Aging for the stationary Kardar-Parisi-Zhang equation and related models

Abstract

We study aging properties for stationary models in the KPZ universality class. In particular, we show aging for the stationary KPZ fixed point, the Cole-Hopf solution of the stationary KPZ equation, the height function of the stationary TASEP, last-passage percolation with boundary conditions and stationary directed polymers in the intermediate disorder regime. All of these models are shown to display a universal aging behavior characterized by the rate of decay of their correlations. As a comparison, we show aging for models in the Edwards-Wilkinson universality class where a different decay exponent is obtained. The key ingredient to our proofs is a covariance-to-variance reduction which is a characteristic of space-time stationarity and allows to deduce the asymptotic behavior of the correlations of two points by the one of the variance at one point. We formulate several open problems.

Riemann surfaces for KPZ with periodic boundaries

“…The KPZ fixed point on the infinite line x ∈ is described by a random field h(x , t ), see figure 1, which is known to be Markovian in time [261]. The statistics of h(x , t ) for which exact results have been obtained can be divided into three types, by increasing degree of generality (and of difficulty for exact calculations): one-point statistics, which characterize the probability density of h(x , t ) for given x and t , multiple-point statistics, which consider joint distributions at various positions x j but the same time t , and finally joint statistics at multiple times and positions, which gives the most general characterization of the random field h(x , t ), and are in particular relevant for the issue of aging of KPZ fluctuations [262][263][264]. As explained in section 2.1.3, the time variable can be eliminated by rescaling in the first two cases, when h(x , t ) is only observed at a single time t , and one can consider instead h(x ) = h(x , 1) without loss of generality.…”

Integration by Parts and the KPZ Two-Point Function

“…To prove Theorem 2 we link the variance of h 1 (x) with the covariance between h 1 (x) and h 0 (x) by a simple calculation, and compute this covariance in terms of the Malliavin derivative of h 1 (x). It is also remarkable that this simple relation between the variance and the covariance (covariance-variance reduction) was combined in [16] with tools from Malliavin calculus for concentration bounds to study aging for the stationary KPZ equation and related models.…”