Symmetries of stochastic colored vertex models

  title={Symmetries of stochastic colored vertex models},
  author={Pavel Galashin},
  journal={The Annals of Probability},
  • Pavel Galashin
  • Published 13 March 2020
  • Mathematics
  • The Annals of Probability
Author(s): Galashin, Pavel | Abstract: We discover a new property of the stochastic colored six-vertex model called flip invariance. We use it to show that for a given collection of observables of the model, any transformation that preserves the distribution of each individual observable also preserves their joint distribution. This generalizes recent shift-invariance results of Borodin-Gorin-Wheeler. As limiting cases, we obtain similar statements for the Brownian last passage percolation, the… 

Observables of Stochastic Colored Vertex Models and Local Relation

We study the stochastic colored six vertex (SC6V) model and its fusion. Our main result is an integral expression for natural observables of this model -- joint q-moments of height functions. This

Hidden invariance of last passage percolation and directed polymers

Last passage percolation and directed polymer models on $\mathbb{Z}^2$ are invariant under translation and certain reflections. When these models have an integrable structure coming from either the

Critical varieties in the Grassmannian

We introduce a family of spaces called critical varieties. Each critical variety is a subset of one of the positroid varieties in the Grassmannian. The combinatorics of positroid varieties is

Positroids, knots, and $q,t$-Catalan numbers.

We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $\mathbb{C}$ and point counts over $\mathbb{F}_q$) to Khovanov--Rozansky

Hidden Symmetries of Weighted Lozenge Tilings

It is proved that this weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) is symmetric for large families of regions.

Shock fluctuations in TASEP under a variety of time scalings

We consider the totally asymmetric simple exclusion process (TASEP) with two different initial conditions with shock discontinuities, made by block of fully packed particles. Initially a second class

Shift invariance of half space integrable models

. We formulate and establish symmetries of certain integrable half space models, analogous to recent results on symmetries for models in a full space. Our starting point is the colored stochastic six

Orthogonal polynomial duality and unitary symmetries of multi--species ASEP$(q,\boldsymbol{\theta})$ and higher--spin vertex models via $^*$--bialgebra structure of higher rank quantum groups

We propose a novel, general method to produce orthogonal polynomial dualities from the ∗ –bialgebra structure of Drinfeld–Jimbo quantum groups. The ∗ –structure allows for the construction certain

Cutoff profile of the Metropolis biased card shuffling

We consider the Metropolis biased card shuffling (also called the multi-species ASEP on a finite interval or the random Metropolis scan). Its convergence to stationary was believed to exhibit a

Joint $q$-moments and shift invariance for the multi-species $q$-TAZRP on the infinite line

This paper presents a novel method for computing certain particle locations in the multi–species q – TAZRP (totally asymmetric zero range process). The method is based on a decomposition of the



Shift‐invariance for vertex models and polymers

We establish a symmetry in a variety of integrable stochastic systems: certain multi‐point distributions of natural observables are unchanged under a shift of a subset of observation points. The

Coloured stochastic vertex models and their spectral theory

This work is dedicated to $\mathfrak{sl}_{n+1}$-related integrable stochastic vertex models; we call such models coloured. We prove several results about these models, which include the following:

Hidden invariance of last passage percolation and directed polymers

Last passage percolation and directed polymer models on $\mathbb{Z}^2$ are invariant under translation and certain reflections. When these models have an integrable structure coming from either the

Brownian analogues of Burke's theorem

Colored five‐vertex models and Lascoux polynomials and atoms

We construct an integrable colored five‐vertex model whose partition function is a Lascoux atom based on the five‐vertex model of Motegi and Sakai and the colored five‐vertex model of Brubaker, the

Renormalization Fixed Point of the KPZ Universality Class

The one dimensional Kardar–Parisi–Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a

Positroid varieties: juggling and geometry

Abstract While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat

The Kardar-Parisi-Zhang Equation and Universality Class

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or

From multiline queues to Macdonald polynomials via the exclusion process

abstract:Recently James Martin introduced {\it multiline queues}, and used them to give a combinatorial formula for the stationary distribution of the multispecies asymmetric simple exclusion process

The directed landscape

The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage