark 4. Theorem1for a special choice of driving processesX(·;n)was conjectured in[17].

Remark 5. Theorem1is known to hold for certain projections of the processY.Namely,in the literature one can find proofs of the convergence towards the joint distribution ofW_i^N,i=1, . . . , N([4]and[22]give explicit formulas for the transition probabilities of both the prelimit and the limiting processes,then an application of Stirling’s formula yields the convergence),the joint distribution of W_iⁱ, i=1, . . . , N and the joint distribution of W₁ⁱ,i=1, . . . , N ([21]and[3]provide formulas for the transition probabilities of the prelimit processes,while[22]gives them for the limiting process;the convergence of the former to the latter can be established via well-known techniques,see e.g. [6], Section7),as well as the distribution ofW at anyfixedmoment of time(this again follows from the explicit formulas for the relevant distributions given in[4]and[22]).

Using the well-known convergence of the standard Poisson process to the standard Brownian motion, we conclude from Theorem1that the laws of the processes(Y (nt )−t n)/√

n,t≥0 converge in the limitn→ ∞to the law of the processW, started from 0. Similarly, applying Theorem1 to the interlacing dynamics ofN (N +1)/2 independent simple random walks and using the convergence of the latter to standard Brownian motions, we prove the conjecture in [17] on the convergence of the particle dynamics related to the shuffling algorithm for the domino tilings of the Aztec diamond to the processW started from 0.

The proof of Theorem1 is based on the following idea. The limit process W can be constructed inductively using Skorokhod reflection maps in time-dependent intervals recently introduced in [9], applied to the independent Brownian motions driving the particles. Moreover, the prelimit processes can be seen to be the images of the respective driving processesX(·;n)under similar Skorokhod maps. Putting these facts together with the joint continuity of the Skorokhod map in the driving path and the time-dependent boundaries, one obtains Theorem1.

It is worth noting that regularity properties of Skorokhod maps in polyhedral domains with oblique directions of reflection has been studied before in [13] and [10], where sufficient algebraic conditions for the continuity of the Skorokhod map were established using the so-called reflection mapping technique. However, it seems thatGT^{(N )}is not one of the domains for which the regularity properties are known. For instance, the results of [13], Theorem 1, and [10], Theorem 2.2, are restricted to the situation, where the number of constraints coincides with the dimension of the state space of the unconstrained process. This is not the case here, with the number of constraints beingN (N−1) and the dimension beingN (N +1)/2. Therefore, to the best of our knowledge the continuity of the Skorokhod map inGT^{(N )}, that we analyze here, is novel and might be of interest by itself (allowing for example to construct strong solutions of stochastic differential equations with reflection onGT^{(N )}).

There are two directions in which Theorem1might be generalized. First, it seems plausible that the condition on the prelimit processes to have unit steps in Theorem1can be weakened. There are examples of the dynamics on

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Gelfand–Tsetlin patterns in the literature (for example, interlacing dynamics driven by sums of geometric random variables as in [4,23], or the shuffling algorithm for boxed plane partitions as in [2,5,7]), for which the fixed time distribution is known to converge to the “GUE-minors process.” Therefore, it is reasonable to expect that the laws of the paths of these discrete processes also converge to the law ofW. One might also want to study what happens when the different components of the driving process have different asymptotic behavior, that is, when the sequences {a_n(·)}n∈N,{b_n}n∈Nin Theorem1may also depend on the particles. Some particular results into this direction were obtained in Section 7 of [6].

In addition, Theorem1allows to recover many properties of the processW (for example, the ones studied in [22]) from the corresponding properties of the discrete prelimit processes, which are sometimes easier to prove (see e.g.

[4]). For some more details in this direction, please see Remark11below.

It is worth noting that there is another natural way to construct a stochastic dynamics onGT^{(N )}which is based on the Robinson–Schensted–Knuth correspondence, see [18,19]. This dynamics is driven byNrandom walks as opposed toN (N +1)/2 in our case. While the evolution itself is very much different from the one considered in the present article due to strong correlations between components, the fixed time distributions (and also some other projections) are similar and can be made the same by an appropriate choice of the driving processes.

The rest of the article is organized as follows. In Section2we introduce stochastic processes studied throughout the paper, in Section3we explain the relation of these processes to the Skorohod reflection map, and in Section4we prove Theorem1.

2. Processes on interlacing particle configurations

In this section, we give rigorous definitions of the stochastic dynamics on interlacing particle configurations that we study. To this end, letX(t ):= {X_i^j(t), j=1, . . . , N, i=1, . . . , j},t≥0 be a random process taking values in Z^{N (N}⁺^1)/2. We impose the followingregularityconditions onX:

1. The trajectories ofXare càdlàg, that is, for everyt≥0, the limitsX(t−):=lims↑tX(s)andX(t+):=lims↓tX(s) exist, andX(t )=X(t+).

2. Each coordinateX^j_i has unit increments, that is, for everyt≥0, we have|X^j_i(t)−X_i^j(t−)| ≤1.

Note that the regularity conditions imply, in particular, that on any time interval[0, T], the trajectory ofXhas finitely many points of discontinuity.

Given a regular processX, we construct theinterlacing dynamicsX(t),t≥0 driven byX and taking values in GT^{(N )}as follows. The initial valueX(0)can be anyGT^{(N )}-valued random variable. IfXis constant on some time interval[t1, t2], then so isX. If at≥0 is a point of discontinuity inX, then the value ofX(t)depends only onX(t−) andX(t )−X(t−), and is given by the followingsequential update. First, we defineX₁¹(t), thenX_i²(t),i=1,2, then X_i³(t),i=1,2,3, etc. To start with, we set

X1¹(t)=X1¹(t−)+X₁¹(t)−X¹₁(t−);

in other words, the increments of the processX₁¹coincide with those ofX₁¹. Subsequently, once the values ofX_i^j(t), j =1, . . . , k−1,i=1, . . . , j are determined, we defineX_i^k(t)for eachi=1, . . . , kindependently by the following procedure, in which each type of update is performed only if the conditions for the previous types arenotsatisfied.

1. Ifi >1 andX_i^k(t−)=X_i^k₋⁻₁¹(t)−1, then we say that particleX_i^kispushedbyX_i^k₋⁻₁¹and setX_i^k(t)=X_i^k(t−)+1.

2. Ifi < kandX_i^k(t−)=X_i^k⁻¹(t), then we say that particleX_i^kispushedbyX_i^k⁻¹and setX_i^k(t)=X_i^k(t−)−1.

3. IfX^k_i(t)−X_i^k(t−)=1, then we check whetheri < kandX_i^k⁻¹(t)=X_i^k(t−)+1. If so, then we say that particle X_i^kisblockedbyX_i^k⁻¹and setX_i^k(t)=X_i^k(t−); otherwise, we setX_i^k(t)=X_i^k(t−)+1.

4. IfX^k_i(t)−X^k_i(t−)= −1, then we check whetheri >1 andX_i^k₋⁻₁¹(t)=X_i^k(t−). If so, then we say that particleX_i^k isblockedbyX_i^k₋⁻₁¹and setX_i^k(t)=X_i^k(t−); otherwise, we setX_i^k(t)=X_i^k(t−)−1.

5. IfX_i^k(t)−X^k_i(t−)=0, then we setX_i^k(t)=X_i^k(t−).

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The above definition can be also adapted to discrete-time processes. IfX(t ),t=0,1,2, . . .is such a process, then we defineX(t ),¯ t≥0 by

X(t )¯ =X(n) forn≤t < n+1, n=0,1,2, . . . (2)

and letX(t),t=0,1,2, . . .be the restriction of the interlacing dynamics driven byX¯ to integer times.

The following proposition enlists the properties ofXwhich are immediate from the construction.

Proposition 6. X(t),t ≥0 is a well-defined stochastic process taking values in GT^{(N )}.The paths of the process X depend only on the increments of the driving processX,but not on its initial value X(0).Suppose that X has independent increments, that is,for any T >0, the processX(t )−X(T−),t ≥T is independent of the process X(t )−X(0), 0≤t < T.ThenX is a Markov process.

Examples.

1. IfXis a family ofN (N+1)/2independent standard Poisson processes,thenX is the Markov processY defined in theIntroduction.

2. If{ξ_i^j(t): j=1, . . . , N, i=1, . . . , j, t=0,1,2, . . .}is an array of i.i.d.Bernoulli random variables with parame- terpandX^j_i(t)=_t₋₁

s=0ξ_i^j(s),thenX gives the dynamics studied in[17]and[4],which is related to the shuffling algorithm for sampling random domino tilings of the Aztec diamond.

Next we aim to introduce a continuous-space analogue of the above discrete processes. This processW was first defined and studied in [22]. It takes values in the continuum version ofGT^{(N )}, which we denote byGT^{(N )}c . An element ofGT^{(N )}_c is an array of realsx_i^j,j=1, . . . , N,i=1, . . . , j satisfying the inequalities

x_i^j₋₁≤x^j_i₋⁻₁¹≤x_i^j. (3)

The processW taking values inGT^{(N )}_c is defined level by level. Forj =1, we letW₁¹ be a standard Brownian motion. Forj >1, we define the componentW_i^j as a standard Brownian motion (independent of those used in the previous steps)reflectedon the trajectories ofW_i^j₋⁻₁¹andW_i^j⁻¹. In other words,

W_i^j(·)=B_i^j(·)+Λ^j_i(·)− ˜Λ^j_i(·), (4)

whereB_i^j is a new standard Brownian motion,Λ^j_i is the local time accumulated at zero byW_i^j −W_i^j₋⁻₁¹andΛ˜^j_i is the local time accumulated at zero byW_i^j⁻¹−W_i^j. In particular,W_j^jandW₁^j are reflected on only one trajectory. We can startW from any deterministic or random initial condition with the only restriction being thatW (0)belongs to GT^{(N )}with probability 1. For more details, please see [22], Section 2, and [22], Section 4.

Properties. As shown in[22],the processWstarted from0possesses certain interesting properties.In particular,the projectionW_i^N,i=1, . . . , N evolves according to the Dyson’s Brownian motion of dimensionN.Also,for any fixed t,the distribution ofW (t )is given by the distribution of the eigenvalues of corners of anN×N Gaussian Hermitian matrix(see[16,20]).

Another intriguing property is the following Markovity: Fix anN∈Nand0< t₁< t₂<· · ·< t_N and consider the processZwhich is defined to coincide with theR^k-valued process(W_i^k: i=1, . . . , k)on the time interval[t_k₋₁, t_k) (here, we use the conventiont₀=0). Then the processZ(t ),0≤t≤t_Nis Markovian and its transitional probabilities can be given explicitly, see [22], proof of Proposition 6, and also [4], Proposition 2.5, for a similar statement in discrete settings.

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3. Skorokhod maps

The proof of our main result relies on the observation that the limiting processWis given by the image of the vector of driving Brownian motions, started inW (0), under an appropriate Skorokhod reflection map, whereas the prelimit processes X(·;n), n∈N are given by images of the driving processes X(·;n),n∈N, started in X(0;n),n∈N, respectively, under slightly modified Skorokhod maps. Our construction relies on [9], Theorem 2.6, which we state in the following proposition for the convenience of the reader.

Proposition 7. Letl,r be two right-continuous functions with left limits on[0,∞)taking values in[−∞,∞)and (−∞,∞],respectively.Suppose further thatl(t )≤r(t ),t≥0.Then,for every right-continuous functionψwith left limits on[0,∞)taking values inR,there exists a unique pair of functionsφ,ηin the same space satisfying

1. For everyt∈ [0,∞),φ(t)=ψ (t )+η(t )∈ [l(t ), r(t )];

2. For every0≤s≤t <∞,

η(t )−η(s)≥0 ifφ(u) < r(u)for allu∈(s, t], (5)

η(t )−η(s)≤0 ifφ(u) > l(u)for allu∈(s, t]; (6)

3. For every0≤t <∞,

η(t )−η(t−)≥0 ifφ(t) < r(t), (7)

η(t )−η(t−)≤0 ifφ(t) > l(t), (8)

whereη(0−)is to be interpreted as0.

Moreover,the mapΓ,which sends the triple(l, r, ψ )to the functionφis jointly continuous with respect to the topology of uniform convergence on compact sets.

We will refer toΓ (l, r,·)as theextended Skorokhod mapin the time-dependent interval[l(t ), r(t )],t≥0. It will not be important in the following, but it is worth mentioning that the mapΓ can be given explicitly (see [9], Theorem 2.6).

3.1. Skorokhod maps for the prelimit processes

Since the definition ofX(·;n)does not depend on the initial valueX(0;n)we may assume thatX(0;n)=X(0;n).

In this subsection we show that in this case each of the prelimit processesX(·;n),n∈Ncan be constructed as the image of the vector of driving processes under a deterministic map, which we will refer to as the discrete Skorokhod map.

In our construction we fix ann∈Nand proceed by induction over the number of levelsN. ForN =1, we set X˜(·;n)=X(·;n)=X(·;n). ForN≥2, we may assume that the processX˜(·;n)with(N−1)levels, with initial condition being the restriction ofX(0;n)to the first(N−1)levels, has already been constructed. We fix a path of this process and will construct the corresponding paths of the particles on levelN.

We consider first an 1< i < N and defineX˜_i^N(·;n)as the image of X_i^N(·;n), started in X_i^N(0;n), under the extended Skorokhod map in the time-dependent interval

X˜_i^N₋⁻₁¹(t;n),X˜_i^N⁻¹(t;n)−1

, t≥0 (9)

in the sense of Proposition7above. Next, leti=1. In this case, we defineX˜₁^N(·;n)as the image ofX^N₁(·;n), started inX₁^N(0;n), under the extended Skorokhod map in the time-dependent interval

−∞,X˜₁^N⁻¹(t;n)−1

, t≥0 (10)

in the sense of Proposition7. Similarly, fori=N, we defineX˜_N^N(·;n)as the image ofX_N^N(·;n), started inX_N^N(0;n), under the extended Skorokhod map in the time-dependent interval

X˜_N^N₋⁻₁¹(t;n),∞

, t≥0 (11)

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in the sense of Proposition7. This finishes the construction.

We remark at this point that the paths of the processesX˜(·;n),n∈Nare given by images of the paths of the driving processesX(·;n),n∈N, started inX(0;n),n∈N, respectively, under adeterministicmapΦ^SKdepending only on N(and not onn):

X˜(·;n)=Φ^SK X(·;n)

, n∈N. (12)

We will refer toΦ^SKas thediscrete Skorokhod map. Finally, for eachn∈N, we defineY˜(·;n)as the process constructed from ^X(^·;ⁿ⁾_b⁻^aⁿ⁽^·⁾

n by employing the discrete Skorokhod map in the rescaled coordinates; that is, repeating the procedure above, but rescaling all initial conditions, processes involved and the constant 1 in (9), (10) according to the change of coordinates

[0,∞)×Z^{N (N}⁺^1)/2→ [0,∞)×R^{N (N}⁺^1)/2, (t, x)→

t,x−a_n(t) b_n

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The following proposition shows thatX˜(·;n),n∈Nare in fact the processes defined in Theorem1, and thatY˜(·;n), n∈Ncoincide with the corresponding rescaled processes.

Proposition 8. Suppose that the processesX˜(·;n), n∈N and X(·;n),n∈Nare constructed by using the same driving processesX(·;n),n∈N.Then,

X˜(·;n)=X(·;n), n∈N, (14)

Y˜(·;n)=X(·;n)−a_n(·)

b_n , n∈N (15)

with probability1.

Proof. Fix ann∈N. We start with the proof of (14) for thatnand proceed by induction over the number of levelsN.

ForN=1, the statement (14) follows directly from the definition ofX˜(·;n). Now, letN≥2 and suppose that (14) holds for allN< N. Then, the paths of the particles on the first (N−1)levels of X˜(·;n)must coincide with the paths of the particles on the first(N−1)levels ofX(·;n). Now, consider the trajectories of the particles on theNth level in the two processes. If the trajectories coincide up to some timet₁, then they clearly coincide up to timet₂> t₁ as long as no particles are pushed or blocked inX(·;n)in the time interval[t₁, t₂].

In case that a particle on levelNinX(·;n)is pushed, the trajectory of the pushing particle must have a jump of size 1 at the moment of the push and the other particle is pushed accordingly. The definitions ofΦ^SKand of the extended Skorokhod map in a time-dependent interval incorporated inΦ^SKshow that the value ofX˜(·;n)=Φ^SK(X(·;n))after the push must coincide with the value ofX(·;n)after the push.

Similarly, when a particle on levelN inX(·;n)is blocked, the blocking particle must be at distance 1 from the blocked particle at the moment, when it blocks the other particle, and the trajectory of the component inX(·;n) driving the blocked particle must have a jump of size 1 at the same moment. By the definitions ofΦ^SK and the extended Skorokhod map in a time-dependent interval, the value ofX˜(·;n)=Φ^SK(X(·;n))after a particle is blocked must coincide with the value ofX(·;n)at the same moment of time.

The statement (15) can be shown by arguing as in the proof of the statement (14), but working in the coordinates

obtained by the rescaling (13).

3.2. Skorokhod map for the limiting process

In this subsection we will employ Proposition7to show that the processW can be viewed as the image of the vector of driving Brownian motions, started inW (0), under a deterministic map, which we will refer to as thecontinuum Skorokhod map. To this end, we letB:= {B_i^j, j=1, . . . , N, i=1, . . . , j}be the collection of independent standard Brownian motions driving the processW and will construct an auxilliary GT^{(N )}c -valued process W˜ on the same probability space, whicha posterioriwill turn out to coincide withW.

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The construction proceeds by induction over the number of levelsN. ForN=1, we setW˜ =B₁¹=W. ForN≥2, we may assume that the paths of the particles on the first(N−1)levels of the processW˜, with initial condition being the restriction of W (0)to the first(N−1)levels, have already been constructed. Then, for 1< i < N, we define the path of the particleW˜_i^N as the image of the driving Brownian motionB_i^N, started inW_i^N(0), under the extended Skorokhod map in the time-dependent interval

W˜_i^N₋⁻₁¹(t),W˜_i^N⁻¹(t)

, t≥0 (16)

in the sense of Proposition7. Fori=1, we define the path of the particleW˜₁^N as the image of the driving Brownian motionB₁^N, started inW₁^N(0), under the extended Skorokhod map in the time-dependent interval

−∞,W˜₁^N⁻¹(t)

, t≥0 (17)

defined as in Proposition 7. Finally, for i=N, we define the path of the particleW˜_N^N as the image of the driving Brownian motionB_N^N, started inW_N^N(0), under the extended Skorokhod map in the time-dependent interval

W˜_N^N₋⁻₁¹(t),∞

, t≥0 (18)

in the sense of Proposition7.

We note that the processW˜ is given by the image of the Brownian motionB, started inW (0), under adeterministic mapΨ^SK:

W˜ =Ψ^SK(B). (19)

We will refer toΨ^SKas thecontinuum Skorokhod map. The following proposition shows thatW˜ coincides withW.

Proposition 9. Let the processesW˜ and W be defined on the same probability space and be driven by the same collectionBof independent standard Brownian motions.Then,with probability1:

W (t )˜ =W (t ) for allt≥0. (20)

Proof. We proceed by induction over the number of levelsN. ForN=1, the statement of the proposition follows directly from the definition ofW˜. Now, consider anN ≥2 and assume that the proposition holds for allN< N.

Since the paths of the particles on the first(N−1)levels inW˜ andW coincide with the paths of the particles in the respective processes with(N−1)levels, we haveW˜_i^j=W_i^jfor allj=1, . . . , N−1,i=1, . . . , jwith probability 1.

Now, fix an 1< i < N and consider the particleW˜_i^N on theNth level inW˜ and the corresponding particleW_i^N inW. The equation (4), the fact that the local time processesΛ^N_i ,Λ˜^N_i are non-decreasing and their properties

∞ 0

1_{_WN

i (t)−W_i^N₋⁻₁¹(t)>0}dΛ^N_i (t)=0,

∞ 0

1_{_W^N−1

i (t)−W_i^N(t)>0}dΛ˜^N_i (t)=0

show thatW_i^N solves the extended Skorokhod problem in the time-dependent interval W_i^N₋⁻₁¹(t), W_i^N⁻¹(t)

=W˜_i^N₋⁻₁¹(t),W˜_i^N⁻¹(t)

, t≥0 (21)

for the Brownian motionB_i^N, started inW_i^N(0). SinceW˜_i^N solves the same extended Skorokhod problem by definition, the uniqueness statement of Proposition7implies that it must holdW˜_i^N=W_i^N with probability 1. In the cases that i=1 ori=N, one can argue in the same manner, invoking the uniqueness statement of Proposition7for the

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extended Skorokhod maps in the time-dependent intervals −∞,W˜₁^N⁻¹(t)

=

−∞, W₁^N⁻¹(t)

, t≥0, (22)

W˜_N^N₋⁻₁¹(t),∞

=

W_N^N₋⁻₁¹(t),∞

, t≥0, (23)

respectively, to conclude thatW˜₁^N=W₁^NandW˜_N^N=W_N^Nalmost surely. This finishes the proof of the proposition.

Remark 10. Although this will not play a role in the following,we remark that due to Proposition9and the findings in[22]the particles on the same level inW˜ never collide after time0.Thus,if at time0no two of these particles are at the same location, [9],Corollary2.4,shows that the extended Skorokhod maps used in the construction ofW˜ are in fact regular Skorokhod maps in the sense of[9],Definition2.1.

4. Limiting procedure

We now come to the proof of the main result of the paper, stating that the processW serves as a universal limiting object for discrete processes with block/push interactions.

Proof of Theorem1. We proceed by induction over the number of levelsN. ForN=1, there is nothing to show.

We now letN ≥2 be fixed and assume that the theorem holds for all N< N. Since the first (N−1)levels of the processes ^X⁽^·;ⁿ⁾_b⁻^aⁿ⁽^·⁾

n ,n∈Nevolve as the corresponding processes with(N−1)levels, we conclude that their laws converge to the law of{W_i^j: j=1, . . . , N−1, i=1, . . . , j}onD([0,∞),R^(N⁻^1)N/2), started according to the restriction ofW (0)to the first(N−1)levels. By virtue of the Skorokhod Representation Theorem in the form of [12], Theorem 3.1.8, and the fact that the limit processes are almost surely continuous, one can define the processes on the first(N−1)levels ofX(·;n)together with the processesX^N_i (·;n),i=1, . . . , N, started inX_i^N(0;n),i=1, . . . , N, respectively, for all values ofn∈Non the same probability space such that

X_i^j(·;n)−a_n(·) bn

n→∞−→W_i^j, j=1, . . . , N−1, i=1, . . . , j, (24) X^N_i (·;N )−a_n(·)

bn

n→∞−→B_i^N, i=1, . . . , N (25)

almost surely, where {W_i^j: j =1, . . . , N −1, i=1, . . . , j} is a GT^(N_c ⁻¹⁾-valued process of the law described in Theorem1and(B₁^N, . . . , B_N^N)is anN-dimensional standard Brownian motion started in(W₁^N(0), . . . , W_N^N(0)).

Now, recall from Proposition8and the definition of the processesY˜(·;n),n∈Nthat the paths of the processes

Xi^N(·;n)−a_n(·)

b_n ,i=1, . . . , N,n∈Nare given by the images of the driving processes ^X

N

i(·;n)−a_n(·)

b_n ,i=1, . . . , N,n∈N under the extended Skorokhod maps in the appropriate time-dependent intervals (we will refer to the latter asI_i^N(n)).

Moreover, from Proposition9 and the definition of the process W˜ we know that the paths of the processes W_i^N, i=1, . . . , N are given by the images of the driving Brownian motionsB₁^N, . . . , B_N^N under the extended Skorokhod maps in the appropriate time-dependent intervals (we will refer to the latter asI_i^N(∞)).

It now suffices to observe that, for all 1≤i≤N, the left (resp. right) boundaries of the time-dependent intervals I_i^N(n)converge in the limit asn→ ∞to the left (resp. right) boundaries of the time-dependent intervalsI_i^N(∞)in D([0,∞),R)with probability 1. We remark that here the assumptionbn→ ∞asn→ ∞is used. By applying the continuity statement in Proposition7we conclude that the paths of the processes^X

N

i (·;n)−an(·)

b_n ,i=1, . . . , Nconverge in the limit asn→ ∞to the paths of the processesW_i^N,i=1, . . . , N almost surely inD([0,∞),R). This finishes

the proof.

Remark 11. Theorem1shows that one can reprove the properties of the processW and its projections stated at the end of Section2by using the corresponding properties of the processY and taking limits.Indeed,one can reexpress

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the transition probabilities of the processWas limits of the corresponding transition probabilities of the appropriately rescaled versions of the processY.

Acknowledgements

The work on this article started while the authors were staying at MSRI and we would like to thank the organizers of the program “Random Spatial Processes”. We are also grateful to A. Borodin and I. Corwin for useful discussions.

V. G. was partially supported by RFBR-CNRS grants 10-01-93114 and 11-01-93105.

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