# Road Layout in The KPZ Class

## TL;DR

This paper uses a mathematical model to study road layouts and traffic patterns, comparing them to known theoretical models in physics.

## Contribution

The paper introduces a novel traffic model using exponential last passage percolation in the KPZ universality class.

## Key findings

- Traffic trajectories coalescing are modeled using geodesics from last passage percolation.
- Exponential LPP is shown to be suitable for modeling road networks due to its KPZ class properties.
- The model's predictions align with observed characteristics of real-world road networks.

## Abstract

We propose a road layout and traffic model, based on last passage percolation (LPP). An easy naïve argument shows that coalescence of traffic trajectories is essential to be considered when observing traffic networks around us. This is a fundamental feature in first passage percolation (FPP) models where nearby geodesics naturally coalesce in search of the easiest passage through the landscape. Road designers seek the same in pursuing cost savings, hence FPP geodesics are straightforward candidates to model road layouts. Unfortunately no detailed knowledge is rigorously available on FPP geodesics. To address this, we use exponential LPP instead to build a stochastic model of road traffic and prove certain characteristics thereof. Cars start from every point of the lattice and follow half-infinite geodesics in random directions. Exponential LPP is known to be in the KPZ universality class and it is widely expected that FPP shares very similar properties, hence our findings should equally apply to FPP-based modelling. We address several traffic-related quantities of this model and compare our theorems to real life road networks.