Edgeworth-type expansion for the one-point distribution of the KPZ fixed point with a large height at a prior location

We consider the Kardar-Parisi-Zhang (KPZ) fixed point $\mathrm{H}(x,\tau)$ with the narrow-wedge initial condition and investigate the distribution of $\mathrm{H}(x,\tau)$ conditioned on a large height at an earlier space-time point $\mathrm{H}(x',\tau')$. As $\mathrm{H}(x',\tau')$ tends to infinity, we prove that the conditional one-point distribution of $\mathrm{H}(x,\tau)$ in the regime $\tau>\tau'$ converges to the Gaussian Unitary Ensemble (GUE) Tracy-Widom distribution and that the next two lower-order error terms can be expressed as derivatives of the Tracy-Widom distribution. The lowe order expansion here is analogue to the Edgeworth expansion in the central limit theorem. These KPZ-type limiting behaviors are different from the Gaussian-type ones obtained in \cite{Liu-Wang22} where they study the finite-dimensional distribution of $\mathrm{H}(x,\tau)$ conditioned on a large height at a later space-time point $\mathrm{H}(x',\tau')$. They show, with the narrow-wedge initial condition, that the conditional random field $\mathrm{H}(x,\tau)$ in the regime $\tau<\tau'$ converges to the minimum of two independent Brownian bridges modified by linear drifts as $\mathrm{H}(x',\tau')$ goes to infinity. The two results stated above provide the phase diagram of the asymptotic behaviors of a conditional law of KPZ fixed point in the regimes $\tau>\tau'$ and $\tau<\tau'$ when $\mathrm{H}(x',\tau')$ goes to infinity.