The Busemann Process and Steep Highways in Directed First Passage Percolation

TL;DR

The work develops a comprehensive framework for the Busemann process in planar directed FPP, proving unconditional limit existence under mild regularity and deriving exact, integrable-distribution results via a multi-line fixed-point construction. By introducing and analyzing boundary-update maps ($A$, $V$, $H$) and their fixed points, the authors connect Busemann limits to semi-infinite geodesics, competition interfaces, highways, and convoy phenomena, and they link SWFPP to the SJR model to obtain a decomposition of the limit shape. They further show that these fixed points yield invariant measures for parallel TASEP and establish detailed structure of highways and branch points along the axis, including renewal-type descriptions and density results. The combination of queueing-inspired update maps, reversibility, and intertwining provides a robust toolkit for precisely describing the geometry of geodesics and the associated particle-system dynamics in integrable regimes. Overall, the paper advances both the probabilistic understanding of Busemann functions in directed FPP and their connections to exact solvable models and discrete-time interacting particle systems, with explicit consequences for limit shapes, competition interfaces, and highway-like structures.

Abstract

We consider the Busemann process in planar directed first passage percolation. We extend existing techniques to establish the existence of the process in our setting and determine its distribution in a number of integrable models. As examples of their utility, we show how these explicit distributions may be used to quantify the semi-infinite geodesics passing through thin rectangles, and the clustering phenomenon observed in competition interface angles. There is a natural connection with various particle systems, and in particular we obtain the multi-class invariant distributions for discrete-time TASEP with parallel updates.

Figures (6)

• Figure 1: An illustration of the paths considered in strict-weak first passage percolation and the rule for assigning passage times. Namely, one looks at all up-right directed paths which connect two vertices and sums over the horizontal weights collected, taking the path which minimises this quantity. Observe that there is a pronounced asymmetry between the directions, which does not appear in the more familiar last passage percolation.
• Figure 2: The movement of material within our store at time step $k$. The store has some initial quantity $X_k$, which is added to by input $I_k$. Then an attempt is made to output the desired amount $W_k$. In this instance, there was enough material to cover the demand. Here the new store size is $X_{k + 1} = (X_k + I_k - W_k)^+$ and the output is $I'_k = (X_k + I_k) \wedge W_k$.
• Figure 4: A time step under the sequential TASEP dynamics, with the particles initially stacked up at the origin. The particles each receive a jump size (weight), and will attempt to jump that distance unless blocked by the particle in front of them.
• Figure 5: An illustration of a particle interpretation of the $H$ map. On the left are the arguments to the map. The inter-particle distances on the top row are given by the sequence $I$ and on the bottom by $W$. The particles on the bottom row are matched to a particle ahead of and above them. The unmatched particles on the top row stay where they are, and the matched particles move back, switching places with their match. The result is $H(I, W)$. The leftover particles are themselves interesting and give the output of a G/G/1 queue. Compare with ferr-mart-mtasep.
• Figure 6:

Theorems & Definitions (161)

• Remark 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.5
• Remark 2.6
• Theorem 2.7: Theorem 7.1 of mart-prab-fixed
• Theorem 2.8
• Proposition 2.9
• Corollary 2.10
• Theorem 2.11