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, 2024 • 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.

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

To rewrite history with each passing day.

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.

In the Jack limit $q=t^{\beta/2}\to1$ the absorbing wall disappears, and the Macdonald noncolliding walks turn into the $\beta$-noncolliding random walks studied by Huang [arXiv:1708.07115]. Taking $q=0$ (Hall-Littlewood degeneration) and further sending $t\to 1$, we obtain a continuous time particle system on $\mathbb{Z}_{\ge0}$ with inhomogeneous jump rates and absorbing wall at zero.

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