A Probabilistic Parking Process and Labeled IDLA

TL;DR

This paper introduces a probabilistic parking process related to IDLA, analyzing its stationary distribution, completion time, and statistical properties of parking functions, revealing new insights into parking protocols and their probabilistic behavior.

Contribution

It presents a novel probabilistic parking process linked to IDLA, computes its stationary distribution, expected completion time, and explores statistical properties of parking functions.

Findings

Computed the stationary distribution for the process.

Determined the expected parking completion time.

Proved negative correlation in some cases.

Abstract

In 1966, Konheim and Weiss [33] introduced a now classical parking protocol. The deterministic process and its resultant objects, known as parking functions, have since become a favorite object of study in enumerative combinatorics. In our work, we introduce and study a probabilistic variant of the classical parking protocol, which is closely related to Internal Diffusion Limited Aggregation, or IDLA, introduced in 1991 by Diaconis and Fulton [19]. In particular, we compute the stationary distribution of this process when initiated with a particular class of initial preferences, of which weakly increasing parking functions are a subset. Furthermore, we compute the expected time it takes for the protocol to complete assuming all of the cars park, and prove that, in some cases, the parking process is negatively correlated. In addition, we study statistics of uniformly random weakly increasing parking functions such as the distribution of the last entry, the probability that a specific set of cars is lucky, and the expected number of lucky cars.