# A Stochastic Approach to Eulerian Numbers

## TL;DR

This paper explores the behavior of a one-dimensional random walk model called Internal Diffusion Limited Aggregation, revealing that the distribution of occupied sites follows an Eulerian probability distribution and analyzing its expected run time.

## Contribution

It establishes a novel connection between the aggregate behavior of the model and Eulerian probability distributions, and computes the expected run time using generating functions.

## Key findings

- Occupied sites follow an Eulerian probability distribution.
- Derived the probability of a certain number of positive sites.
- Computed the expected run time of the aggregation process.

## Abstract

We examine the aggregate behavior of one-dimensional random walks in a model known as (one-dimensional) Internal Diffusion Limited Aggregation. In this model, a sequence of $n$ particles perform random walks on the integers, beginning at the origin. Each particle walks until it reaches an unoccupied site, at which point it occupies that site and the next particle begins its walk. After all walks are complete, the set of occupied sites is an interval of length $n$ containing the origin. We show the probability that $k$ of the occupied sites are positive is given by an Eulerian probability distribution. Having made this connection, we use generating function techniques to compute the expected run time of the model.