Dynamics of Finite Inhomogeneous Particle Systems with Exclusion Interaction

We study finite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. We show that the particle jump rates determine explicitly a unique partition of the system into maximal stable sub-systems, and that this partition can be obtained by a linear-time algorithm using only elementary arithmetic. The internal configuration of each stable sub-system possesses an explicit product-geometric limiting distribution, and the location of each stable sub-system obeys a strong law of large numbers with an explicit speed; the characteristic parameters of each stable sub-system are simple functions of the rate parameters for the corresponding particles. For the case where the entire system is stable, we provide a central limit theorem describing the fluctuations around the law of large numbers. Our approach draws on ramifications, in the exclusion context, of classical work of Goodman and Massey on partially-stable Jackson queueing networks.


Introduction
The exclusion process is a prototypical model of non-equilibrium statistical mechanics, representing the dynamics of a lattice gas with hard-core interaction between particles, originating in the mathematical literature with [53] and in the applied literature with [42].In the present paper, we consider systems of N +1 particles on the one-dimensional integer lattice Z, performing continuous-time, nearest-neighbour random walks with exclusion interaction, in which each particle possesses arbitrary finite positive jumps rates.The configuration space of the system is X N +1 , where, for n ∈ N := {1, 2, 3, . ..}, (1.1) The exclusion constraint means that there can be at most one particle at any site of Z at any given time.The dynamics of the interacting particle system are described by a timehomogeneous, continuous-time Markov chain on X N +1 , specified by non-negative rate parameters a 1 , b 1 , . . ., a N +1 , b N +1 .The ith particle (enumerated left to right) attempts to make a nearest-neighbour jump to the left at rate a i .If, when the corresponding exponential clock rings, the site to the left is unoccupied, the jump is executed and the particle moves, but if the destination is occupied by another particle, the attempted jump is suppressed and the particle does not move (this is the exclusion rule).Similarly, the ith particle attempts to jump to the right at rate b i , subject to the exclusion rule.The exclusion rule ensures that the left-to-right order of the particles is always preserved.See Figure 1 for a schematic.Let X(t) = (X 1 (t), . . ., X N +1 (t)) ∈ X N +1 be the configuration of the Markov chain at time t ∈ R + := [0, ∞), started from a fixed initial configuration X(0) ∈ X N +1 .Denote the number of empty sites (holes) between particles i and i + 1 at time t ∈ R + by η i (t) := X i+1 (t) − X i (t) − 1, for i ∈ [N ] := {1, 2, . . ., N }, (1.2) so η i (t) ∈ Z + := {0, 1, 2, . ..}.An equivalent description of the system is captured by the continuous-time Markov chain ξ := (ξ(t)) t∈R + , where ξ(t) := (X 1 (t), η 1 (t), . . ., η N (t)) ∈ Z × Z N + . (1.3) An important fact is that η := (η(t)) t∈R + , where η(t) := (η 1 (t), . . ., η N (t)) ∈ Z N + , is also a continuous-time Markov chain, which can be represented via a Jackson network of N queues of M/M/1 type (we explain this is Section 3).This is the reason that we take N +1 particles.The process η can also be interpreted as a generalization, with site-dependent rates and emigration and immigration at the boundaries, of the zero-range process on the finite set [N ], with η(t) being the vector of particle occupancies at time t.
Figure 1: Schematic of the model in the case of N + 1 = 6 particles, illustrating some of the main notation.Filled circles represent particles, empty circles represent unoccupied lattice sites, and directed arcs represent admissible transitions, with exponential rates indicated.
The main contribution of the present paper is to characterize the long-term dynamics of the particle system.In particular, we give a complete and explicit classification with regards to stability (which subsets of particles are typically relatively mutually close) and law of large numbers behaviour (particle speeds), as well as some results on fluctuations in the case where the whole system is stable.Our main results are stated formally in Section 2 below; their content is summarized as follows.
Theorem 2.1 shows that there is a unique partition of the system, determined by the parameters a i , b i , into maximal stable sub-systems, which we call stable clouds.Theorem 2.1 shows that the internal configuration of each stable cloud possesses an explicit product-geometric limiting distribution (this is a precise sense in which the cloud is 'stable'), while distances between particles in different clouds diverge in probability.Theorem 2.1 also shows that the location of each stable cloud obeys a strong law of large numbers with an explicit speed; speeds of clouds are non-decreasing left to right.The cloud partition is characterized by a finite non-linear system, which is the translation to our setting of the classical Goodman and Massey [26] equations for Jackson networks.Moreover, in Theorem 2.3 we show that the cloud partition can be obtained by a lineartime algorithm of elementary steps, streamlining the Jackson algorithm from [26] by exploiting the additional structure of our model.For the case where the entire system is stable, i.e., there is a single cloud, Theorem 2.13 gives a central limit theorem describing the fluctuations around the law of large numbers; this time, the foundational result for Jackson networks is a general central limit theorem of Anantharam and Konstantopoulos [4].Section 2 presents these results, and several corollaries and examples, after introducing the necessary notation.First we indicate some relevant previous work.
Many aspects of interacting particle systems and their links to (finite and infinite) queueing networks are well understood, with much attention on systems of infinitely many particles.The exclusion and zero-range processes were introduced in a fundamental paper of Spitzer [53].The earliest explicit mention of the link between the exclusion process and Jackson's results for queueing networks that we have seen is by Kipnis [38,p. 399], in the case of homogeneous rates, where the connection to the zero-range process is also given; see also [2,10,23].Versions of these connections have facilitated much subsequent progress.For example, connections between variations on the totally asymmetric exclusion processes (TASEP), in which particles can only move to the right, and tandem queueing networks, in which customers are routed through a sequence of queues in series, are explored in [22,39,51].Other relevant work includes [5,37,54].Aspects of exclusion systems on Z with finitely many particles have been studied in the symmetric case [3], and in totally asymmetric systems with different jump rates [47]; exclusion interaction on finite lattice segments has also been studied (see e.g.[16]).
The authors are not aware of previous work on the decomposition into stable subsystems of finite exclusion systems on Z, with general nearest-neighbour transition rates.While the connection between the exclusion process and Jackson networks is well known, we have not seen the important work of Goodman and Massey [26] interpreted in the exclusion context before.In the queueing context, the results of [26] characterize systems in which some but not all queues are stable; this phenomenon has subsequently become known in the queueing literature as 'partial stability' [1,8,11].Our main result can thus be seen as a classification of partial stability for finite exclusion systems.
Yet another interpretation of η is as a random walk on the orthant Z N + with boundary reflections.For N ∈ {1, 2}, there are exhaustive criteria for classifying stability of such walks (see e.g.[18,19]) in terms of readily accessible quantities (first and second moments of increments).For N ∈ {3, 4} the generic classification is available, but requires precise knowledge of quantities which are hard to compute, namely stationary distributions for lower-dimensional projections [19,30].For N ≥ 5, additional complexity arising from the structure of high-dimensional dynamical systems means that the generic case is intractable [24].In the present paper, we demonstrate that the dynamics of the particle system admits a complete, and explicit, stability description for any N , demonstrating the special place of these models in the general theory.
In a continuum setting, there is an interesting comparison between our results and an extensive existing literature on diffusions with rank-based interactions, including the Atlas model of financial mathematics and its relatives [9,14,28,29,34,45,49,50,55].As we explain in Section 6, where we describe the continuum setting in more detail, the classification of the asymptotic behaviour of these continuum models has so far been limited to deciding whether the entire system is stable (i.e., a single stable cloud).We believe that the ideas in the present paper indicate a route to obtaining more detailed information (such as the full cloud decomposition) in these continuum models.We aim to address some of these questions in future work.A direct comparison between our model and the continuum model is not obvious, as there is a sense in which the collision mechanism in the continuum model is elastic, whereas the exclusion mechanism is inelastic, but we describe in Section 6 an elastic modification of the particle system that bears a closer analogy with the continuum model.In addition, aspects of the comparison go beyond analogy, as it is known that in certain parameter and scaling regimes, the continuum model serves as a scaling limit for certain discrete models; see [34, §3], and our Section 6 below.Many of these models admit versions with infinitely-many particles (e.g.[49,50]), which in certain cases reside in the famous KPZ universality class [56].

Main results
To state our main results, we define some quantities associated with our system, depending on (subsets of) the parameters a i , b i .For our main results, we will assume that at least all the b i are positive.
Note that, by reversing the ordering of the particles (i.e., mapping Z to −Z), we can swap the roles of the a i and b i in (A) and in the formulas that follow.(2.1) here and throughout the paper, the convention is that an empty sum is 0 and an empty product is 1.Then define 2) The quantities α, β defined in (2.1) arise as solutions of certain balance equations associated with a dual random walk derived from the dynamics of a tagged empty site of Z (a hole).To describe this walk it is more convenient to work in the queueing setting, where the hole corresponds to a customer in the queueing network, and we can impose a priority service policy to make sure the tagged customer is always served; the resulting routing matrix P given by (2.10) below is then the source of the formulas for α, β.The interpretation of v(I) will be as a speed for a putative stable sub-collection of particles.For example, v(ℓ; 1) = b ℓ − a ℓ is the intrinsic speed of a singleton particle in free space; whether its long-term dynamics matches this speed depends on its interaction with the rest of the system.For general m ∈ N, definition (2.2) gives the formula where α, β are given by (2.1), and v is given by (2.2).The interpretation of ρI (j) is as a stability parameter; it corresponds in the language of queueing theory to a workload for the queue associated with gap j in the system I. See Section 3 for elaboration of the queueing interpretation.An ordered n-tuple Θ = (θ 1 , . . ., θ n ) of discrete intervals ; and (iii) for every ℓ, r ∈ [n] with ℓ < r, every i ∈ θ ℓ and j ∈ θ r satisfy i < j.Here n =: |Θ| is the number of parts in the partition.Write θ ∈ Θ if θ is one of the parts of Θ.Given an ordered partition Θ = (θ 1 , . . ., θ n ), we write Θ ⋆ := (θ j : |θ j | ≥ 2) for the ordered non-singletons; if |θ j | = 1 for all j, we set Θ ⋆ := ∅.
For Part of our result will be to identify an ordered partition Θ = (θ 1 , . . ., θ n ) of [N + 1] in which each θ ∈ Θ is a stable cloud in the long-term dynamics of the particle system.Stability means that the relative displacements of the particles indexed by θ are ergodic, in a sense made precise in the statement of Theorem 2.1 below, which includes an explicit limit distribution for η θ (t); moreover, each cloud travels at an asymptotic speed.To describe the limit distribution for the displacements within each cloud, we define, for where ρI is defined at (2.4).For A ⊆ Z k + , set ϖ I (A) := z∈A ϖ I (z).Then, if 0 < ρI (j) < 1 for every j, ϖ I is a non-vanishing probability distribution on Z k + .The stable cloud decomposition is encoded in a general traffic equation, arising from the connection between our model and a Jackson network of queues in series with nearestneighbour routing.We describe this connection in detail in Section 3; for now we introduce the notation needed to formulate the general traffic equation (2.11).Define (2.8) If N = 1, define λ 1 := a 1 + b 2 ; else, for N ≥ 2, define , and λ j := 0 for 2 ≤ j ≤ N − 1. (2.9) Also define the matrix P := (P i,j ) i,j∈[N ] by , for 2 ≤ i ≤ N ; (2.10) and P i,j := 0 for all i, j with |i − j| ̸ = 1.For vectors x = (x i ) i∈[N ] and y = (y i ) i∈[N ] in R N , write x ∧ y for the vector (x i ∧ y i ) i∈ [N ] .The general traffic equation is the matrix-vector equation for a vector ν ∈ R N given by ν = (ν ∧ µ)P + λ, (2.11) where µ = (µ i ) i∈[N ] and λ = (λ i ) i∈ [N ] .In (2.11) and elsewhere we view vectors as column vectors when necessary.In the Jackson network context, the quantities λ, µ, and P represent arrival rates of customers, service rates, and routing probabilities, respectively; see Section 3 for a precise description.Now we can state our first main result.
Theorem 2.1 describes the asymptotic behaviour of the particle system through the cloud partition Θ and the formulas (2.3) and (2.4) for the v and ρ.Partition Θ is characterized via the solution ν of the non-linear system (2.11); those i for which ρ i ≥ 1 mark boundaries between successive parts of Θ.While it is not, in principle, a difficult task for a computer to solve the system (2.11), we present below an algorithm for obtaining Θ via a sequence of comparisons involving only applications of the formula (2.3), without directly computing the solution to (2.11).
There is a closely related algorithm due to Goodman and Massey [26], in the general Jackson network context.In our setting we can exploit the linear structure to produce an algorithm that needs only the formula (2.3), involving elementary arithmetic (a similar simplification takes place for the algorithm of [26]).The structure of our algorithm (and Theorem 4.4 below from which it is derived) is of additional interest, as it provides some intuition as to how stable clouds are formed.Roughly, the algorithm goes as follows: start from the ordered partition of all singletons as candidate stable clouds, and successively merge any candidate stable clouds in which the intrinsic speeds, as computed by (2.3), are such that the speed of the candidate cloud to the left exceeds the speed of the candidate cloud to the right.In the Jackson network context, the intuition-bearing quantities in the algorithm of [26] are the net growth rates of the queues, which are differences of our speeds; specifically, v i+1 − v i = (ν i − µ i ) + is the net growth rate of queue i (see Lemma 3.4 below).1. Initialize with κ = 0 and Θ 0 := ({1}, {2}, . . ., {N + 1}), the partition of singletons.

Given the ordered partition Θ
, where |Θ κ | = n, then set Θ = Θ κ and stop.4. Otherwise, perform the following update procedure to get Θ κ+1 .Let J Choose some j ∈ J. Then define and take Θ κ+1 = (θ κ+1 (c) One can in fact perform the merger in Step 4 at every j ∈ J, rather than picking just one; this follows from Theorem 4.4 below.However, for simplicity of presentation we only perform a single pairwise merger per iteration in the description above.
(d) The algorithm of Goodman and Massey [26, p. 863] is similar, in that it also requires at most N steps, while each step requires inverting a matrix, which, in our case, is tridiagonal and so provides formulas comparable to (2.3); the sequence of cloud mergers in Algorithm 2.4 is a little more adapted to the linear structure of our setting.
The proofs of Theorem 2.1 and 2.3 are given in Section 4 below.Next we state several corollaries to these two results: the proofs of these also appear in Section 4. The first two corollaries pertain to the extreme cases of a single stable cloud, and a system in which each particle constitutes its own stable cloud.
Corollary 2.6.Let N ∈ N and suppose that (A) holds.The partition Θ consists only of singletons if and only if (2.15) In the case where (2.15) holds, we have that Recall that ρ is defined by (2.4).From Theorem 2.1, we have that Θ = ([N + 1]) consists of a single stable cloud if and only if , where ρ i = ν i /µ i is defined in terms of the solution ν to the general traffic equation (2.11); see (2.12).Furthermore, if ρ i < 1 for all i, then (2.11) reduces to the linear equation ν = νP +λ, which can be solved explicitly (see Section 3 below), to give the formula (2.16) below.Thus we will sometimes also, with a small abuse of notation, refer to (2.16) as a definition of ρ i in the stable case.
Corollary 2.7.Let N ∈ N and suppose that (A) holds.The partition Θ has a single part if and only if ρ i < 1 for all i ∈ [N ], where, for i ∈ [N ], and where vm := v(1; m) is given by the ℓ = 1 version of (2.3), i.e., (2.17) ) , where vN+1 is given by (2.17). Remarks which expresses the stability condition in terms of the putative intrinsic speed associated with the sub-system of particles {1, 2, . . ., i} compared to the putative intrinsic speed of the whole system.See Section 6 for related results in a diffusion context.
In the case of a single stable cloud, Theorem 2.13 below gives a central limit theorem to accompany Corollary 2.7.First we present some illustrative examples.(2.15) clearly holds, and so Corollary 2.6 applies, and each particle is a singleton cloud with lim t→∞ t −1 X i (t) = u, a.s.Moreover, for every i ∈ , where F t = σ(X(s), s ≤ t).It follows from this, or the corresponding supermartingale condition for the discrete-time embedded Markov chain, that lim inf t→∞ (X i+1 (t) − X i (t)) = 1, a.s.(combine, for example, Thm. 3.5.8and Lem.3.6.5 from [43]).Thus although the particles form singleton clouds, nevertheless every pair of consecutive particles meets infinitely often (particles meeting means that they occupy adjacent sites).One can ask whether, say, three or four consecutive particles meet infinitely often or not; we do not address such 'recurrence or transience' questions systematically here, but cover some cases in Example 2.19 below.△ Example 2.10 (Totally asymmetric case).The assumption (A) permits a i = 0; here is one such example.Suppose that . This is the totally asymmetric case in which particles can jump only to the right, but may do so at different rates: see e.g.[47] and references therein.Corollary 2.7 shows that the system is stable if and only if b i > b N +1 for all i ∈ [N ] and, moreover, if this latter condition is satisfied, lim 11 (Small systems).Suppose that N = 1 (two particles).Corollary 2.7 implies that the system is stable if and only if b Suppose that N = 2 (three particles).Consider the four inequalities (2.23) In this case, the complete classification of the system is as follows.
( To verify (a), note for example that (2.20) and (2.21) imply (b To see how the classification laid out here follows from the stated results, note that, with ρ i as defined at (2.16), inequalities (2.22) and (2.23) are equivalent to ρ 1 < 1 and ρ 2 < 1, respectively.Thus Corollary 2.7 yields (i).On the other hand, Corollary 2.6 gives (iv).For (ii), notice that (2.20) implies that a valid first step for Algorithm 2.4 is to merge {1} and {2}, but then failure of (2.23) Finally, note that in the stable case (i), the limiting speed is by (2.17).△ Here is one more corollary, which gives a necessary and sufficient condition for all the speeds v i in Theorem 2.1(iii) to be positive.Corollary 2.12.Let N ∈ N and suppose that (A) holds.The following are equivalent.
We next turn to fluctuations.For simplicity, we assume that the system is stable, i.e., the ρ i given by (2.16) satisfy ρ i < 1 for all i ∈ [N ].The following central limit theorem is a consequence of results of Anantharam and Konstantopoulos [4] for counting processes of transitions in continuous-time Markov chains.Let N (µ, σ 2 ) denote the normal distribution on R with mean µ ∈ R and variance σ 2 ∈ (0, ∞).Theorem 2.13.Let N ∈ N and suppose that (A) holds.Suppose that the ρ i given by (2.16) satisfy ρ i < 1 for all i ∈ [N ].Let vN+1 be given by (2.17).Then there exists σ 2 ∈ (0, ∞) such that, for all w ∈ R and z 1 , . . ., z N ∈ Z + , where Φ is the cumulative distribution function of N (0, 1).In particular, if Remarks 2.14.(a) Theorem 2.13 covers the case of a single stable cloud.In the general (partially stable) case, we anticipate that there is a joint central limit theorem in which the components associated with clouds with different speeds are independent, while those associated with clouds of the same speed are correlated; in the case in which all clouds are singletons, one case of such a result is known in the context of the scaling limit results of [34]: see Section 6.3 below.
(b) In the case N ≥ 2, Theorem 2.13 is deduced from Thm. 10 of [4], which gives a central limit theorem for counting processes in stable Jackson networks; when N ≥ 2, Thm.11 of [4] also gives an expression for the limiting variance σ 2 , which is in principle computable, but somewhat involved.For N = 1, Theorem 2.13 does not fall directly under the scope of Thms. 10 and 11 of [4], since one must distinguish between the two different types of arrivals and departures (to the left or to the right); nevertheless, the result can be deduced from the general approach of [4] in the N = 1 case too (use e.g.their Thm.5).The main additional ingredient needed to deduce Theorem 2.13 from [4] is the asymptotic independence between the fluctuations of the location of the cloud (X 1 ) and the internal displacements (η), which is a consequence of a separation of time scales: see Section 5 below.As mentioned in Remarks 2.14(b), the N = 1 case of Theorem 2.13 is not treated explicitly in [4].Thus it is of interest to give an explicit computation of σ 2 in that case.We will use the properties of the M/M/1 queue to do so: this is the next result.
Theorem 2.15.Suppose that N = 1, that (A) holds, and a 1 + b 2 < a 2 + b 1 .Then the central limit theorem in Theorem 2.13 holds, with We finish this section with some further examples.
Example 2.16 (1 dog and N sheep; lattice Atlas model).Assume that a 1 = a ∈ (0, 1), and That is, the leftmost particle ('the dog') has an intrinsic drift to the right, while all other particles ('the sheep') have no intrinsic drift.In this case ρ j given by (2.16) satisfies By Corollary 2.7 the whole system is stable, and, by (2.17), the speed of the cloud is vN+1 = 1−a N +1 .By (2.13) and (2.27), the long-run expected size of the particle cloud satisfies as N → ∞.In [10, pp. 191-192] the closely related model where a 1 = 0, b 1 = 1, and ) in [10] recovers a version of this example.This is a lattice relative of the continuum Atlas model [9,29], in which the leftmost diffusing particle (Atlas) carries the weight of the rest on its shoulders: see Section 6 below for a discussion of such continuum models.△ Example 2.17 (Sheep between two dogs, symmetric case).We now set (two particles), then the argument in Example 2.9 shows that the system is recurrent, in the sense that lim inf t→∞ (X 2 (t) − X 1 (t)) = 1, a.s.In the case N = 2 (three particles), it can be shown that the system is also recurrent, i.e., lim inf t→∞ (X 3 (t) − X 1 (t)) = 2, a.s.; it is, however, 'critically recurrent' in that excursion durations are heavy-tailed.This follows by considering (η 1 (t), η 2 (t)) t∈R + as a random walk with boundary reflection in the quarter-plane Z 2 + and applying results of Asymont et al. [7].To apply these results, it is most convenient to work with the corresponding discrete-time embedded process.Then, in the notation of [7, p. 942], the entries in the interior covariance matrix are where r := 3 i=1 (a i + b i ), while the reflection vector components are which in this case amounts, after a little simplification, to On the other hand, if N = 3, we believe that lim t→∞ (X 4 (t) − X 1 (t)) = ∞, a.s.; when b i ≡ a i ≡ 1 for all i ∈ [4], it seems possible to approach this by reducing the problem to a random walk on Z 3 that is zero drift everywhere, and homogeneous unless one or more coordinates is zero.We conjecture that the same is true for any four (or more) consecutive particles in a constant-drift system with N ≥ 4, but, since such questions appear rather delicate and are not directly relevant for the main phenomena of the present paper, we do not explore them further here.△ It is of interest to extend the questions posed in Example 2.19 to the setting where there are several non-singleton stable clouds with the same speed.For example, if Θ = (θ 1 , θ 2 ) consists of two stable clouds with v(θ 1 ) = v(θ 2 ), then we believe strongly that the system is recurrent, i.e., lim inf t→∞ (X N +1 (t) − X 1 (t)) < ∞, a.s.A proof of this might be built on the central limit theorem, Theorem 2.13, as follows.Suppose the contrary, that the system is transient, i.e., the two stable clouds drift apart.Then the clouds evolve essentially independently, so each should satisfy Theorem 2.13 with the same speed and variance on the same scale (with different constants), which suggests they would be in the wrong order with positive probability, providing a contradiction.The following problem deals with the case of more than two clouds, and settling it seems harder.
The mathematical development to prove the results stated above begins in Section 3, where we explain the connection between our model and an appropriate Jackson queueing network, and build on the classical work of Goodman and Massey [26] to give some fundamental results on stability and dynamics.Section 4 develops these ideas further to examine the structure underlying Algorithm 2.4, and here we give the proofs of Theorems 2.1 and 2.3, and their corollaries.Section 5 is devoted to the central limit theorem in the stable case, and presents the proofs of Theorems 2.13 and 2.15.Finally, in Section 6 we discuss the relationship between the lattice model that we study here and a family of continuum models that have been studied in the literature, and mention some open problems in that direction.

Representation as a Jackson network
Consider a system of N ∈ N queues, labelled by [N ].The parameters of the system are arrival rates λ = (λ i ) i∈[N ] , service rates µ = (µ i ) i∈[N ] and P = (P ij ) i,j∈[N ] , a sub-stochastic routing matrix.Exogenous customers entering the system arrive at queue i ∈ [N ] via an independent Poisson process of rate λ i ∈ R + .Queue i ∈ [N ] serves customers at exponential rate µ i ∈ R + .Once a customer at queue i is served, it is routed to a queue j with probability P ij , while with probability Q i := 1 − j∈[N ] P ij the customer departs from the system.

Figure 2:
Schematic of a Jackson network on N nodes.The λ i are the exogenous arrival rates, the µ i are the service rates, and P is the (substochastic) routing matrix by which customers are redirected following service.From queue i, particles depart the system at rate The process η from (1.2) for the particle system described in Section 1 can be interpreted as the queue-length process for a Jackson network with parameters given by (2.8), (2.9) and (2.10).
Provided i∈[N ] λ i > 0 and i∈[N ] Q i > 0, customers both enter and leave the system, and it is called an open Jackson network.We assume that every queue can be filled, meaning that, for every i ∈ [N ], there is a j ∈ [N ] and k ∈ Z + for which λ j > 0 and (P k ) ji > 0, and that every queue can be drained, meaning that, for every i ∈ [N ], there is a j ∈ [N ] and k ∈ Z + for which Q j > 0 and (P k ) ij > 0. The process that tracks the number of customers in each queue at time t ∈ R + is a continuous-time Markov chain on Z N + .Jackson networks are named for early contributions of R.R.P. Jackson [33] and J.R. Jackson [31,32].For a general overview see e.For i ∈ [N ] recall from (1.2) that η i (t) = X i+1 (t)−X i (t)−1, the number of empty sites between consecutive particles at time t ∈ R + .We claim that the process η = (η i ) i∈[N ] is precisely the queue-length process for a corresponding Jackson network, namely, the Jackson network with parameters λ, µ and P given as functions of (a i , b i ) i∈[N +1] through formulas (2.8), (2.9) and (2.10).
To see this, observe that 'customers' in the queueing network correspond to unoccupied sites between particles in the particle system.Exogenous customers enter the network only when the leftmost particle jumps to the left (rate a 1 ) or the rightmost particle jumps to the right (rate b N +1 ).Customers at queue i are 'served' if either the particle at the left end of the interval jumps right (rate b i ) or if the particle at the right end of the interval jumps left (rate a i+1 ).If the particle at the left end of the interval jumps right, the customer is routed to queue i − 1 (if i ≥ 2, at rate b i = µ i P i,i−1 ) or leaves the system.Similarly in the other case.Customers leave the system only when the leftmost particle jumps to the right (rate µ 1 Q 1 = b 1 ) or the rightmost particle jumps to the left (rate For some of the results in this section, we will relax assumption (A) to the following; but see the remark below (A) about generality.
If (A ′ ) holds, then the parameters λ, µ and P satisfy the conditions to ensure that the system η is an open Jackson network in which every queue can be filled and drained: for example, if b i > 0 for all i, then customers can enter the network at queue N (since λ N > 0), progress in sequence through queues N − 1, N − 2, . .., and exit through queue 1 (since . The following statement summarizes the Jackson representation. Proposition 3.1.Let N ∈ N and suppose that (A ′ ) holds.The evolution of the process η = (η i ) i∈[N ] can be described by an open Jackson network in which every queue can be filled and drained, with η i (t) being the number of customers at queue i ∈ [N ] at time t ∈ R + .The exogenous arrival rates λ are given by (2.9), service rates µ are given by (2.8), and routing matrix P is given by (2.10).
Remark 3.2.We believe that Proposition 3.1 is known, but we could not find the statement explicitly in the literature, and are not confident of an attribution.For simple exclusion with homogeneous jump rates one can find the result expressed by Kipnis [38, pp. 398-9].Spitzer [53, pp. 280-1] shows that the gaps of empty sites in exclusion process viewed from a tagged particle have a product-geometric stationary distribution, but queues are not mentioned; the closest we have been able to find in [53] to an identification of exclusion and Jackson dynamics (rather than just stationary distributions) is Spitzer's report of an observation by Kesten [53, p. 281] that Poisson streams can be identified in exclusion processes.
The Jackson representation given by Proposition 3.1 enables us to obtain the following result, which is, to a substantial degree, a translation to our context of a classical result for Jackson networks due to Goodman and Massey [26], which characterizes the stable subset S of the queues in the network: when S = [N ] the network is stable, when S = ∅ it is unstable, and otherwise it is partially stable (cf.[1]).Proposition 3.3.Let N ∈ N and suppose that (A ′ ) holds.The general traffic equation (2.11) has a unique solution, which we denote by ν = (ν i ) i∈ [N ] .Write ρ i := ν i /µ i for all i ∈ [N ] and S := {i ∈ [N ] : ρ i < 1}.For every z = (z i ) i∈S with z i ∈ Z + , we have where the first equality holds a.s. and in L 1 .Moreover, for each fixed η(0) ∈ Z N + , there exists δ > 0 such that, , for all i ∈ S and all z ∈ R + . (3.2) On the other hand, for where the first equality holds a.s. and in L 1 .
Proof.Existence and uniqueness of the solution to (2.11) under the stated conditions follows from Thm. 1 of Goodman and Massey [26], which also gives the rightmost equalities in (3.1) and (3.3).
The convergence of the time-averages in (3.1) and (3.3) we deduce from the construction of Goodman and Massey.For a stable system, the Markov chain ergodic theorem applies directly; in the general case U ̸ = ∅, the idea is to construct modified processes (by adjusting relevant Jackson network parameters) that are stable and satisfy appropriate stochastic comparison inequalities with components of η, and again appeal to the ergodic theorem.For convenience, we recapitulate the main steps in the construction here, pointing to [26] for some more details; we start with the upper bound.
Following [26, p. 865-6], we construct a continuous-time Markov chain η + with components η + i (t), t ∈ R + , i ∈ [N ], by re-routing certain customers.Specifically, for every i ∈ U and j ∈ S with P ij > 0, we (i) declare that customers, instead of flowing from i to j, will depart the system, and (ii) add an exogenous arrival stream into j with rate equal to µ i P ij , the maximal flow rate from i to j.In other words, the modified system η + is a Jackson network with parameters µ + i = µ i , Since no customers flow from U to S in η + , the process η + S := (η + i ) i∈S observed only on S is itself the queue-length process of a Jackson network, with parameters λ + j = λ j + i∈U µ i P ij , j ∈ S, and P + ij = P ij , i, j ∈ S. The solution to the general traffic equations for η + S coincides with the solution ν to the equations for η restricted to S, and hence, by definition of S, η + S is stable.Then we apply the ergodic theorem for irreducible, continuous-time Markov chains (see e.g.Thm.3.8.1 of [44, p. 126]) to obtain lim t→∞ Moreover, we can construct η and η + on the same probability space, so that η i (t) ≤ η + i (t) for all i ∈ [N ] and all t ∈ R + : see [26, p. 867-8] for details.Hence we conclude that lim sup In the other direction, it is shown in [26, pp. 866-7], that for any ε > 0, by increasing the rate of departures from the system for every node in U , one can construct a process η ε which is stable, for which η i (t) ≥ η ε i (t) for all i ∈ [N ] and all t ∈ R + .In more detail, if ν = (ν i ) i∈[N ] is the solution to (2.11) for η, then η ε is a Jackson system with parameters λ ε , µ ε , P ε , where, for ε ≥ 0, λ ε i := λ i , and For i ∈ U , we have ν i ≥ µ i , and hence Q ε i := 1 − j∈[N ] P ε ij > 0 for ε > 0, so customers served at states in U may now depart the system.Note that, by (3.5), lim ε→0 µ ε = µ 0 and lim ε→0 P ε = P 0 .We also note that, for every ε ≥ 0, P ε is an irreducible substochastic matrix with at least one row sum strictly less than 1, so the matrix power (P ε ) k tends to 0 as k → ∞.
In particular, from (3.5), we have that for i ∈ S, lim ε→0 ρ ε i = lim ε→0 ν ε i /µ i = ρ i < 1, while for i ∈ U , lim ε→0 ρ ε i = lim ε→0 ν ε i /µ ε i = 1.Hence, by another application of the ergodic theorem, lim inf Taking ε → 0 in (3.6) and combining with (3.4), we obtain the a.s.convergence in (3.1); the L 1 convergence follows from the bounded convergence theorem.On the other hand, if i ∈ U , then, for any z ∈ N, Here, by Fatou's lemma, a.s., lim inf and then taking ε → 0 gives the a.s.convergence (3.3); again, bounded convergence gives the L 1 case.It remains to prove (3.2).Fix η(0) ∈ Z N + .The Goodman and Massey construction furnishes the coupling of η and η + such that η i (t) ≤ η + i (t) for all i ∈ [N ] and all t ∈ R + , where η + S is an ergodic Jackson network over S, i.e., max i∈S ρ + i < 1.Hence, by (3.1), there exists δ > 0 such that lim t→∞ P(η i (t) ≥ z) ≤ lim t→∞ P(η + i (t) ≥ z) ≤ e −δz for all i ∈ S and all z ∈ R + .Moreover, since a stable Jackson network is exponentially ergodic, as is established in Thm.2.1 of [21] via a Lyapunov function approach (see also [20]), or in [41] via a spectral gap approach, it follows that there exists δ > 0 such that P(η i (t) ≥ z) ≤ P(η + i (t) ≥ z) ≤ e −δz +e −δt for all i ∈ S and all t, z ∈ R + .Thus we can conclude that, for any ε > 0 there exists δ > 0 such that sup t≥εz P(η i (t) ≥ z) ≤ e −δz for all z ∈ R + and all i ∈ S. In addition, if ζ t denotes the total number of exogenous arrivals in the Jackson network η over time [0, t], then max i∈  Using Proposition 3.3, we show in Theorem 3.5 that each particle satisfies a law of large numbers with a deterministic asymptotic speed.Before stating that result, we state an algebraic result on the quantities that will play the roles of the speeds.Lemma 3.4.Let N ∈ N and suppose that (A ′ ) holds.Suppose that ν = (ν i ) i∈[N ] is the unique solution to the general traffic equation (2.11), whose existence is guaranteed by Proposition 3.3.Write ρ i := ν i /µ i , and adopt the convention that ρ 0 = ρ N +1 = 1.Define (3.7) Then, writing x + := x1{x ≥ 0} for x ∈ R, we have Here is our result on existence of speeds that underlies much of our analysis.
Theorem 3.5.Let N ∈ N and suppose that (A ′ ) holds.Then there exist is the unique solution to the general traffic equation (2.11), and ρ i := ν i /µ i , then the v i in (3.9) are the quantities defined by (3.7).
We first prove the lemma.
Proof of Theorem 3.5.Suppose that and N + i (t), t ∈ R + , be two independent homogeneous Poisson processes, of rates a i and b i , respectively.For all t ∈ R + , set η 0 (t) := +∞ and η N +1 (t) := +∞.Then we have the stochastic integral representation (3.11) since the attempted jumps are suppressed if the corresponding 'queues' are empty.Here, as usual η i (s−) := lim u↑s η i (u) for s > 0, and Then M ± i are continuous-time, square-integrable martingales, and we can re-write (3.11) as (3.12) Recall from the statement of Proposition 3.3 that S denotes the set of i ∈ [N ] such that ρ i < 1.Then we have from Proposition 3.3 that, for i ∈ [N ], a.s., lim Note that (3.13) also holds, trivially, for i = 0, N + 1.Let Then H + i is left continuous with right limits, and hence [46, p. 63] using [46, pp. 75-76], with the fact that [ is the quadratic ation of the (compensated) Poisson process [46, p. 71 2 is a non-negative, right-continuous submartingale.By an appropriate maximal inequality [35, p. 13], for any p > 1, Applying the Borel-Cantelli lemma with (3.14) along subsequence t = t n := 2 n , it follows that, a.s., for all but finitely many n ∈ Z + , sup for n = n(t) ∈ Z + , and so, a.s., for all t ∈ R + sufficiently large, It follows that, for any p ′ > p > 1, t −p ′ /2 |Y + i (t)| → 0, a.s., as t → ∞.Together with the analogous argument involving M − i , we have thus shown that, a.s., lim Combining (3.12) with (3.13) and (3.15), we conclude that X i (t)/t → v i , a.s., where This completes the proof of (3.9), with the v i given by (3.7), establishing the theorem.
A consequence of Proposition 3.3 is that S = [N ] (i.e., the system is stable) if and only if the (unique) solution ν to (2.11) satisfies ρ i := ν i /µ i < 1 for every i ∈ [N ].Any such ν thus solves also ν = νP + λ, i.e., ν(I − P ) = λ, (3.16)where I is the N by N identity matrix.We call (3.16) the stable traffic equation; note that unlike the general traffic equation (2.11), the system (3.16) is linear.The system (3.16) in fact classifies whether or not the system is stable, as the following result shows.Since at this point we invoke the formulas (2.16) and (2.17), we now need to assume (A).(i) The process η is stable if and only if ρ i < 1 for all i ∈ [N ].Equivalently, the process is stable if and only if λ(I − P ) −1 < µ, componentwise.
(ii) The ρ i are given by the explicit formula (2.16), where vN+1 is given by (2.17).
Proof.Part (i) is essentially just a specialization of the characterization of stable Jackson networks (which goes back to Jackson [31,32]; see e.g.[15, §2.1] or [19, §3.5]) to our setting; we give a proof using Proposition 3.3.Consider the stable traffic equation (3.16),where λ and P are given by (2.9) and (2.10) respectively.The matrix I − P is an M-matrix, and hence invertible (see Lemma 7.1 of [15]) and hence the solution ν = λ(I − P ) −1 of (3.16) exists and is unique.Proposition 3.3 says that if the system is stable, then there is a solution to (3.16) with ν i < µ i for all i.Conversely, if there exists a solution to (3.16) for which ν i < µ i for all i ∈ [N ], this solution is necessarily the (unique) solution to the general traffic equation (2.11), and the system is stable, by Proposition 3.3.This argument proves part (i).

Monotonicity, restriction, and mergers
In this section we study the relationship between the stability characterization of the whole system, in terms of appropriate traffic equations, as derived from the Jackson representation and presented in Propositions 3.3 and 3.6 above, with the traffic equations associated with certain sub-systems of the full system.This will allow us to characterize the maximal stable sub-systems, and show that their characteristic parameters can be expressed in terms of their intrinsic parameters only, and hence prove our main stability results, Theorems 2.1 and 2.3, and their corollaries presented in Section 2.
Let I = [ℓ; m] ⊆ [N + 1] be a discrete interval with m ≥ 2; recall that Also, recalling the definition of µ i from (2.8), define the matrix (P I i,j ) i,j∈I • by , for ℓ + 1 ≤ i ≤ ℓ + m − 2; with λ and P [N +1] = P defined by (4.1) and (4.2) coincide with the definitions given previously at (2.9) and (2.10).Given I ⊆ [N + 1], we call the system the reduced traffic equation corresponding to I.For a solution ν I = (ν I i ) i∈I • to (4.3), we write ρ I (i) := ν I i /µ i for all i ∈ I • .Then, similarly to (3.10), the reduced traffic equation can be written in terms of (ρ I i ) i∈I • as with the convention that ρ I min I−1 = ρ I max I = 1.Analogously to (3.7), we then define again with the convention ρ I min I−1 = ρ I max I = 1.Note that v I satisfies an appropriate version of the algebraic Lemma 3.4.
The next result concerns solutions of the reduced traffic equation (4.3).
Lemma 4.1.Let N ∈ N and suppose that (A) holds.Let I ⊆ [N + 1] be a discrete interval with |I| ≥ 2. There is a unique solution ν I to (4.3); equivalently, there is a unique solution ρ I to (4.4).Moreover, if ρ I (j) ≤ 1 for all j ∈ I • , then ρ I = ρI as defined at (2.4) and, for all i ∈ I, v I i = v(I) with the definitions at (4.5) and (2.2).Proof.Existence and uniqueness of the solution to (4.3) follows from the results of [26], exactly as existence and uniqueness of the solution to the general traffic equation (2.11) in Proposition 3.3.If ρ I (j) ≤ 1 for all j ∈ I • , then (4.3) coincides with the system ν I = ν I P I +λ I , a reduced version of the stable traffic equation (3.16), and the argument of Proposition 3.6 implies that ρ I and v I satisfy the appropriate versions of (2.16) and (2.17), which establishes that ρ I = ρI as defined at (2.4) and v I i = v(I) as defined at (2.2).
Given a discrete interval I ⊆ [N + 1], we say I is a candidate stable cloud if the solution to the reduced traffic equation (4.3), or, equivalently, the system (4.4),gives ρ I i < 1 for all i ∈ I • .To facilitate the proof of Theorem 2.3, verifying the correctness of Algorithm 2.4, we need to identify when candidate stable clouds are genuine stable clouds.Here the key property is that a candidate stable cloud is necessarily either a stable cloud, or a subset of a stable cloud; hence we need to test whether a candidate stable cloud is maximal, or whether it can be extended to a larger candidate stable cloud.The next two lemmas present results in this direction.
Lemma 4.2 is a consistency result, which shows, in particular, that the solution ν to the general traffic equation (2.11), when restricted to a stable cloud θ ∈ Θ, coincides with ν θ , the solution to the reduced traffic equation (4.3) for I = θ.In other words, the ρ I i solve exactly the system (4.4) over I = I 0 , which is also the system solved by the ρ I 0 i .By uniqueness of solutions to the reduced traffic equation (4.3) (see Lemma 4.1), it follows that ν I i = ν I 0 i for all i ∈ I • 0 .Lemma 4.3 gives conditions when a candidate stable cloud is further stabilized by the external system, and will allow us to extend a candidate stable cloud either to the left or to the right.The proof uses similar restriction ideas to the proof of Lemma 4.2.
Lemma 4.3.Let N ∈ N and suppose that (A) holds.Let I ⊆ [N + 1] be a discrete interval with |I| ≥ 2, and let I 0 ⊆ I be such that ρ I 0 i ≤ 1 for all i ∈ I • 0 .(i) Suppose that min I 0 = min I, max I 0 < max I, and ρ I max I 0 < 1.Then v I i < v I 0 i for all i ∈ I • 0 .
(ii) Suppose that min I 0 > min I, max I 0 = max I, and ρ I min Moreover, if either of the conditions in (i) or (ii) hold, then ρ I i < ρ I 0 i for all i ∈ I • 0 .Proof.Suppose that the conditions in (i) hold; for convenience, let K = max I 0 .Since ν I 0 solves (4.3) for I 0 , and ρ I 0 i ≤ 1 for all i ∈ I • 0 , it follows from Lemma 4.1 that ρ I 0 i = ρI 0 (i), i ∈ I • 0 , for ρI 0 as defined at (2.4).Since ν I solves (4.K < b K , it follows that ρ I j = ρ′ I 0 (j) < ρI 0 (j) = ρ I 0 j for all j ∈ I • 0 , as claimed.In particular, ρ I j < 1 for all j ∈ I • 0 .Let L = min I = min I 0 .Then, from (4.5) and the fact that ρ I L < ρ and, by the appropriate version of (3.8), v I 0 j = v I 0 L > v I L = v I j for all j ∈ I • 0 .This completes the proof of the lemma in the case where the hypotheses of (i) hold.The argument in the case that the hypotheses in (ii) hold is similar.
For statement (iii), existence of the v i follows from Theorem 3.5.Lemma 3.4 shows that, for each θ ∈ Θ, v i is the same for all i ∈ θ • ; call the common value v(θ).The fact that v(θ) satisfies (2.2) and (2.3) follows from Lemma 4.1.This completes the proof of the theorem.
The following theorem will be our main tool to prove Theorem 2.3: it takes the results of Lemmas 4.1-4.3 and presents them in terms of pairwise comparisons of adjacent candidate stable clouds, tailored to the structure of Algorithm 2.4.In particular, Theorem 4.4 shows that if one has two adjacent candidate stable clouds, in which the cloud to the left has a greater intrinsic speed, then the union of the two candidate stable clouds is itself a candidate stable cloud.
Recall the definition of v from (2.2).Let I := I 1 ∪ I 2 , which is also a discrete interval subset of [N + 1].Then the following hold.
(i) If v(I 1 ) > v(I 2 ), then ρ I i < 1 for all i ∈ I • , and v(I 1 ) > v(I) > v(I 2 ).Moreover, Consider also the solution ν I to the reduced traffic equation (4.3) over I := I 1 ∪ I 2 .Suppose first that ρ I (K) ≥ 1, i.e., ν I K ≥ µ K .Then Lemma 4.2, applied first with I 0 = I 1 and second with I 0 = I 2 , implies that (4.8) holds.From (4.5), this means that v I i = v I 1 i for all i ∈ I 1 and v I i = v I 2 i for all i ∈ I 2 .Theorem 3.5 and (3.8) (appropriately adapted) imply that if ρ I (K) = 1, we have v I K = v I K+1 , while if ρ I K > 1, we have v I K < v I K+1 .Thus it follows from (4.9) that ρ I K > 1 implies v(I 1 ) < v(I 2 ); (4.10) On the other hand, Lemma 4.3 shows that For ease of notation, write X 1 (t) := X 1 (t) − vN+1 t.As described in Remarks 2.14(b), from the results of [4] one can directly deduce the marginal central limit theorem for a constant σ 2 ∈ (0, ∞).Take s, t with 0 < t − s < t < ∞, and let E s,t (w Since ξ and η defined by (1.2) and (1.3) are both Markov, we have that P(η(t) = z | ξ(t − s)) depends on ξ(t − s) only through η(t − s), and not X 1 (t − s).Hence Then we can choose s 0 large enough (depending on ε and z) such that, for all s, t with s ≥ s 0 and t − s ≥ s 0 , P[η(t − s) ∈ A] ≥ 1 − 2ε, and The preceding argument, together with a similar argument for a lower bound, shows that for every ε > 0, z ∈ Z N + , and w ∈ R, we may choose and fix s (large) such that, for all t with t − s large enough, Now from (5.1) we have that (σ 2 (t−s)) −1/2 X 1 (t−s) converges in distribution to N (0, σ 2 ) as t → ∞.For fixed s, lim t→∞ t t−s = 1, and so lim t→∞ P[E s,t (w)] = Φ(w), the standard normal distribution function.Moreover, it is not hard to see that, since X 1 has uniformly bounded jump rates with increments ±1, for fixed s.Hence, for fixed z ∈ Z N + , for all t large enough it holds that using continuity of Φ. Combining these bounds we conclude that, for every ε > 0 and all t sufficiently large This completes the proof.
Finally, we turn to the proof of Theorem 2.15, the main part of which is to compute the limiting variance σ 2 in the N = 1 case of Theorem 2.13.
Hence, using the total variance formula and collecting the above computations, we get From here, we verify (5.2), to conclude the proof.
6 Discussion and connection to continuum models

Diffusions with rank-based interactions
Continuum (diffusion) models, in which N + 1 particles perform Brownian motions with drifts and diffusion coefficients determined by their ranks, have been extensively studied in recent years, and include the Atlas model and its relatives: see [9,14,28,29,34,45,49,50,55] and references therein.An informative recent overview of the literature is given in [50, §2].In this section we describe these models using some stochastic differential equations (SDEs) rather informally, so that we can make comparisons; precise formulations, existence and uniqueness results can be found in the references cited.In the early versions of these models (e.g.[9,45]), the particles move independently, and are allowed to pass through each other, but exchange drift and diffusion parameters when they do so.For t ∈ R + , denote by y i (t) the location of particle with label i ∈ [N +1], and let x i (t) denote the location of the particle with rank i (i = 1 being the leftmost particle), so x 1 (t) ≤ • • • ≤ x N +1 (t) is an ordering of y 1 (t), . . ., y N +1 (t).Also let r i (t) denote the rank of particle with label i.Let u 1 , . . ., u N +1 ∈ R and σ 1 , . . ., σ N +1 ∈ (0, ∞) be collections of drift and volatility parameters.Let W 1 , . . ., W N +1 be independent, standard R-valued Brownian motions.Suppose that i.e., particle i performs Brownian motion with drift and volatility parameters determined by its rank; if u j ≡ u and σ j ≡ σ are constant, the particles are independent, otherwise, there is interaction mediated by the ranks.The Atlas model [9,29] has where L i (t) is the local time at 0 of x i − x i−1 up to time t, with the convention L 1 ≡ L N +2 ≡ 0. The local-time terms in (6.2) maintain the order of the x i , while each particle x i has its own intrinsic drift and diffusion coefficients, so the continuum process x = (x 1 , . . ., x N +1 ) shares key features with our particle process X = (X 1 , . . ., X N +1 ) described in Section 1.However, it turns out that there are important differences inherent in the reflection mechanisms; before expanding on this, we introduce an extension of (6.2) to more general reflections.
As far as we are aware, long-time stability of the continuum system described above has been studied only as far as classifying whether or not the whole system is stable, rather than identifying a full cloud decomposition as we do in Theorem 2.1.In the symmetric case, the criterion for stability was obtained in [45,Thm. 8] (for the case of constant diffusion coefficients) and [9,29], and in the stable case the asymptotic speed was derived; we summarize these results as follows, referring to e.g.[50, Prop.2.2] and [29, Prop.2] for precise formulations.The stability condition here is comparable to (2.19), but different in detail; we elaborate on this in the next section.In the general (asymmetric) setting of (6.4), the criterion for stability is also known: see e.g.[50, Prop.2.1], and involves the inverse of a tridiagonal matrix, reminiscent of the Jackson criterion (see Remarks 2.8).In the next section, we turn to the relationship between the continuum model and lattice models, which is multifaceted but has been partially elaborated in [34, §3].

Particle models with elastic collisions
It turns out that a suitable discrete-space parallel to the continuum model in Section 6.1 is obtained by modifying the collision mechanism for the model in Section 1 from the exclusion rule to an elastic collision rule.We sketch here only the main idea; a pertinent discussion can also be found in [34, §3].
In the case N = 1, two particles have jump rates a 1 , b 1 and a 2 , b 2 .Under the exclusion rule, if the particles are adjacent, the total activity is reduced (particle 1 jumps left at rate a 1 , while particle 2 jumps right at rate b 2 ).Suppose instead the total activity is maintained, by particles transferring 'momentum' to their neighbours when they attempt to jump, so that particle 1 jumps left at rate a 1 + a 2 , while particle 2 jumps right at rate b 1 +b 2 .We call this the elastic collision rule; see Figure 3 for an illustration for a 6-particle  configuration.A slight variation on this is to modify the process so that 'collisions' mean that two particles occupy the same site, rather than adjacent sites.The elastic interaction is then equivalent to the two particles being permitted to pass each other, exchanging jump parameters as they do so (similarly to the diffusion model described in Section 6.1).The elastic model is more homogeneous than the exclusion model, in the following sense: regardless of the present configuration, the rate at which a single coordinate changes by +1 is constant (namely B N +1 := N +1 i=1 b i ), as is the rate at which a single coordinate changes by −1 (A N +1 := N +1 i=1 a i ).Another way of saying this is that the centre of mass of the particle system performs a continuous-time simple random walk on (N + 1) −1 Z with left jump rate A N +1 and right jump rate B N +1 ; thus it has speed U N +1 := (B N +1 − A N +1 )/(N + 1).Furthermore, considering just the leftmost k ∈ [N + 1] particles, their centre of mass has jump rates A k , B k , and speed U k = (B k − A k )/k.By a similar reasoning to Algorithm 2.4, it becomes very plausible that the condition for stability of the system should be U k > U N +1 for all k ∈ [N ], exactly as in Theorem 6.1.It is natural to conjecture: Conjecture 6.2.In the lattice model with elastic collisions, the system consists of a single stable cloud if and only if U k > U N +1 for all k ∈ [N ].Moreover, if the system is stable, there is a strong law of large numbers with limiting speed U N +1 .
As this model is not the focus of the present paper, we do not attempt to establish Conjecture 6.2 here.Additionally, we anticipate that a similar algorithm to Algorithm 2.4, based on the (somewhat simpler) algebra of cloud speeds in the elastic case, can be developed to obtain the full cloud decomposition for the elastic model.This stability criterion in Conjecture 6.2 coincides with ours in Corollary 2.7 when N = 1 (but not for N ≥ 2), and even for N = 1, the limiting speed is different.As an example, take N =

Scaling limits and heavy traffic
We indicate briefly, by an example, how the diffusion models of Section 6.1 arise as scaling limits of certain near-critical particle system models, and that the limits are elastic even when the particle system is not: see [34, §3] for more detail.As an elementary starting point, suppose that X (s) t , t ∈ R + is a continuous-time, simple random walk on Z with jump rate 1 to the left and 1 + s −1/2 u to the right, where u ∈ R and s ∈ (0, ∞) is a scale parameter.Then E(X

Theorem 2 . 3 .
Let N ∈ N and suppose that (A) holds.Then Algorithm 2.4 below produces the unique partition Θ featuring in Theorem 2.1.Algorithm 2.4.The algorithm takes as input data N ∈ N and the parameters a i , b i ∈ (0, ∞), i ∈ [N + 1], and outputs an ordered partition Θ of [N + 1].

1 ,
. . ., θ κ+1 n−1 ).Iterate κ → κ + 1 and return to Step 2. Remarks 2.5.(a) If |Θ κ | = 1 at any point, then Step 3 will terminate the algorithm; thus whenever Step 2 is executed, one has n = |Θ κ | ≥ 2. Similarly, the set J in Step 4 will always be non-empty.(b) The merger executed in Step 4 reduces by one the number of parts in the partition, so Algorithm 2.4 will terminate in at most N iterations.
e., now the first and the last particles are dogs with drifts directed inside, and the particles between those are sheep.Then, we readily find ρ k = a for k = 1, . . ., N and vN+1 = 0 (the last fact is clearly a consequence of symmetry).In particular, the expected size of the cloud is linear in N , unlike in Example 2.16.△ Example 2.18 (Sheep between two dogs, asymmetric case).Assume that a 1 = a and b N +1 = b with 0 < a ≤ b < 1, and b 1 = a 2 = b 2 = • • • = a N +1 = 1; this is a generalization of Example 2.17, in which the drift of the left dog is permitted to be stronger than the right.Here, (2.16) and (2.17) imply that vN+1 = b−a N +1 ≥ 0 and ρ k = a + (b−a)k N +1 .△ Example 2.19 (Recurrence of small systems with constant drifts for N ≤ 2).Consider again the setting of Example 2.9, where b g. [15, Ch. 2 & 7] or [52, Ch. 1].
and this decays exponentially in z, by standard Poisson tail bounds.Combining these bounds yields (3.2).

Lemma 4 . 2 .
Let N ∈ N and suppose that (A) holds.Let I ⊆ [N +1] be a discrete interval with |I| ≥ 2, and let I 0 ⊆ I be such that (i) either min I 0 = min I or ρ I min I 0 −1 ≥ 1, and (ii) either max I 0 = max I or ρ I max I 0 ≥ 1.Then ν I i = ν I 0 i for all i ∈ I • 0 .Proof.Consider the solution ν I to the reduced traffic equation (4.3) over I, and let ρ I i := ν I i /µ i .Since ν I solves (4.3) for I, we have from (4.4) that the ρ I i , i ∈ I • 0 , satisfy ) with boundary conditions ρ I min I 0 −1 = 1 (because either min I 0 = min I, in which case ρ I min I 0 −1 = ρ I min I−1 = 1, the left boundary condition for (4.4) over I, or min I 0 > min I and ρ I min I 0 −1 ≥ 1, so 1 ∧ ρ I min I 0 −1 = 1) and ρ I max I 0 = 1 (analogously).

Theorem 4 . 4 .
Let N ∈ N and suppose that (A) holds.Let I 1 and I 2 be disjoint discrete interval subsets of [N + 1], with K := max I 1 = min I 2 − 1.Let ν I k be the solution to the reduced traffic equation (4.3) with I = I k , k ∈ {1, 2}.Define ρ I k I 1 and I 2 be disjoint discrete interval subsets of [N + 1], with K := max I 1 = min I 2 − 1.Let ν I k be the solution to the reduced traffic equation (4.3) with I = I k , k ∈ {1, 2}.Define ρ I k i := ν I k i /µ i , and define v I k by (4.5).Suppose that ρ I 1

Theorem 6 . 1 .
For the system with symmetric collisions defined by (6.2), writeU k := k −1 k j=1 u j for k ∈ [N +1].The system is stable if and only if U k > U N +1 for all k ∈ [N ].Moreover, if stable, there is a strong law of large numbers with limiting speed U N +1 .

Figure 3 :
Figure3: Schematic of a lattice model with N + 1 = 6 particles and an elastic collision rule.Filled circles represent particles, empty circles represent unoccupied lattice sites.Within each block of adjacent particles, only the particles at the extreme left or right can jump, but their jump rates are the sum of the intrinsic rates for all particles in the block.An essentially equivalent model is obtained by permitting particles to pass through each other, exchanging parameters as they do so.
1 and a 1 = a, b 1 = b, a 2 = b 2 = 1.Then the stable case of the exclusion model (b > a) has asymptotic speed v2 = (b − a)/(1 + b), while the stable case of the elastic model has asymptotic speed U 2 = (b − a)/2.
a.s., and ζ t is Poisson with mean Ct for C