# Stochastic six-vertex model

@article{Borodin2016StochasticSM, title={Stochastic six-vertex model}, author={Alexei Borodin and Ivan Corwin and Vadim Gorin}, journal={Duke Mathematical Journal}, year={2016}, volume={165}, pages={563-624} }

We study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line. We show that the random height function of the model converges to an explicit deterministic limit shape as the mesh size tends to 0. We further prove that the one-point fluctuations around the limit shape are asymptotically governed by the GUE Tracy-Widom distribution. We also explain an equivalent formulation of our model as an interacting particle system, which can be viewed as a discrete time…

## 108 Citations

### Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model

- MathematicsCommunications in Mathematical Physics
- 2019

In this paper we consider the stochastic six-vertex model on a cylinder with arbitrary initial data. First, we show that it exhibits a limit shape in the thermodynamic limit, whose density profile is…

### Limit Shapes of the Stochastic Six Vertex Model

- MathematicsCommunications in Mathematical Physics
- 2018

It is shown that limit shapes for the stochastic 6-vertex model on a cylinder with the uniform boundary state on one end are solutions to the Burger type equation. Solutions to these equations are…

### Stochastic PDE Limit of the Six Vertex Model

- MathematicsCommunications in Mathematical Physics
- 2020

We study the stochastic six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter $$\Delta \rightarrow 1^+$$ Δ → 1 + so as to zoom into the ferroelectric/disordered phase…

### 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…

### EQUATION FROM THE SIX-VERTEX MODEL By

- Mathematics
- 2019

A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are…

### The two-point correlation function in the six-vertex model

- PhysicsJournal of Physics A: Mathematical and Theoretical
- 2022

We study numerically the two-point correlation functions of height functions in the six-vertex model with domain wall boundary conditions. The correlation functions and the height functions are…

### Two-point convergence of the stochastic six-vertex model to the Airy process

- Mathematics
- 2020

In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time $T$, it is known that the one-point height function fluctuations are of…

### Color-position symmetry in interacting particle systems

- MathematicsThe Annals of Probability
- 2019

We prove a color-position symmetry for a class of ASEP-like interacting particle systems with discrete time on the one-dimensional lattice. The full space-time inhomogeneity of our systems allows to…

### A stochastic telegraph equation from the six-vertex model

- MathematicsThe Annals of Probability
- 2019

A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are…

### Stochastic symplectic ice

- MathematicsLetters in Mathematical Physics
- 2022

In this paper, we construct solvable ice models (six-vertex models) with stochastic weights and U-turn right boundary, which we term “stochastic symplectic ice”. The models consist of alternating…

## References

SHOWING 1-10 OF 81 REFERENCES

### Large time asymptotics of growth models on space-like paths I: PushASEP

- Mathematics
- 2007

We consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied,…

### The 6-vertex model with fixed boundary conditions

- Mathematics
- 2007

We study the 6-vertex model with fixed boundary conditions. In the thermodynamical limit there is a formation of the limit shape. We collect most of the known results about the analytical properties…

### Height Fluctuations in the Honeycomb Dimer Model

- Mathematics
- 2004

We study a model of random surfaces arising in the dimer model on the honeycomb lattice. For a fixed “wire frame” boundary condition, as the lattice spacing ϵ → 0, Cohn, Kenyon and Propp [3] showed…

### The asymmetric six-vertex model

- Physics
- 1992

The exact solution of the asymmetric six-vertex model, published nearly without derivation by Sutherlandet al. in 1967, is rederived in detail. The transfer matrix method and the Bethe Ansatz…

### One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality.

- Mathematics, PhysicsPhysical review letters
- 2010

The solution confirms that the KPZ equation describes the interface motion in the regime of weak driving force, and provides a determinantal formula for the probability distribution function of the height h(x,t) for all t>0.

### Domino tilings and the six-vertex model at its free-fermion point

- Mathematics
- 2006

At the free-fermion point, the six-vertex model with domain wall boundary conditions (DWBC) can be related to the Aztec diamond, a domino tiling problem. We study the mapping on the level of complete…

### Non-equilibrium behaviour of a many particle process: Density profile and local equilibria

- Mathematics
- 1981

SummaryOne considers a simple exclusion particle jump process on ℤ, where the underlying one particle motion is a degenerate random walk that moves only to the right. One starts with the…

### Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

- Mathematics
- 2013

We construct a family of stochastic growth models in 2 + 1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1 + 1 dimensional growth models in the…

### Asymptotics in ASEP with Step Initial Condition

- Mathematics
- 2008

In previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition, particles beginning at the positive integers. There it…

### Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field

- Mathematics
- 2012

We study large-scale height fluctuations of random stepped surfaces corresponding to uniformly random lozenge tilings of polygons on the triangular lattice. For a class of polygons (which allows…