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

@article{Aggarwal2019LimitSA,
  title={Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model},
  author={Amol Aggarwal},
  journal={Communications in Mathematical Physics},
  year={2019},
  volume={376},
  pages={681-746}
}
  • A. Aggarwal
  • Published 28 February 2019
  • Mathematics
  • Communications in Mathematical Physics
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 given by the entropy solution to an explicit, non-linear conservation law that was predicted by Gwa–Spohn (Phys Rev Lett 68:725–728, 1992) and by Reshetikhin–Sridhar (Commun Math Phys 363:741–765, 2018). Then, we show that the local statistics of this model around any continuity point of its limit… 
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References

SHOWING 1-10 OF 87 REFERENCES
Stochastic six-vertex model
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
Limit Shapes of the Stochastic Six Vertex Model
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 Higher Spin Vertex Models on the Line
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
Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian.
  • Gwa, Spohn
  • Physics
    Physical review letters
  • 1992
TLDR
It is proved that the dynamical scaling exponent for kinetic roughening is z=3/2 in 1+1 dimensions and diagonalize it using the Bethe ansatz and predict the large-scale asymptotic behavior of the vertical polarization correlations.
A stochastic telegraph equation from the six-vertex model
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
Local statistics of lattice dimers
Limit Shapes for the Asymmetric Five Vertex Model
We compute the free energy and surface tension function for the five-vertex model, a model of non-intersecting monotone lattice paths on the grid in which each corner gets a positive weight. We give
Transversal Fluctuations of the ASEP, Stochastic Six Vertex Model, and Hall-Littlewood Gibbsian Line Ensembles
We consider the ASEP and the stochastic six vertex model started with step initial data. After a long time, T, it is known that the one-point height function fluctuations for these systems are of
The 6-vertex model with fixed boundary conditions
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
Dimers in Piecewise Temperleyan Domains
We study the large-scale behavior of the height function in the dimer model on the square lattice. Richard Kenyon has shown that the fluctuations of the height function on Temperleyan discretizations
...
...