# KPZ equation correlations in time

@article{Corwin2019KPZEC,
title={KPZ equation correlations in time},
author={Ivan Corwin and Promit Ghosal and Alan Hammond},
journal={arXiv: Probability},
year={2019}
}
• Published 22 July 2019
• Mathematics
• arXiv: Probability
We consider the narrow wedge solution to the Kardar-Parisi-Zhang stochastic PDE under the characteristic $3:2:1$ scaling of time, space and fluctuations. We study the correlation of fluctuations at two different times. We show that when the times are close to each other, the correlation approaches one at a power-law rate with exponent $2/3$, while when the two times are remote from each other, the correlation tends to zero at a power-law rate with exponent $-1/3$. We also prove exponential-type…

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## References

SHOWING 1-10 OF 72 REFERENCES
Slow decorrelations in Kardar?Parisi?Zhang growth
For stochastic growth models in the Kardar–Parisi–Zhang (KPZ) class in 1+1 dimensions, fluctuations grow as t1/3 during time t and the correlation length at a fixed time scales as t2/3. In this work
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
Universality of slow decorrelation in KPZ growth
• Mathematics
• 2012
There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar-Parisi-Zhang (KPZ) universality class. A proper rescaling of time should introduce a
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
The KPZ fixed point
• Mathematics
Acta Mathematica
• 2021
An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process with arbitrary initial condition. The
Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison
• Mathematics
• 2019
We consider directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the
The Kardar-Parisi-Zhang Equation and Universality Class
Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or
Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica
• Physics
• 2018
We consider the fluctuations in the stochastic growth of a one-dimensional interface of height $h(x,t)$ described by the Kardar-Parisi-Zhang (KPZ) universality class. We study the joint probability
Renormalization Fixed Point of the KPZ Universality Class
• Mathematics
• 2011
The one dimensional Kardar–Parisi–Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a
Exponents governing the rarity of disjoint polymers in Brownian last passage percolation
• A. Hammond
• Mathematics
Proceedings of the London Mathematical Society
• 2019
In last passage percolation models lying in the KPZ universality class, long maximizing paths have a typical deviation from the linear interpolation of their endpoints governed by the two‐thirds