# Random Walks on Intersecting Geometries

## TL;DR

This paper analytically studies symmetric random walks on a plane with intersecting lines, revealing how the walker's long-term behavior transitions from line to plane dominance and how small biases influence this crossover.

## Contribution

It introduces an exact analytical approach for random walks on crossing geometries and demonstrates the impact of small drift perturbations on the asymptotic behavior.

## Key findings

- Large time behavior is dominated by the plane after a crossover time proportional to n_l^2.
- Weak bias can shift the probability density to favor the lines at long times.
- Simulation results confirm the analytical predictions.

## Abstract

We present an analytical approach to study simple symmetric random walks (RWs) on a crossing geometry consisting of a plane square lattice crossed by $n_l$ number of lines that all meet each other at a single point (the origin) on the plane. The probability density to find the walker at a given distance from the origin either in a line or in the plane geometry is exactly calculated as a function of time t. We find that the large time asymptotic behavior of the walker for any arbitrary number $n_l$ of lines is eventually governed by the plane geometry after a crossover time approximately given by $t_c\propto n_l^2$. We show that this competition can be changed in favor of the line geometry by switching on an arbitrarily small perturbation of a drift term in which even a weak biased walk is able to drain the whole probability density into the line at long time limit. We also present the results of our extensive simulations of the model which perfectly support our analytical predictions. Our method can, however, be simply extended to other crossing geometries with a single common point.