# Analytical and simulation studies of pedestrian flow at a crossing with random update rule

## TL;DR

This paper investigates pedestrian flow dynamics at crossings using a 2D lattice model with a random update rule, revealing phase transitions, jamming behavior, and validating analytical predictions with simulations.

## Contribution

It introduces a mean field analytical approach to pedestrian flow with a random update rule, identifying phase behavior and jamming phenomena under different boundary conditions.

## Key findings

- Existence of an intermediate phase with border-moving pedestrians.
- Analytical results match simulation data for velocity, density, and flow rate.
- Critical injection probability varies nontrivially with forward moving probability.

## Abstract

The intersecting pedestrian flow on the 2D lattice with random update rule is studied. Each pedestrian has three moving directions without the back step. Under periodic boundary conditions, an intermediate phase has been found at which some pedestrians could move along the border of jamming stripes. We have performed mean field analysis for the moving and intermediate phase respectively. The analytical results agree with the simulation results well. The empty site moves along the interface of jamming stripes when the system only has one empty site. The average movement of empty site in one Monte Carlo step (MCS) has been analyzed through the master equation. Under open boundary conditions, the system exhibits moving and jamming phases. The critical injection probability $\alpha_c$ shows nontrivially against the forward moving probability $q$. The analytical results of average velocity, the density and the flow rate against the injection probability in the moving phase also agree with simulation results well.