A Peano curve from mated geodesic trees in the directed landscape

TL;DR

The paper constructs and analyzes a natural space-filling Peano curve $\eta$ that threads between the upward geodesic tree and its dual interface portrait in the directed landscape, drawing inspiration from the Brownian web and the Tóth–Werner curve. It establishes a KPZ-consistent, area-parametrized parametrization, proves almost-sure local $1/5$-Hölder regularity in the intrinsic metric while showing sharp failure of $1/5$-Hölder regularity in the Euclidean sense, and demonstrates a finite $5$th variation with a precise probabilistic normalization. A central technical innovation is a volume-accumulation framework for geodesics, built via a percolation-like argument, barrier events, and FKG-type decorrelation, which yields both tail bounds and Hölder control. The work also develops a detailed fractal-geometry correspondence: under mild conditions the intrinsic Hausdorff dimension of plane sets translates to a scaled dimension on the $\eta$-preimage, connecting KPZ geometry with fractal transport properties. Together, these results illuminate the intricate coupling structure of the directed landscape, provide a robust method to extract regularity and variation properties of space-filling interfaces, and lay groundwork for conditional independence results and future KPZ-geometric constructions.

Abstract

For the directed landscape, the putative universal space-time scaling limit object in the (1+1) dimensional Kardar-Parisi-Zhang (KPZ) universality class, consider the geodesic tree -- the tree formed by the coalescing semi-infinite geodesics in a given direction. As shown in Bhatia '23, this tree comes interlocked with a dual tree, which (up to a reflection) has the same marginal law as the geodesic tree. Analogous examples of one ended planar trees formed by coalescent semi-infinite random paths and their duals are objects of interest in various other probability models, a classical example being the Brownian web, which is constructed as a scaling limit of coalescent random walks. In this paper, we continue the study of the geodesic tree and its dual in the directed landscape and exhibit a new space-filling curve traversing between the two trees that is naturally parametrized by the area it covers and encodes the geometry of the two trees; this parallels the construction of the Tóth-Werner curve between the Brownian web and its dual. We study the regularity and fractal properties of this Peano curve, exploiting simultaneously the symmetries of the directed landscape and probabilistic estimates obtained in planar exponential last passage percolation, which is known to converge to the directed landscape in the scaling limit. On the way, we develop a novel coalescence estimate for geodesics, and this has recently found application in other work.

Figures (15)

• Figure 1: A simulation of the Peano curve for exponential LPP: The blue paths form the tree of semi-infinite geodesics in the $(1,1)$ direction, while the red paths form the dual tree, which we note has the same law as the geodesic tree up to a reflection Pim16. The green curve drawn in between the two mated trees is a portion of the Peano curve, and the corresponding curve in the directed landscape is space-filling.
• Figure 2: A large simulation showing a portion of the Peano curve corresponding to the tree formed by $(1,1)$ directional semi-infinite geodesics in exponential LPP
• Figure 3: Proof of Lemma \ref{['lem:1']}: In the first panel, we have the case when $p_1\neq p_2$ and we see that for points $p$ enclosed between $(\Upsilon_{p_1},\Gamma_{p_1})$ and $(\Upsilon_{p_2},\Gamma_{p_2})$, the corner $\zeta=(p,\Upsilon_p,\Gamma_p)$ satisfies $\zeta_1\leq \zeta\leq \zeta_2$. In the panel to the right, we show an example of the case when $p_1=p_2$ but $\zeta_1\neq \zeta_2$. Again, we see that for points $p$ enclosed between $(\Upsilon_{p_1},\Gamma_{p_1})$ and $(\Upsilon_{p_2},\Gamma_{p_2})$ (or equivalently, between $\Gamma_{p_1},\Gamma_{p_2}$), we have $\zeta_1\leq \zeta\leq \zeta_2$.
• Figure 4: Proof of Proposition \ref{['p:htail']}, case 1: In this case, there exist $0\le v_1<v_0\le 1$ such that $x(\eta(v_0))=y$, $z=\eta(v_1)\in \Gamma_0$ and $t(z) \le y^{3/2-\delta}, x(z)\ge y/2$. This implies that the geodesic $\Gamma_0$ has atypically large transversal fluctuation at or below height $y^{3/2}-\delta$ and hence this case is unlikely.
• Figure 5: Proof of Proposition \ref{['p:htail']}, case 2: In this case, there exist $0\le v_1<v_0\le 1$ such that $x(\eta(v_0))=y$, $z=\eta(v_1)\in \Gamma_0$ and $t(z) \le y^{3/2-\delta}, x(z)\le y/2$. This implies that the geodesic $\Gamma_{\eta(v_0)}$ has atypically large transversal fluctuation at or below height $y^{3/2}-\delta$. To handle this we divide the line segment $\{y\}\times [-y^{3/2-\delta},y^{3/2-\delta}]$ into segments of length $1$ each and call these segments $I_i$. For each $I_{i}$, the event that there exists a point $p\in I_{i}$ such that $\Gamma_{p}$ has atypically large transversal fluctuations is unlikely and we get the result by a union bound over $i$.

Theorems & Definitions (113)

• Proposition 1: Bha23
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Proposition 7
• Proposition 8
• Lemma 9
• proof