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Some recent progress on the periodic KPZ equation

Yu Gu, Tomasz Komorowski

TL;DR

This work analyzes the periodic KPZ equation on a torus, focusing on central limit behavior for the height and the winding number of the associated directed polymer in a periodic environment. It develops two complementary approaches: (i) a homogenization framework based on solving a Poisson equation for a corrector $\chi$ and decomposing the height into co-boundary/martingale terms, and (ii) a Clark–Ocone/Malliavin calculus route yielding explicit variance formulas in terms of stationary endpoint densities. A striking result, in the 1+1 spacetime white-noise case, is that the limiting variances can be written using independent Brownian bridges, enabling explicit small-$\beta$ expansions and a law of the iterated logarithm for $h$. The analysis links the growth of the KPZ interface to an effective diffusivity, providing a rigorous route from subdiffusive to diffusive scaling via a diffusive infrared cutoff on large tori, and supplies an explicit corrector and variance structure to guide future work. Open questions include open KPZ extensions, the asymptotics of the torus-length-dependent diffusivity $\Sigma_L(\beta)$, and point-to-point polymer limits in periodic settings.

Abstract

We review recent progress on the study of the Kardar-Parisi-Zhang (KPZ) equation in a periodic setting, which describes the random growth of an interface in a cylindrical geometry. The main results include central limit theorems for the height of the interface and the winding number of the directed polymer in a periodic random environment. We present two different approaches for each result, utilizing either a homogenization argument or tools from Malliavin calculus. A surprising finding in the case of a $1+1$ spacetime white noise is that the effective variances for both the height and the winding number can be expressed in terms of independent Brownian bridges. Additionally, we present two new results: (i) the explicit expression of the corrector used in the homogenization argument, and (ii) the law of the iterated logarithm for the height function.

Some recent progress on the periodic KPZ equation

TL;DR

This work analyzes the periodic KPZ equation on a torus, focusing on central limit behavior for the height and the winding number of the associated directed polymer in a periodic environment. It develops two complementary approaches: (i) a homogenization framework based on solving a Poisson equation for a corrector and decomposing the height into co-boundary/martingale terms, and (ii) a Clark–Ocone/Malliavin calculus route yielding explicit variance formulas in terms of stationary endpoint densities. A striking result, in the 1+1 spacetime white-noise case, is that the limiting variances can be written using independent Brownian bridges, enabling explicit small- expansions and a law of the iterated logarithm for . The analysis links the growth of the KPZ interface to an effective diffusivity, providing a rigorous route from subdiffusive to diffusive scaling via a diffusive infrared cutoff on large tori, and supplies an explicit corrector and variance structure to guide future work. Open questions include open KPZ extensions, the asymptotics of the torus-length-dependent diffusivity , and point-to-point polymer limits in periodic settings.

Abstract

We review recent progress on the study of the Kardar-Parisi-Zhang (KPZ) equation in a periodic setting, which describes the random growth of an interface in a cylindrical geometry. The main results include central limit theorems for the height of the interface and the winding number of the directed polymer in a periodic random environment. We present two different approaches for each result, utilizing either a homogenization argument or tools from Malliavin calculus. A surprising finding in the case of a spacetime white noise is that the effective variances for both the height and the winding number can be expressed in terms of independent Brownian bridges. Additionally, we present two new results: (i) the explicit expression of the corrector used in the homogenization argument, and (ii) the law of the iterated logarithm for the height function.
Paper Structure (23 sections, 10 theorems, 244 equations)

This paper contains 23 sections, 10 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Setup and notations
  3. Organization of the paper
  4. Polymer endpoint density as the projective process
  5. Invariant measure and exponential mixing
  6. A nonlocal nonlinear SPDE, a PDE hierarchy and the Lyapunov exponent
  7. A reaction-diffusion SPDE
  8. A PDE hierarchy
  9. Lyapunov exponent
  10. Fluctuations of the height function
  11. Homogenization of additive functional of projective process
  12. Poisson equation, corrector, martingale decomposition
  13. Clark-Ocone formula
  14. Fluctuations of the winding number
  15. Sum of weakly dependent random variables
  16. ...and 8 more sections

Key Result

Theorem 2.1

The Markov process $\{\rho_{\mathrm{f}}(t,\cdot;0,\nu)\}_{t\geq0}$, taking values in ${\mathcal{M}}_1(\mathbb{T}^d)$, has a unique invariant probability measure, denoted by $\pi_\infty$. In the case of a $1+1$ space-time white noise, the invariant measure is given by the law of the ${\mathcal{M}}_1( and $W$ is a standard Brownian bridge connecting $(0,0)$ and $(L,0)$, i.e. a continuous trajectory,

Theorems & Definitions (20)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 10 more