# Convergence of exclusion processes and the KPZ equation to the KPZ fixed point

@article{Quastel2020ConvergenceOE,
title={Convergence of exclusion processes and the KPZ equation to the KPZ fixed point},
author={Jeremy Quastel and Sourav Sarkar},
journal={arXiv: Probability},
year={2020}
}
• Published 14 August 2020
• Mathematics
• arXiv: Probability
We show that under the 1:2:3 scaling, critically probing large space and time, the height function of finite range asymmetric exclusion processes and the KPZ equation converge to the KPZ fixed point, constructed earlier as a limit of the totally asymmetric simple exclusion process through exact formulas. Consequently, based on recent results of \cite{wu},\cite{DM20}, the KPZ line ensemble converges to the Airy line ensemble. For the KPZ equation we are able to start from a continuous function…

### Exceptional times when the KPZ fixed point violates Johansson's conjecture on maximizer uniqueness

• Mathematics
• 2021
In 2002, Johansson conjectured that the maximum of the Airy2 process minus the parabola x is almost surely achieved at a unique location [Joh03, Conjecture 1.5]. This result was proved a decade later

### Non-Stationary KPZ equation from ASEP with slow bonds

We make progress on the weak-universality conjecture for the KPZ equation by proving the height functions associated to a class of non-integrable and non-stationary generalizations of ASEP converge

### Law of Iterated Logarithms and Fractal Properties of the KPZ Equation

• Mathematics
• 2021
We consider the Cole-Hopf solution of the (1 + 1)-dimensional KPZ equation started from the narrow wedge initial condition. In this article, we ask how the peaks and valleys of the KPZ height

### Cutoff profile of ASEP on a segment

• Mathematics
Probability Theory and Related Fields
• 2022
This paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length N and finds that for particle densities in (0, 1), the total-variation cutoff window of ASEP is N 1 / 3 and the cutoff profile is 1-F GUE, where F GUE is the Tracy-Widom distribution function.

### KPZ on torus: Gaussian fluctuations

• Mathematics
• 2021
In this paper, we study the KPZ equation on the torus and derive Gaussian fluctuations in large time.

### Cutoff profile of the Metropolis biased card shuffling

We consider the Metropolis biased card shuﬄing (also called the multi-species ASEP on a ﬁnite interval or the random Metropolis scan). Its convergence to stationary was believed to exhibit a

### KPZ-type fluctuation exponents for interacting diffusions in equilibrium

• Mathematics
• 2020
We consider systems of N diffusions in equilibrium interacting through a potential V . We study a “height function” which for the special choice V pxq “ e, coincides with the partition function of a

### Cutoff profile of ASEP on a segment

• Materials Science
Probability Theory and Related Fields
• 2022
This paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length N. Our main result is that for particle densities in (0, 1), the total-variation cutoff

### Scaling limit of the TASEP speed process

• Mathematics
• 2022
. We show that the multi-type stationary distribution of the totally asymmetric simple exclusion process (TASEP) scales to a nontrivial limit around the Bernoulli measure of density 1 / 2. This is

### One-point asymptotics for half-flat ASEP

• Mathematics
• 2022
. We consider the asymmetric simple exclusion process (ASEP) with half-ﬂat initial condition. We show that the one-point marginals of the ASEP height function are described by those of the Airy 2 → 1

## References

SHOWING 1-10 OF 35 REFERENCES

### Crossover distributions at the edge of the rarefaction fan

• Mathematics
• 2010
We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $\rho_-<\rho_+$ so that macroscopically one has a

### Transversal Fluctuations of the ASEP, Stochastic Six Vertex Model, and Hall-Littlewood Gibbsian Line Ensembles

• Mathematics
• 2017
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

### Large-distance and long-time properties of a randomly stirred fluid

• Mathematics
• 1977
Dynamic renormalization-group methods are used to study the large-distance, long-time behavior of velocity correlations generated by the Navier-Stokes equations for a randomly stirred, incompressible

### Stochastic Burgers and KPZ Equations from Particle Systems

• Mathematics
• 1997
Abstract: We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that,

### Introduction to KPZ

This is an introductory survey of the Kardar-Parisi-Zhang equation (KPZ). The first chapter provides a non-rigorous background to the equation and to some of the many models which are supposed to lie

### Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions

• Mathematics
• 2011
We consider the solution of the stochastic heat equation $$\partial_T {\cal Z} = {{1}\over{2}} \partial_X^2 {\cal Z} - {\cal Z} \dot{\cal{W}}$$ with delta function initial condition {\cal Z}

### Replica approach to the KPZ equation with the half Brownian motion initial condition

• Mathematics
• 2011
We consider the one-dimensional Kardar–Parisi–Zhang (KPZ) equation with the half Brownian motion initial condition, studied previously through the weakly asymmetric simple exclusion process. We

### 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