Stochastic PDE Limit of the Six Vertex Model

@article{Corwin2020StochasticPL,
  title={Stochastic PDE Limit of the Six Vertex Model},
  author={Ivan Corwin and Promit Ghosal and Hao Shen and Li-Cheng Tsai},
  journal={Communications in Mathematical Physics},
  year={2020},
  volume={375},
  pages={1945-2038}
}
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 critical point) its height function fluctuations converge to the solution to the Kardar–Parisi–Zhang (KPZ) equation. We also prove that the one-dimensional family of stochastic Gibbs states for the symmetric six vertex model converge under the same scaling to the stationary solution to the… 

Irreversible Markov Dynamics and Hydrodynamics for KPZ States in the Stochastic Six Vertex Model

We introduce a family of Markov growth processes on discrete height functions defined on the 2-dimensional square lattice. Each height function corresponds to a configuration of the six vertex model

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

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

The KPZ Equation, Non-Stationary Solutions, and Weak Universality for Finite-Range Interactions

We study the weak KPZ universality problem by extending the KPZ universality results for weakly asymmetric exclusion processes to non-simple variants under deterministic initial data with constant

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

Time evolution of the Kardar-Parisi-Zhang equation

The use of the non-linear SPDEs are inevitable in both physics and applied mathematics since many of the physical phenomena in nature can be effectively modeled in random and non-linear way. 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:

Nonexistence and uniqueness for pure states of ferroelectric six‐vertex models

  • A. Aggarwal
  • Mathematics
    Proceedings of the London Mathematical Society
  • 2022
In this paper, we consider the existence and uniqueness of pure states with some fixed slope (s,t)∈[0,1]2$(s, t) \in [0, 1]^2$ for a general ferroelectric six‐vertex model. First, we show there is an

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

Dualities in quantum integrable many-body systems and integrable probabilities. Part I

Abstract In this study we map the dualities observed in the framework of integrable probabilities into the dualities familiar in a realm of integrable many-body systems. The dualities between the

References

SHOWING 1-10 OF 116 REFERENCES

Random Matrices and the Six-Vertex Model

This book provides a detailed description of the Riemann-Hilbert approach (RH approach) to the asymptotic analysis of both continuous and discrete orthogonal polynomials, and applications to random

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

Height Fluctuations for the Stationary KPZ Equation

We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data H(0,X)=B(X)$\mathcal {H}(0,X)=B(X)$, for B(X) a two-sided standard Brownian motion) and show that

Lectures on Integrable Probability: stochastic vertex models and symmetric functions

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

Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems

We introduce what we call the second-order Boltzmann–Gibbs principle, which allows one to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear

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

Higher spin six vertex model and symmetric rational functions

We consider a fully inhomogeneous 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

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

  • A. Aggarwal
  • Mathematics
    Communications 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

Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

We consider the weakly asymmetric simple exclusion process on the discrete space $\{1,...,n-1\}$, in contact with stochastic reservoirs, both with density $\rho\in{(0,1)}$ at the extremity points,

KPZ equation, its renormalization and invariant measures

The Kardar–Parisi–Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because of the inconsistency between the nonlinearity and the roughness of the forcing noise.
...