Non-existence of three non-coalescing infinite geodesics with the same direction in the directed landscape

Abstract

It is believed that for metric-like models in the KPZ class the following property holds: with probability one, starting from any point, there are at most two semi-infinite geodesics with the same direction that do not coalesce. Until now, such a result was only proved for one model - exponential LPP (Coupier 11') using its inherent connection to the totally asymmetric exclusion process. We prove that the above property holds for the directed landscape, the universal scaling limit of models in the KPZ class. Our proof reduces the problem to one on line ensembles and therefore paves the way to show similar results for other metric-like models in the KPZ class. Finally, combining our result with the ones in (Busani, Seppalainen,Sorensen 22', Bhatia 23') we obtain the full qualitative geometric description of infinite geodesics in the directed landscape.

Figures (6)

• Figure 1.1: The main steps behind the proof of Theorem \ref{['thm:5']}
• Figure 3.1: An illustration of a typical realization of the event $A^{U,M,l}$. By order of geodesics, the infinite geodesics $\pi_{L}^{\xi^{i-1}}$ and $\pi_{R}^{\xi^{i+1}}$ sandwich all infinite geodesics starting from $\mathcal{R}$ and going in direction $\xi^i$. Our main result will show that contrary to the illustration, with high probability, there are no more than two distinct geodesics going in direction $\xi^i$ and that are disjoint between time $M$ and time $U$.
• Figure 4.1: Given the two disjoint geodesics $\pi_{a_1}^{b_1}$ and $\pi_{a_2}^{b_2}$, we construct two geodesics, $\pi_{a_1}^{u_l}$ and $\pi_{a_2}^{u_r}$, whose upper endpoints are $\phi$ close to one another.
• Figure 5.1: The Bouquet construction.
• Figure 5.2: Two aspects of the event $\mathrm{Disj}^3_{\mathcal{C}_\text{scl}}$

Theorems & Definitions (55)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5: Solution to the N3G problem in the DL
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• proof
• Theorem 2.1: Dauvergne-Zhang-2021