Invariant measures and shocks in the KPZ fixed point

TL;DR

This paper constructs and characterizes invariant measures for the KPZ fixed point, revealing their structure as Brownian motions plus Bessel processes, and establishes their limits and extremal properties.

Contribution

It introduces a family of invariant measures parameterized by , describes their structure, and proves they are the extremal invariant measures under standard recentering.

Findings

Invariant measures are supported on functions with specific asymptotic slopes.

These measures are limits of measures from the open KPZ fixed point.

All extremal invariant measures are Brownian motions with drift.

Abstract

We construct a family of invariant measures from the perspective of a shock in the KPZ fixed point. These measures are parameterized by a positive number $\theta > 0$, and are supported on functions $f$ satisfying $\lim_{|x| \to \infty} \frac{f(x)}{|x|} = 2\theta$. Each can be described as the sum of a Brownian motion and an independent Bessel-$3$ process with drift. We show that these measures appear as the $L \to \infty$ limit of the measures constructed by Barraquand, Corwin, and Yang for the conjectural open KPZ fixed point on $[0,L]$, after recentering by an appropriately defined shock location. Furthermore, we show that, with respect to the standard, deterministic recentering at $x = 0$, all extremal invariant measures for the KPZ fixed point are Brownian motions with drift. To do this, we first show that any extremal invariant measures must be supported on functions having fixed asymptotic slopes at $\pm \infty$. Using a one-force-one-solution principle from the work of Busani, Sepp\"al\"ainen, and the second author, this rules out all other invariant measures except those having left slope $-2\theta$ and right slope $+2\theta$ for some $\theta > 0$. To handle this case, we derive the limiting fluctuations of the shock for a special choice of initial condition. Additionally, we derive the limiting fluctuations of the shock for the case of the invariant measure from the perspective of a shock, and for the case of initial data $f(x) = 2\theta|x|$.