arXiv:2210.04999v2 [math.PR] 20 Oct 2022 Asymptotics of the one-point distribution of the KPZ fixed point conditioned on a large height at an earlier point Ron Nissim, Ruixuan Zhang Abstract We consider the Kardar-Parisi-Zhang (KPZ) fixed point H(x, τ ) with the step initial condition and investigate the distribution of H(x, τ ) conditioned on a large height at an earlier space-time point H(x ′ ,τ ′ ). When the height H(x ′ ,τ ′ ) tends to infinity, we prove that the conditional one- point distribution of H(x, τ ) in the regime τ>τ ′ converges to the Gaussian Unitary Ensemble (GUE) Tracy-Widom distribution, and the next three lower order error terms can be expressed as the derivatives of of the GUE Tracy-Widom distribution. These KPZ-type limiting behaviors are different from the Gaussian-type ones obtained in [LW22] where they study the finite dimensional distribution of H(x, τ ) conditioned on a large height at a later space-time point H(x ′ ,τ ′ ). They prove, with the step initial condition, the conditional random field H(x, τ ) in the regime τ<τ ′ converges to the minimum of two independent Brownian bridges modified by linear drifts as H(x ′ ,τ ′ ) 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 τ>τ ′ and τ<τ ′ when H(x ′ ,τ ′ ) goes to infinity. 1 Introduction 1.1 Background on the KPZ universality class and KPZ fixed point. The Kardar-Parisi-Zhang (KPZ) equation, a nonlinear stochastic PDE, ∂ t H = λ∂ 2 x H + ν (∂ x H) 2 + ξ, (1.1) where H : (0, ∞) × R → R and ξ : (0, ∞) × R → R is the space-time white noise, was introduced by Kardar, Parisi and Zhang in [KPZ86] to capture the evolution of the interface of a broad class of random growth model. Several physics paper [FNS77, vBKS85, KPZ86] predicted that the height function H(t, x), up to scaling factors, should converge to a model-independent universal random field. In the last two decades, a huge class of models in (1 + 1) space-time dimension, the KPZ universality class, have been exactly solved and then shown to share the same scaling limit function. For instance, the longest increasing subsequence [BDJ99], the directed last passage percolation [Joh00], the polynuclear growth model [Joh03], the asymmetric simple exclusion process (ASEP) [TW08, TW09], the directed random polymer [ACQ11, Sep12] and the KPZ equation [QS22] etc. The limit space-time field H(x, τ ), where x ∈ R,t ≥ 0, of the KPZ universality class, is called the KPZ fixed point, which depends on the initial condition H(x, 0) = h 0 (x) for a function h 0 in the space of upper-semi-continuous functions. It was first constructed by Matetski, Quastel and Remenik in [MQR21] recently, as a Markov process with explicit transition probability by analysing the totally asymmetric simple exclusion process (TASEP). Due to the complexity of the formula for transition probability, it is challenging to derive explicit formulas for the distribution of H(x, τ ). Pioneering works have been done in the past two decades. [BDJ99, Joh00] prove that the marginal one point distribution of H(x, τ ) is governed by the Tracy-Widom distribution. The multi-point distribution along the spatial direction, namely the law of (H(x 1 ,τ ), ··· , H(x n ,τ )) for τ fixed, is characterized by the finite dimensional distribution of the Airy process and its analogs [PS02, Joh03, IS04, BFM07, BFPS07, BFS08, MQR21]. The general explicit formulas for joint distributions with possible different temporal points (H(x 1 ,τ 1 ), ··· , H(x n ,τ n )) were obtained recently by Johansson and Raham [JR21], and Liu independently [Liu22a]. 1