Symmetries of stochastic colored vertex models

Galashin, Pavel

doi:10.48550/arxiv.2003.06330

Cited by 8 publications

(28 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [BGW19,Gal20], certain shift-and flip-invariance are proved for the colored six-vertex model. Our colored TASEP can be obtained from the colored six-vertex model, by passing the parameters there to a certain limit.…”

Section: Colored Tasep Identitiesmentioning

confidence: 99%

“…The shift-invariance of passage times (of Theorem 1.1 and 1.2) is effectively an equal in distribution concerning the configurations in different times, thus does not follow from these results. On the other hand, it seems that there is no corresponding arguments as those in the proofs in [BGW19,Gal20] in the degenerated setting of the colored TASEP model, since the full power of the colored stochastic six-vertex model would be necessary in their proofs.…”

Section: Colored Tasep Identitiesmentioning

confidence: 99%

“…It is asked as [BGW19, Conjecture 1.5] whether such shift-invariance holds in more generality, with ordering imposed only for endpoints where relative shift happens. This has been answered in [Gal20] and [Dau20] in different settings. It would be natural to ask if similar extension holds for our results, i.e., are some of the ordering inequalities in Theorem 1.1 not necessary for the shift-invariance to hold?…”

Section: Colored Tasep Identitiesmentioning

confidence: 99%

“…We shall take an input from [Gal20], which is a distributional identity of the colored six-vertex model, and we pass it to the colored TASEP limit. Similar inputs from the colored six-vertex model to the colored TASEP have appeared in [BGR20].…”

Section: Input From the Colored Six-vertex Model And Proof Strategymentioning

confidence: 99%

“…In particular, we avoid working with formulae from algebraic combinatorics or representation theory. Our starting point is a crucial input of a shift-invariance property of the colored six-vertex model, proved in [Gal20]. Our central idea is to use analyticity of the distribution functions, combined with independence properties of the colored TASEP in different space-time areas.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Shift-Invariance of the Colored TASEP and Finishing Times of the Oriented Swap Process

Zhang¹

2021

Preprint

View full text Add to dashboard Cite

We prove a new shift-invariance property of the colored TASEP. It is in the same spirit as recent results of Borodin-Gorin-Wheeler, Dauvergen, and Galashin, while out of the scope of the generality of their methods. Our proof takes shift-invariance of the colored six-vertex model as an input, and uses analyticity of the probability functions and induction arguments. We apply our shift-invariance to prove a distributional identity between the finishing times of the oriented swap process and the point-to-line passage times in exponential last-passage percolation, which is conjectured by Bisi-Cunden-Gibbons-Romik and Bufetov-Gorin-Romik, and is also equivalent to a purely combinatorial identity in relation with the Edelman-Greene correspondence. With known results from last-passage percolation, this also implies new asymptotic results on the finishing times of the oriented swap process.

show abstract

Section: Colored Tasep Identitiesmentioning

confidence: 99%

Section: Colored Tasep Identitiesmentioning

confidence: 99%

Section: Colored Tasep Identitiesmentioning

confidence: 99%

Section: Input From the Colored Six-vertex Model And Proof Strategymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Shift-Invariance of the Colored TASEP and Finishing Times of the Oriented Swap Process

Zhang¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

Hidden diagonal integrability of q-Hahn vertex model and Beta polymer model

Korotkikh

2022

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

We study a new integrable probabilistic system, defined in terms of a stochastic colored vertex model on a square lattice. The main distinctive feature of our model is a new family of parameters attached to diagonals rather than to rows or columns, like in other similar models. Because of these new parameters the previously known results about vertex models cannot be directly applied, but nevertheless the integrability remains, and we prove explicit integral expressions for q-deformed moments of the (colored) height functions of the model. Following known techniques our model can be interpreted as a q-discretization of the Beta polymer model from (Probab Theory Relat Fields 167(3):1057–1116 (2017). arXiv:1503.04117) with a new family of parameters, also attached to diagonals. To demonstrate how integrability with respect to the new diagonal parameters works, we extend the known results about Tracy–Widom large-scale fluctuations of the Beta polymer model.

show abstract

The geometric Burge correspondence and the partition function of polymer replicas

Bisi,

O'Connell,

Zygouras

2020

Preprint

View full text Add to dashboard Cite

We construct a geometric lifting of the Burge correspondence as a composition of local birational maps on generic Young-diagram-shaped arrays. We prove a fundamental link with the geometric Robinson-Schensted-Knuth correspondence and with the geometric Sch ützenberger involution. We also show a number of properties of the geometric Burge correspondence, also specializing them to the case of symmetric input arrays. In particular, our construction shows that such a mapping is volume preserving in log-log variables. As an application, we consider a model of two polymer paths of given length constrained to have the same endpoint, known as polymer replica. We prove that the distribution of the polymer replica partition function in a log-gamma random environment is a Whittaker measure, and deduce the corresponding Whittaker integral identity. For a certain choice of the parameters, we notice a distributional identity between our model and the symmetric log-gamma polymer studied by O'Connell, Seppäläinen, and Zygouras

show abstract

Symmetries of stochastic colored vertex models

Cited by 8 publications

References 34 publications

Shift-Invariance of the Colored TASEP and Finishing Times of the Oriented Swap Process

Shift-Invariance of the Colored TASEP and Finishing Times of the Oriented Swap Process

Hidden diagonal integrability of q-Hahn vertex model and Beta polymer model

The geometric Burge correspondence and the partition function of polymer replicas

Contact Info

Product

Resources

About