When the geodesic becomes rigid in the directed landscape

Recently, there has been a huge development on the study of the Kardar-Parisi-Zhang universality class [BDJ99, Joh00, Joh03, BFPS07, TW08, TW09, BC14, MQR17, DOV18, JR19, Liu19]. Very recently, a four-parameter random field, the so-called directed landscape, was constructed [DOV18]. It is believed that the directed landscape is the limiting law for all the models in the Kardar-Parisi-Zhang universality class, and this has been confirmed for several classic models [DV21]. For the directed landscape, a lot of information is known, such as the finite-dimensional distributions or the distribution of the spatial process. On the other hand, it is less known about the geodesic. There are studies on the properties of the geodesic [BSS17, Ham20, HS20, BHS18, BGH21, BGH19, BF20, DSV20, CHHM21, DV21]. However, the explicit one-point distribution of the point-to-point geodesic was only obtained in [Liu21] very recently. The goal of this paper is to investigate one property of the geodesic in the directed landscape using the formula obtained in [Liu21]. Let L(x, s; y, t) be the directed landscape. In this paper, we will fix the point (0, 0; 0, 1) and denote L = L(0, 0; 0, 1). Denote Π(s), 0 < s < 1, the geodesic from (0, 0) to (0, 1). It is known that Π is almost surely unique [DOV18]. We also denote L(s) = L(0, 0;Π(s), s) for 0 < s < 1. We remark that the fact Π(s) is on the geodesic implies L(s) + L(Π(s), s; 0, 1) = L. The main result of this paper is about the fluctuations of Π(s) and L(s) when L becomes large. Theorem 1.1 (Rigidity of the geodesic). For any fixed s ∈ (0, 1) and x1, x2, l1, l2 ∈ R satisfying x1 < x2 and l1 < l2, we have