The directed landscape

Abstract

The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that last passage geodesics converge to random functions with Holder-2/3- continuous paths. This work completes the construction of the central object in the Kardar-Parisi-Zhang universality class, the directed landscape.

Figures (6)

• Figure 1: An example of last passage across three functions. The purple path is the last passage path from $(x, 3)$ to $(y, 1)$. It can be viewed as either maximizing the sum of increments along the path, or minimizing the sum of gaps. We will always think of our sequences of functions as being labelled so that $f_{i}$ sits above $f_{i+1}$. Our notions of 'right' and 'left' in the paper are with respect to this picture.
• Figure 2: An illustration of the metric composition law for the directed landscape. The geodesic from $(x, r)$ to $(y, t)$ passes through a point $(z^*, s)$ for some $z^* \in \mathbb{R}$, giving that the right hand side of \ref{['E:land-metric']} is greater than or equal the left. See \ref{['E:geodesic-definition']} and surrounding discussion for a precise definition of a geodesic in $\mathcal{L}$. Also, for any $z \in \mathbb{R}$, a concatenation of the geodesic from $(x, r)$ to $(z, s)$ with the geodesic from $(z, s)$ to $(y, t)$ gives a candidate for a maximizing path from $(x, r)$ to $(y, t)$, yielding the opposite inequality. The original Brownian lines that give rise to $\mathcal{L}(\cdot, r; \cdot, s)$ and $\mathcal{L}(\cdot, s; \cdot, t)$ are independent. This allows us to build up $\mathcal{L}$ at any finite set of times using metric composition with independent increments, and hence construct the directed landscape from independent Airy sheets analogously to the construction of Brownian motion.
• Figure 3: A sketch of a 'Brownian melon' from time $0$ to just after time $1$. If we zoom in around the location $(1, 2\sqrt{n})$ on a $O(n^{-1/3})\times O(n^{-1/6})$ parallelogram with slope $\sqrt{n}$, then we get the Airy line ensemble.
• Figure 4: An illustration of the proof of Lemma \ref{['L:k-line-prior']}. We want to understand the last passage value in the melon from $(x, n)$ to $(z, k)$; here $n=5$ and $k =3$. To do this, note that this is the bottom path in a collection of $k-1$ disjoint last passage paths from $(0, n)$ to $(z, 1)$ and one path from $(x, n)$ to $(z, 1)$: this is Figure \ref{['fig:melontrans']} (a). We can then 'demelonize' (Figure \ref{['fig:melontrans']} (b)) and then 'remelonize' at $z$ after reversing the original lines (Figure \ref{['fig:melontrans']} (c)). These transformations leave the appropriate last passage values (i.e the sum of the lengths of the coloured paths) unchanged. In Figure \ref{['fig:melontrans']} (c), the last passage paths use the top 3 lines up to time $z$, except for the graph of one increasing path from $(z-x, 1)$ to $(z, 3)$: this path gives the first passage value that appears in the statement of the lemma. This is only a sketch, i.e. Figures \ref{['fig:melontrans']} (a) and (c) are not truly the melon and reverse melon of Figure \ref{['fig:melontrans']} (b).
• Figure 5: An illustration of the bridge representation $\mathcal{B}$ of the Airy line ensemble. Figure \ref{['fig:Bridge']}(a) is the Airy line ensemble, with points at three times identified. Points with the same time coordinate are grouped together if they are close. To sample the bridge representation on this grid, we erase all lines between the specified points and resample independent Brownian bridges that are conditioned not to intersect each other if either of their endpoints are close (i.e. have the same colour). The result is Figure \ref{['fig:Bridge']}(b). If we only look at the top half of the lines in the bridge representation, then with high probability they do not intersect each other and resemble the Airy line ensemble.

Theorems & Definitions (118)

• Theorem \oldthetheorem
• Definition \oldthetheorem
• Theorem \oldthetheorem
• Remark 1.1
• Definition \oldthetheorem
• Remark 1.2
• Theorem \oldthetheorem: Full scaling limit of Brownian last passage percolation
• Proposition \oldthetheorem
• Theorem \oldthetheorem: Continuity of directed geodesics
• Theorem \oldthetheorem: Convergence of last passage paths