Multi-Class Invariant Distributions in TASEP

• The paper presents a detailed derivation of multi-class invariant distributions using tandem queueing and last-passage percolation techniques.
• It demonstrates that sequential updates (R1/R2) yield universal, homogeneous product measures while sublattice parallel updates (R3) produce spatially inhomogeneous, β-dependent measures.
• The work analyzes how update parameters influence collision probabilities and asymptotic particle speeds, highlighting duality between speed processes and invariant measures.

The multi-class invariant distributions for discrete-time TASEP provide a rigorous characterization of the stationary (invariant) measures for totally asymmetric simple exclusion processes (TASEP) with several particle types under various discrete-time update rules. The structure of these invariant measures is fundamentally connected to the interplay between stochastic particle dynamics, queuing constructions, last-passage percolation, and update protocols. This article surveys the core mathematical principles, methodologies, and results underlying these invariant distributions, with a focus on update-scheme dependence, speed process duality, and explicit probabilistic constructions.

1. Discrete-Time TASEP Rules and Invariant Measures

Discrete-time TASEP differs from the standard continuous-time process by the way jump attempts are scheduled and executed. Three principal update rules are central to the classification of invariant measures:

• Sequential Updates (R1, R2):
  • R1: Updates processed from right to left.
  • R2: Updates processed from left to right.
  • Both admit only translation-invariant stationary measures that are Bernoulli product measures $\nu_{(\rho)} = \bigotimes_{x} \text{Ber}(\rho)$, $\rho \in [0,1]$, with each site independently occupied with probability $\rho$. Strikingly, these measures are independent of $\beta$, the per-site jump attempt probability.
• Sublattice Parallel Update (R3):
  • Even sites are updated, then odd sites.
  • The invariant measure splits: even sites retain marginal $\text{Ber}(\rho)$, but odd sites have marginal

$\frac{\rho(1-\beta)}{1-\rho\beta},$

resulting in a spatially inhomogeneous invariant measure. Here, $\beta$ directly influences the invariant measure, producing a two-layer structure not present in the continuous-time or R1/R2 models.

For multi-type (multi-class) TASEP, the construction of invariant measures employs coupled product measures on single-type configurations—a hierarchical arrangement where the density of the $i$-th class is $r_i - r_{i-1}$ (with $r_0=0$). Under the tandem-queue construction, independent Bernoulli processes $\nu_{(\rho_1)} \times \cdots \times \nu_{(\rho_n)}$ with ordered parameters $\rho_1 < \cdots < \rho_n$ are transformed into the unique translation-invariant stationary distribution for the multi-type TASEP (Martin et al., 2010).

2. Comparison of Discrete and Continuous-Time Multi-Class Invariant Distributions

The similarities and differences between discrete- and continuous-time TASEP are dictated by the update rules:

• Universality for R1/R2:

The invariant product measures $\nu_{(\rho)}$ for R1 and R2 are identical to the continuous-time case. This “universality” implies that invariant distributions—including multi-class constructions—do not depend on $\beta$, and collision/overtaking statistics are also $\beta$-independent.

• Sublattice-Parallel (R3) Distinction:

R3's stationary measure depends on both $\rho$ and $\beta$, diverging sharply from the continuous-time case. As $\beta \to 0$, discrete-time R3 interpolates to the continuous-time model (by time-rescaling), making R3 a more general case from which others can be obtained as limits.

3. Construction via Queueing and Percolation Representations

The multi-class invariant distributions are realized using queueing and last-passage percolation constructions:

• Queueing Representation:

Given arrival processes $\alpha_1, \ldots, \alpha_n$, define recursively the departure process $D^{(n)}$, then set $$\eta^k = D^{(n - k + 1)}(\alpha_k, \ldots, \alpha_n)$$. The multi-type configuration is given by

$\xi(x) = n + 1 - \sum_{k=1}^n \eta^k(x).$

The result of this transformation is a stationary measure for the multi-type TASEP, matching the tandem queueing perspective [(Martin et al., 2010), Theorem 3.3].

• Last-Passage Percolation Relation:

Passage times $T(n, k)$ satisfy

$T(n, k) = \max\{T(n-1, k), T(n, k-1)\} + w(n, k),$

where $w(n,k)$ are i.i.d. geometric random variables. Explicit recursions (modified for each rule) underpin the computation of invariant measures and the analysis of the joint behavior of different classes.

4. Speeds of Second-Class Particles and Collision Probabilities

For multi-class systems, the statistics of second-class and higher-class particles are major indicators of underlying invariant structure:

• Asymptotic Speeds:

The second-class particle position $X^{(i)}(t)$ satisfies a law of large numbers,

$\frac{X^{(i)}(t)}{t} \to U^{(i)},$

where $U^{(i)}$ is a random variable with explicit distribution depending on the update rule (e.g., $P(U^{(i)} \leq u) = 1 - f_i(u)$ for $i=0,1,2$).

• Collision (Overtaking) Probabilities:

Explicit values for joint distributions of adjacent particle speeds, e.g.,

$$P(U_0 > U_1) = 1/2, \quad P(U_0 = U_1) = 1/6, \quad P(U_0 < U_1) = 1/3$$

(for continuous-time and for discrete R1/R2) show invariance to $\beta$ and highlight the robust structure of the invariant measure. In R3, overtaking probabilities become $\beta$-dependent and more intricate.

These results derive from coupling arguments and connections to the corner growth/last-passage percolation model, which supply law of large numbers and collision statistics via percolation techniques (Martin et al., 2010).

5. Role of Update Rules and Fine Structure of Invariant Measures

The choice of update protocol fundamentally alters the structure of the invariant measure:

Update Rule	Invariant Measure Type	$\beta$-Dependence	Spatial Homogeneity
R1/R2	Homogeneous product Bernoulli	No	Yes
R3	Spatially inhomogeneous two-layer	Yes	No

For R1, neighboring particles can move together in a single time step; for R2, a particle may execute multiple jumps in a time interval. R3 introduces a persistent even/odd asymmetry in occupied site statistics.

These differences are critical when characterizing not only the site marginals but also the full joint structure of the invariant measure, especially in the multi-type case.

6. Duality and the Speed Process as a Stationary Measure

A central duality reveals that the collection of asymptotic speeds of particles ordered by initial location, $\{U_n\}$, itself forms a stationary ergodic measure for TASEP—this is the "speed process":

• The distribution $\mu^{(i)}$ of the speed process for model $R_i$ is characterized by the unique stationary measure with one-site marginals having distribution function $g_i = 1 - f_i$.
• The orderings of particle speeds and labels encode a permutation structure where the $Y$- and $X$-processes are inverses.
• This duality is directly analogous to the result proved by Amir, Angel, and Valkó for the continuous-time model, here extended to discrete time via explicit construction (Martin et al., 2010).

7. Explicit Formulas and Scaling Considerations

The explicitness of these constructions enables calculations of macroscopic densities and joint speed distributions:

• The invariant product measures and tandem-queue measures are computable for arbitrary densities and types.
• For R3, the spatially inhomogeneous measure allows an explicit formula for odd-site occupation probability.
• As time is rescaled (in the $\beta \to 0$ limit), the discrete-time processes converge to continuous-time TASEP, unifying the invariant distributions in a scaling context.

Hydrodynamic profiles, speed distributions, and collision statistics are directly computable using the recursive and combinatorial framework outlined in the model definitions.

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In summary, the multi-class invariant distributions in discrete-time TASEP are determined by the interplay of queuing constructions, update protocols, and percolation representations. Sequential update rules admit universal (update-parameter-independent) product-form invariant measures, while sublattice parallel updates generate inhomogeneous, parameter-dependent measures. The explicit queueing/percolation construction extends to the multi-class case, giving precise formulas for the densities of each type. Duality results further identify the process of asymptotic particle speeds as the central stationary ergodic measure, revealing deep connections with the continuous-time model and the broader structure of interacting particle systems (Martin et al., 2010).