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. 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 (1, 2, and Bukh-Cox. Our proofs of stationarity use the Yang-Baxter equation and bypass the Matrix Product Ansatz (used for the mASEP by Prolhac-Evans-Mallick).
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.
]]>Solvable lattice models originated in mathematical physics (Ising model, square ice). They found numerous applications to combinatorics (enumerative and algebraic - keywords are nonsymmetric Macdonald polynomials and Hecke algebras), representation theory (keyword - quantum groups), and more recently, probability. We will start with classical results, such as the Izergin-Korepin determinant and the entropy of the square ice. We’ll select further topics based on the audience’s interests.
August 29, 2023
Leo Petrov - Introduction
September 5, 2023
Nick Sweeney - The Alternating Sign Matrix conjecture
after Chapter 1 of the book Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David M. Bressoud
September 12, 2023
Petch Chueluecha - Dodgson condensation
after Section 3.5 of the book Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David M. Bressoud
September 19, 2023
Mikhail Tikhonov - The Yang-Baxter equation and the Izergin-Korepin determinant
after Wheeler-Zinn-Justin 2015 (Section 4.2 with $u=1$) and Petrov 2020 (Yang-Baxter equation is Proposition 2.1; and the determinant is proven in Section 3.4 with $\gamma=1$, $s_0=0$)
September 26, 2023
Leo Petrov - From the Izergin-Korepin determinant to the number of alternating sign matrices
October 10, 2023
Jacob Campbell - Quantim groups and Yang-Baxter equation
October 17, 2023
TBA
October 24, 2023
TBA
October 31, 2023
TBA
November 7, 2023
TBA
November 14, 2023
TBA
November 28, 2023
Zongrui Yang (Columbia) - Probability seminar talk (special day)
TBD • CIRM, Marseille, France • Conference in honor of Richard Kenyon
]]>By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (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.
]]>3/11 - 6/14 • Los Angeles, CA • Geometry, Statistical Mechanics, and Integrability program at the Institute for Pure and Applied Mathematics (IPAM). In residence at IPAM for parts of the program, including March 11-29; April 15-26; May 24-June 14.
22 - 26 • Pasadena, CA • AIM SQuaRE
]]>We show that the trajectory of the noncolliding $q$-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 of $q$-distributed random lozenge tilings of sawtooth polygons.
In the limit as $m\to \infty$, $q=e^{-\gamma/m}$ with $\gamma>0$ 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.
]]>This result arises in our recent work on multiparameter stochastic systems (where the parameters are speeds attached to each car) in which the presence of parameters preserves the quantum integrability. This includes TASEP (totally asymmetric simple exclusion process), its deformations, and stochastic vertex models, which are all integrable through the Yang-Baxter equation (YBE). In the context of car dynamics, we interpret YBEs as Markov operators intertwining the transition semigroups of the dynamics of the processes differing by a parameter swap. We also construct Markov processes on trajectories which “rewrite the history” of the car dynamics, that is, produce an explicit monotone coupling between the trajectories of the systems differing by a parameter swap.
Based on the joint work with Axel Saenz.
]]>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 the $q$-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.
by OpenAI
In stochastic particle systems, there’s a way
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A single particle, its fate made clear,
Can undo what’s been done and make it reappear.
The laws of probability and chaos at play
Can be bent to our will, if we but obey.
The deterministic systems in our control,
Will yield to a new order, as it starts to unfold.
The particles and their interactions will dictate,
The outcome of our systems, no matter their state.
With the tools of integrability, we can rewrite,
The future of our systems with a single bite.