# Stochasticization of Solutions to the Yang–Baxter Equation

@article{Aggarwal2019StochasticizationOS, title={Stochasticization of Solutions to the Yang–Baxter Equation}, author={Amol Aggarwal and Alexei Borodin and Alexey Bufetov}, journal={Annales Henri Poincar{\'e}}, year={2019}, pages={1-60} }

In this paper, we introduce a procedure that, given a solution to the Yang–Baxter equation as input, produces a stochastic (or Markovian) solution to (a possibly dynamical version of) the Yang–Baxter equation. We then apply this “stochasticization procedure” to obtain three new, stochastic solutions to several different forms of the Yang–Baxter equation. The first is a stochastic, elliptic solution to the dynamical Yang–Baxter equation; the second is a stochastic, higher rank solution to the…

## 10 Citations

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## References

SHOWING 1-10 OF 65 REFERENCES

An integrable 3D lattice model with positive Boltzmann weights

- Mathematics, Physics
- 2013

In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run…

Stochastic Higher Spin Vertex Models on the Line

- Mathematics
- 2015

We introduce a four-parameter family of interacting particle systems on the line, which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain…

On the integrability of zero-range chipping models with factorized steady states

- Mathematics
- 2013

The conditions of the integrability of general zero range chipping models with factorized steady states, which were proposed in Evans et al (2004 J. Phys. A: Math. Gen. 37 L275), are examined. We…

A Fredholm Determinant Representation in ASEP

- Mathematics
- 2008

In previous work (Tracy and Widom in Commun. Math. Phys. 279:815–844, 2008) the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer…

Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

- Mathematics, Physics
- 2006

The tetrahedron equation is a three-dimensional generalization of the Yang–Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field…

Dynamical stochastic higher spin vertex models

- Mathematics
- 2017

We introduce a new family of integrable stochastic processes, called dynamical stochastic higher spin vertex models, arising from fused representations of Felder’s elliptic quantum group $$E_{\tau ,…

Lectures on Integrable Probability: stochastic vertex models and symmetric functions

- Mathematics
- 2018

We consider a homogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the…

STAR-SQUARE AND TETRAHEDRON EQUATIONS IN THE BAXTER-BAZHANOV MODEL

- Physics, Mathematics
- 1993

Spatially symmetrical Baxter-Bazhanov model’s Boltzmann weights are proved to satisfy the tetrahedron equation. The latter is a consequence of the “inversion” and “star-square” relations for the…