Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests
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 (1999). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are \emph{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 $4\times4$ problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size $n\ge4$ which appear new for $n\ge 5$.
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.
MATH 3100 • Introduction to Probability (2 sections)
2024 travel
January
3-6 • San Francisco, CA • JMM 2024, organizing session on “AMS-AWM Special Session on Solvable Lattice Models and their Applications Associated with the Noether Lecture”
8-26 • London, UK • Queen Mary University of London
Asymptotics of noncolliding q-exchangeable random walks
We consider a process of noncolliding $q$-exchangeable random walks on $\mathbb{Z}$ making steps $0$ (straight) and $-1$ (down). A single random walk is called $q$-exchangeable if under an elementary transposition of the neighboring steps $ (\textnormal{down},\textnormal{straight}) \to (\textnormal{straight}, \textnormal{down}) $ the probability of the trajectory is multiplied by a parameter $q\in(0,1)$. Our process of $m$ noncolliding $q$-exchangeable random walks is obtained from the independent $q$-exchangeable walks via the Doob’s $h$-transform for a certain nonnegative eigenfunction $h$ with the eigenvalue less than $1$. The system of $m$ walks evolves in the presence of an absorbing wall at $0$.
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.
Rewriting History in Integrable Stochastic Particle Systems
Imagine two cars, slow (S) and fast (F), driving to the right on a discrete 1-dimensional lattice according to some random walk mechanism, and such that the cars cannot pass each other. We consider two systems, SF and FS, depending on which car is ahead. It is known for some time (through connections to symmetric functions and the RSK correspondence) that if at time 0 the cars are immediate neighbors, the trajectory of the car that is behind is the same (in distribution) in both systems. However, this fact fails when the initial locations of the cars are not immediate neighbors. I will explain how to recover the identity in distribution by suitably randomizing the initial condition in one of the systems.
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.
Rewriting History in Integrable Stochastic Particle Systems
Many integrable stochastic particle systems in one space dimension (such as TASEP — Totally Asymmetric Simple Exclusion Process — and its $q$-deformation, the $q$-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 [1], [2], 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 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.
A poem on the topic
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.
