Spatial covariance of KPZ from flat initial profile

TL;DR

This paper derives the exact asymptotic behavior of the spatial covariance of the KPZ equation with flat initial profile, revealing Gaussian decay contrasting with the narrow-wedge case, and provides a formula for the directed polymer's second moment.

Contribution

It introduces the first exact asymptotic formula for the spatial covariance of KPZ with flat initial data and connects it to the stochastic heat equation and directed polymers.

Findings

Covariance decays as Gaussian for large distances

Explicit formula for the second moment of the directed polymer partition function

Contrast with narrow-wedge initial profile results

Abstract

We study the fixed-time spatial covariance of the KPZ equation with flat initial profile. Using Malliavin calculus and a Clark-Ocone representation, we show that as $|x|\to\infty$, $\mathrm{Cov}[h(t,x),h(t,0)]$ is governed by a boundary-layer regime near the initial time and satisfies $\mathrm{Cov}[h(t,x),h(t,0)] \sim \kappa(t) \int_0^t p_{2r}(x) dr = \frac{2\kappa(t)}{\sqrt{\pi}}t^{3/2}|x|^{-2}\exp\left(-\frac{x^2}{4t}\right),$ as $|x|\to\infty$, where $\kappa(t) = \big(\mathbb{E}[Z(t,0)^{-1}]\big)^2$, $Z$ is the flat stochastic heat equation solution, and $p_t$ is the one-dimensional heat kernel. In sharp contrast with the narrow-wedge regime, where Gu-Pu (2025, Theorem 1.1) proved that for each fixed $t>0$, $\mathrm{Cov}\left[h^{\mathrm{nw}}(t,x),h^{\mathrm{nw}}(t,0)\right]\sim \frac{t}{|x|},$ as $|x|\to\infty$, the flat initial profile exhibits Gaussian decay, yielding, to the best of our knowledge, the first exact spatial covariance asymptotic for the KPZ equation under flat initial data. We also establish an explicit closed-form formula for the second moment of the continuum directed random polymer partition function.