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Abstract Many integrable stochastic particle systems in one space dimension (such as TASEP—Totally Asymmetric Simple Exclusion Process—and itsq-deformation, theq-TASEP) remain integrable if we equip each particle with its own speed parameter. In this work, we present intertwining relations between Markov transition operators of particle systems which differ by a permutation of the speed parameters. These relations generalize our previous works (Petrov and Saenz in Probab Theory Relat Fields 182:481–530, 2022), (Petrov in SIGMA 17(021):34, 2021), but here we employ a novel approach based on the Yang-Baxter equation for the higher spin stochastic six vertex model. Our intertwiners are Markov transition operators, which leads to interesting probabilistic consequences. First, we obtain a new Lax-type differential equation for the Markov transition semigroups of homogeneous, continuous-time versions of our particle systems. Our Lax equation encodes the time evolution of multipoint observables of theq-TASEP and TASEP in a unified way, which may be of interest for the asymptotic analysis of multipoint observables of these systems. Second, we show that our intertwining relations lead to couplings between probability measures on trajectories of particle systems which differ by a permutation of the speed parameters. The conditional distribution for such a coupling is realized as a “rewriting history” random walk which randomly resamples the trajectory of a particle in a chamber determined by the trajectories of the neighboring particles. As a byproduct, we construct a new coupling for standard Poisson processes on the positive real half-line with different rates. 

Abstract We consider a process of noncollidingq-exchangeable random walks on $\mathbb{Z}$making steps 0 (‘straight’) and −1 (‘down’). A single random walk is calledq-exchangeable if under an elementary transposition of the neighboring steps $(\text{down},\text{straight})\to (\text{straight},\text{down})$the probability of the trajectory is multiplied by a parameter $q\in (0,1)$. Our process ofmnoncollidingq-exchangeable random walks is obtained from the independentq-exchangeable walks via the Doob’sh-transform for a nonnegative eigenfunctionh(expressed via theq-Vandermonde product) with the eigenvalue less than 1. The system ofmwalks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is theq-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncollidingq-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel ofq-distributed random lozenge tilings of sawtooth polygons. In the limit as $m\to \mathrm{\infty }$, $q=e^{-\gamma /m}$withγ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel. 

Abstract We obtain a new relation between the distributions$$\upmu _t$$ ${\mu }_t$at different times$$t\ge 0$$ $t\ge 0$of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions$$\upmu _t$$ ${\mu }_t$backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving$$\upmu _t$$ ${\mu }_t$which in turn brings new identities for expectations with respect to$$\upmu _t$$ ${\mu }_t$. The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions. 

Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). In this paper, we present a unified approach to constructing stationary measures for most of the known colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (multispecies ASEP, or mASEP); (2) the q-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the q-Boson particle system; (3) the q-deformed Pushing Totally Asymmetric Simple Exclusion Process (q-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang-Baxter equation. We express the stationary measures as partition functions of new "queue vertex models" on the cylinder. The stationarity property is a direct consequence of the Yang-Baxter equation. For the mASEP on the ring, a particular case of our vertex model is equivalent to the multiline queues of Martin (arXiv:1810.10650). For the colored q-Boson process and the q-PushTASEP on the ring, we recover and generalize known stationary measures constructed using multiline queues or other methods by Ayyer-Mandelshtam-Martin (arXiv:2011.06117, arXiv:2209.09859), and Bukh-Cox (arXiv:1912.03510). Our proofs of stationarity use the Yang-Baxter equation and bypass the Matrix Product Ansatz used for the mASEP by Prolhac-Evans-Mallick (arXiv:0812.3293). On the line and in a quadrant, we use the Yang-Baxter equation to establish a general colored Burke's theorem, which implies that suitable specializations of our queue vertex models produce stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity. 

We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4 × 4 problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size n ≥ 4, which appear new for n ≥ 5. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.