The \(q\)-PushASEP: A New Integrable Model for Traffic in \(1+1\) Dimension

Abstract

We introduce a new interacting (stochastic) particle system \(q\)-PushASEP which interpolates between the \(q\)-TASEP of Borodin and Corwin (Probab Theory Relat Fields 158(1–2):225–400, 2014; see also Borodin et al., Ann Probab 42(6):2314–2382, 2014; Borodin and Corwin, Int Math Res Not 2:499–537, 2015; O’Connell and Pei, Electron J Probab 18(95):1–25, 2013; Borodin et al., Comput Math, 2013) and the \(q\)-PushTASEP introduced recently (Borodin and Petrov, Adv Math, 2013). In the \(q\)-PushASEP, particles can jump to the left or to the right, and there is a certain partially asymmetric pushing mechanism present. This particle system has a nice interpretation as a model of traffic on a one-lane highway. Using the quantum many body system approach, we explicitly compute the expectations of a large family of observables for this system in terms of nested contour integrals. We also discuss relevant Fredholm determinantal formulas for the distribution of the location of each particle, and connections of the model with a certain two-sided version of Macdonald processes and with the semi-discrete stochastic heat equation.

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Notes

  1. 1.

    One needs to additionally justify that \(\mathrm {gap}_{i}\) indeed evolves according to this one-dimensional Markov chain.

  2. 2.

    One should also impose reasonable growth and decay assumptions on the \(a_i\)’s.

  3. 3.

    When \(\mathsf {L}=0\), i.e., for the \(q\)-TASEP, all operations are valid, see [5] and [10].

  4. 4.

    There is no unique way of defining a dynamics on two-dimensional interlacing arrays with these properties. For instance, the “push-block” dynamics may be replaced by the dynamics coming from the \(q\)-version of the Robinson-Schensted column insertion algorithm introduced in [22]. See also [12] for more examples and a general discussion.

  5. 5.

    This mechanism of instantaneous pushes is built into the jump rates. Indeed, if the interlacing is broken, then the higher particles have infinite jump rates due to vanishing denominator. Moreover, if the jump of some \(\lambda ^{(k)}_{j}\) would break the interlacing with lower particles, then the rate assigned to this jump is equal to zero.

  6. 6.

    The \(q=0\) version of the two-sided Macdonald processes (i.e., the two-sided Schur processes) was introduced and investigated in [4].

  7. 7.

    These are our quantities \(G^{(k)}_{k}\) in the description of the scaling.

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Acknowledgments

The authors would like to thank Alexei Borodin for very helpful discussions and remarks. IC was partially supported by the NSF through DMS-1208998 as well as by Microsoft Research through the Schramm Memorial Fellowship, and by the Clay Mathematics Institute through a Clay Research Fellowship. LP was partially supported by the RFBR-CNRS Grant 11-01-93105.

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Correspondence to Ivan Corwin.

Appendices

Appendix 1: Dynamics on Two-Dimensional Interlacing Arrays

Here we briefly explain how the \(q\)-PushASEP arises as a one-dimensional marginal of a certain two-dimensional stochastic Markov dynamics on interlacing arrays of particles. This two-dimensional dynamics may be constructed as an interpolation between the “push-block” dynamics of [5, Sect. 2.3.3] (see also Dynamics 1 in [12, Sect. 5.5]), and the \(q\)-version of the dynamics driven by row insertion RSK algorithm (Dynamics 8 in [12, Sect. 8.2.1]).Footnote 4 Note that the latter dynamics has to be reflected, i.e., the particles under this dynamics must jump to the left instead of jumping to the right. Let us now proceed to the definition of the two-dimensional dynamics.

The state space of the two-dimensional dynamics is the set of triangular arrays of interlacing particles which have integer coordinates (see Fig. 4 for an example):

$$\begin{aligned} \varvec{\lambda }=\{\lambda ^{(k)}_{j}\in \mathbb {Z},\; 1\le j\le k\le N:\lambda ^{(k)}_{j}\le \lambda ^{(k-1)}_{j-1}\le \lambda ^{(k)}_{j-1}\}. \end{aligned}$$

Each particle \(\lambda ^{(k)}_{j}\) can jump either to the right or to the left by one.

Fig. 4
figure4

Particle configuration \(\varvec{\lambda }\) and a visualization of the interlacing property

The right jumps are described as follows. Each particle \(\lambda ^{(k)}_{j}\) has an independent exponential clock with rate

$$\begin{aligned} \mathsf {R}a_k \frac{\left( 1-q^{\lambda ^{(k-1)}_{j-1}-\lambda ^{(k)}_{j}}\right) \left( 1-q^{\lambda ^{(k)}_{j}-\lambda ^{(k)}_{j+1}+1}\right) }{1-q^{\lambda ^{(k)}_{j}-\lambda ^{(k-1)}_{j}+1}}. \end{aligned}$$

When the clock of \(\lambda ^{(k)}_{j}\) rings, the particle jumps to the right by one. If this jump of \(\lambda ^{(k)}_{j}\) would break the interlacing with upper particles, i.e., if \(\lambda ^{(k)}_{j}=\lambda ^{(k+1)}_{j}=\cdots =\lambda ^{(k+m)}_{j}\) (for some \(m\ge 1\)), then all the particles \(\lambda ^{(k+1)}_{j},\ldots ,\lambda ^{(k+m)}_{j}\) are instantaneously pushed to the right by one.Footnote 5

The left jumps are different. Only the leftmost particles \(\lambda ^{(k)}_{k}\) can independently jump to the left by one. At level \(k\) of the array the independent jumps of left particles happen at rate \(\mathsf {L}a_k^{-1}\). When any particle \(\lambda ^{(k-1)}_{j}\) moves to the left by one (independently or due to a push), it instantaneously forces one of its two immediate upper neighbors, \(\lambda ^{(k)}_{j+1}\) or \(\lambda ^{(k)}_{j}\), to move to the left by one with probabilities \(\ell \) and \(1-\ell \), respectively, where

$$\begin{aligned} \ell =q^{\lambda ^{(k-1)}_{j}-\lambda ^{(k)}_{j+1}} \frac{1-q^{\lambda ^{(k)}_{j+1}-\lambda ^{(k-1)}_{j+1}}}{1-q^{\lambda ^{(k-1)}_{j}-\lambda ^{(k-1)}_{j+1}}} \end{aligned}$$

(here \(\lambda ^{(k-1)}_{j}\) denotes the position of the particle before the move).

In the description of the dynamics, all factors of the form \((1-q^{\cdots })\) having nonexistent indices are set to be equal to one. One can readily see that the leftmost particles under this two-sided dynamics on two-dimensional interlacing arrays marginally evolve as a Markov process. In the shifted coordinates \(x_n(t):=\lambda ^{(n)}_{n}(t)-n\), where \(n=1,\ldots ,N\), the evolution of the particles is governed by our \(q\)-PushASEP.

The fixed-time distributions of the two-dimensional dynamics \(\varvec{\lambda }(t)\) described above are probability measures on interlacing arrays. Let the initial configuration be the densely packed one, i.e., \(\lambda ^{(k)}_{j}(0)=0\) for all \(1\le j\le k\le N\). This configuration corresponds to the step initial condition for the \(q\)-PushASEP.

After time \(t\ge 0\), the distribution of \(\varvec{\lambda }(t)\) generalizes the (one-sided) Macdonald processes of [5, 8]. The second Macdonald parameter which is usually denoted by \(t\) is set to zero (so that there is no notational conflict with the time parameter); such Macdonald processes are also referred to as the \(q\)-Whittaker processes.

Put \(a_i\equiv 1\) for simplicity. If \(\mathsf {L}\) is zero, then \(\varvec{\lambda }(t)\) is distributed according to

$$\begin{aligned} {{\mathrm{\mathrm {Prob}}}}\big (\varvec{\lambda }(t)\big )=\frac{1}{Z} P_{\lambda ^{(1)}}(1)P_{\lambda ^{(2)}/\lambda ^{(1)}}(1) \cdots P_{\lambda ^{(N)}/\lambda ^{(N-1)}}(1)Q_{\lambda ^{(N)}}(\rho _{\mathsf {R}t}), \end{aligned}$$
(5.1)

where each \(\lambda ^{(k)}=(\lambda ^{(k)}_{1}\ge \cdots \ge \lambda ^{(k)}_{k})\in \mathbb {Z}^{k}\) is an ordered collection of nonnegative integers, \(P\) and \(Q\) are the (ordinary and skew) Macdonald symmetric functions [19], and \(\rho _{\mathsf {R}t}\) is the so-call Plancherel specialization of \(Q_{\lambda ^{(N)}}\), e.g., see [5, Sect. 2.2.1]. The Plancherel specialization may be defined, e.g., in terms of the generating function for the one-row Macdonald \(Q\) functions (i.e., functions indexed by ordered \(k\)-tuples of integers with \(k=1\)):

$$\begin{aligned} \sum _{n\ge 0}Q_{(n)}(\rho _t)u^{n}=e^{t u}. \end{aligned}$$
(5.2)

On the other hand, for \(\mathsf {R}=0\), the distribution of \(-\varvec{\lambda }(t)\) (this simply means negating all components of the interlacing array) is described by the Macdonald process (5.1) (with \(\mathsf {R}t\) replaced by \(\mathsf {L}t\) in the Plancherel specialization of \(Q_{-\lambda ^{(N)}}\)).

In the general case when \(\mathsf {L}\) and \(\mathsf {R}\) are both positive, we expect that the distribution of \(\varvec{\lambda }(t)\) (started from the packed initial configuration) is given by a certain two-sided version of a Macdonald process. This two-sided version should necessarily have the form

$$\begin{aligned} {{\mathrm{\mathrm {Prob}}}}\big (\varvec{\lambda }(t)\big )=\frac{1}{Z} P_{\lambda ^{(1)}}(1)P_{\lambda ^{(2)}/\lambda ^{(1)}}(1) \cdots P_{\lambda ^{(N)}/\lambda ^{(N-1)}}(1) \mathcal {M}_{N}^{(\mathsf {R}t;\mathsf {L}t)}(\lambda ^{(N)}) \end{aligned}$$
(5.3)

for a suitable nonnegative function \(\mathcal {M}_{N}^{(\mathsf {R}t;\mathsf {L}t)}\) on the \(N\)th floor (cf. (5.1)). Note that here the coordinates \(\lambda ^{(k)}_j\) can be positive or negative (but still must interlace).

Indeed, the product of the \(P\) functions, \(P_{\lambda ^{(1)}}(1)P_{\lambda ^{(2)}/\lambda ^{(1)}}(1) \ldots P_{\lambda ^{(N)}/\lambda ^{(N-1)}}(1)\), corresponds to a certain Gibbs property of Macdonald processes (see [12] for more detail) which is preserved by both the dynamics with \(\mathsf {L}=0\) or \(\mathsf {R}=0\), and thus also by the dynamics with general positive \(\mathsf {R}\) and \(\mathsf {L}\) (this is because the Markov generator of the latter process is a linear combination of the two “pure” right and left generators).

When \(N=1\), the measure (5.3) is simply the convolution of the two “pure” one-sided measures (note that \(P_{\lambda ^{(1)}}(1)=1\)), and so the generating function for \(\mathcal {M}_{1}^{(\mathsf {R}t;\mathsf {L}t)}\) takes the form (cf. (1.4))

$$\begin{aligned} \sum _{n\in \mathbb {Z}}\mathcal {M}_{1}^{(\mathsf {R}t;\mathsf {L}t)}(n) u^{n}=e^{t(\mathsf {R}u+\mathsf {L}u^{-1})}. \end{aligned}$$
(5.4)

Note that in the one-sided case, the one-row functions \(Q_{(n)}\) generate the algebra of symmetric functions to which all the \(Q_{\lambda }\)’s (with \(\lambda \) having nonnegative parts) belong. Thus, identity (5.2) defines \(Q_{\lambda }(\rho _t)\) for all \(\lambda \), and one can proceed to the definition of the one-sided Macdonald processes. In the two-sided case, it is not clear what algebraic structures are responsible for the passage from \(\mathcal {M}_{1}^{(\mathsf {R}t;\mathsf {L}t)}(n)\) (viewed as one-row functions \(Q_{(n)} (\rho _{\mathsf {R}t;\,\mathsf {L}t}^{\text {two-sided}})\)) to the functions \(Q_\lambda \) with \(\lambda \) general. Therefore, at this point we are left to view (5.3) as a defn of the two-sided Plancherel specialization of the general Macdonald symmetric functions \(Q_{\lambda ^{(N)}}(\rho _{\mathsf {R}t;\,\mathsf {L}t}^{\text {two-sided}}) :=\mathcal {M}_{N}^{(\mathsf {R}t;\mathsf {L}t)}(\lambda ^{(N)})\) corresponding to not necessarily one-row \(\lambda \)’s. We do not further develop the theory of two-sided Macdonald processes in the present paper, but note that the desire to understand the distribution of the two-sided dynamics on two-dimensional interlacing integer arrays (5.3), as well as the question of proving Conjecture 1.4, provide some motivation for these objects.Footnote 6

Appendix 2: Formal Scaling Limit as \(q\nearrow 1\)

Consider the scaling of the two-dimensional dynamics described by [5, Thm. 4.1.21]:

Here \(\tau >0\) is the scaled time, \(C(\varepsilon ;\tau )\) represents the global shift of the coordinate system, and \((\mathsf {a}_1,\ldots ,\mathsf {a}_N)\) are the scaled values of the \(a_j\)’s. In the one-sided setting, the Macdonald processes (5.1) converge under this scaling with \(C(\varepsilon ;\tau )=\varepsilon ^{-2}\tau \) to Whittaker processes introduced in [21], see also [5, Ch. 4].

As explained in [5, Sects. 4.1 and 5.2] and [12, Sect. 8.4], the \(q\)-TASEP and the \(q\)-PushTASEP (i.e., the “pure” dynamics corresponding to \(\mathsf {L}=0\) or \(\mathsf {R}=0\)) under this scaling with \(C(\varepsilon ;\tau )=+\varepsilon ^{-2}\tau \) or \(C(\varepsilon ;\tau )=-\varepsilon ^{-2}\tau \), respectively, correspond to stochastic differential equations (SDEs) which describe evolution of the hierarchy of the free energies of the O’Connell–Yor semi-discrete directed polymer [21, 23].Footnote 7 These free energies may also be represented as logarithms of solutions to the semi-discrete stochastic heat equation

$$\begin{aligned} du_j(t)=u_{j-1}(t)-u_j(t)+u_j(t)dB_j(t),\qquad j=1,\ldots ,N;\qquad \qquad u(0,N)=\delta _{1N},\qquad \end{aligned}$$
(6.1)

where \(B_1,\ldots ,B_N\) are independent standard Brownian motions (possibly with linear drifts).

Let us now discuss the formal scaling limit of the two-sided (\(q\)-PushASEP) evolution, i.e., with \(\mathsf {R},\mathsf {L}>0\). Let us scale the \(\mathsf {R}\) and \(\mathsf {L}\) parameters around 1:

$$\begin{aligned} \mathsf {R}=e^{-\varepsilon \mathsf {r}},\qquad \mathsf {L}=e^{-\varepsilon \mathsf {l}}, \end{aligned}$$

where \(\mathsf {r}, \mathsf {l}\in \mathbb {R}\) are the scaled values. Moreover, one should take the global shift \(C(\varepsilon ;\tau )\) to be zero (one should think that the shifts \(\pm \varepsilon ^{-2}\tau \) corresponding to the “pure” right and left dynamics compensate each other).

We will focus only on the leftmost particles \(\lambda ^{(k)}_{k}\), the whole array can be considered in a similar way. The limiting SDEs for the quantities \(G^{(k)}_{k}\) look as (with the agreement that \(G^{(0)}_{0}\equiv 0\))

$$\begin{aligned} dG^{(k)}_{k}=\sqrt{2} \cdot dW_{k} + \left( -2\mathsf {a}_k+\mathsf {l} -\mathsf {r}-e^{G^{(k)}_{k}-G^{(k-1)}_{k-1}} \right) d\tau ,\qquad k=1,\ldots ,N. \end{aligned}$$
(6.2)

Here \(W_1,\ldots ,W_N\) are independent standard driftless Brownian motions.

Remark 6.1

The \(G^{(k)}_{k}\)’s satisfying (6.2) can also be formally interpreted as logarithms of solutions to the semi-discrete stochastic heat equation (6.1). The terms \((-2\mathsf {a}_k+\mathsf {l} -\mathsf {r})\) are absorbed into drifts of the Brownian motions \(B_1,\ldots ,B_N\) in (6.1).

Calculations leading to (6.2) are analogous to what is done in [5, Sect. 5.4.4] and [12, Sect. 8.4.4]. First, note that our scaling dictates

$$\begin{aligned} G^{(k)}_{k}(\tau +d\tau )-G^{(k)}_{k}(\tau )=\frac{\lambda ^{(k)}_{k}(\tau +\varepsilon ^{-2}d\tau ) -\lambda ^{(k)}_{k}(\tau )}{\varepsilon ^{-1}}. \end{aligned}$$
(6.3)

Right jumps of the particle \(\lambda ^{(k)}_{k}\) occur with probability

$$\begin{aligned} \mathsf {R}a_k(1-q^{\lambda ^{(k-1)}_{k-1}-\lambda ^{(k)}_{k}}) =1-\varepsilon (\mathsf {a}_k+\mathsf {r} + e^{G^{(k)}_{k}-G^{(k-1)}_{k-1}})+O(\varepsilon ^{2}). \end{aligned}$$
(6.4)

Left jumps happen at rate

$$\begin{aligned} \mathsf {L}a_k^{-1}=e^{-\varepsilon (\mathsf {l}-\mathsf {a}_k)} =1-(\mathsf {l}-\mathsf {a}_k)\varepsilon +O(\varepsilon ^{2}), \end{aligned}$$
(6.5)

and, moreover, the particle \(\lambda ^{(k)}_{k}\) is pushed to the left by \(\lambda ^{(k-1)}_{k-1}\) with probability \(\varepsilon e^{G^{(k)}_{k}-G^{(k-1)}_{k-1}}+O(\varepsilon ^{2})\). One should multiply this probability by the change in the position of \(\lambda ^{(k-1)}_{k-1}\) during time interval \(t=\varepsilon ^{-2}\tau \), this yields

$$\begin{aligned} \Big (\varepsilon e^{G^{(k)}_{k}-G^{(k-1)}_{k-1}}+O(\varepsilon ^{2})\Big ) \Big ( \varepsilon ^{-1} \big ( G^{(k-1)}_{k-1}(\tau +d\tau ) -G^{(k-1)}_{k-1}(\tau ) \big ) \Big )=O(1). \end{aligned}$$
(6.6)

The constant factors in (6.4) and (6.5) give rise to the change in the position of \(\lambda ^{(k)}_{k}\) (during time interval \(\varepsilon ^{-2}d\tau \)) equal to the difference of two independent Poisson random variables with mean \(\varepsilon ^{-2}d\tau \). In view of (6.3), these summands correspond to the differential of the Brownian motion \(\sqrt{2} \cdot dW_{k}(\tau )\). The summands of order \(\varepsilon \) in (6.4)–(6.5) give rise to constant terms. The constant term (6.6) is multiplied by \(\frac{1}{\varepsilon ^{-1}}\), and thus vanishes.

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Corwin, I., Petrov, L. The \(q\)-PushASEP: A New Integrable Model for Traffic in \(1+1\) Dimension. J Stat Phys 160, 1005–1026 (2015). https://doi.org/10.1007/s10955-015-1218-9

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Keywords

  • Integrable probability
  • Kardar–Parisi–Zhang universality class