A Probabilistic Parking Process and Labeled IDLA

TL;DR

This work introduces a probabilistic parking protocol closely related to IDLA and analyzes two boundary variants (unbounded and open) starting from initial preferences that yield identity outcomes. It derives explicit formulas for the probability that each car parks, the time to complete the process, and negative-correlation properties, using gambler's ruin arguments and Dyck-path/Catalan encodings. The paper also studies statistics of weakly increasing parking functions, obtaining asymptotics for the last-entry distribution and enumerative results for lucky cars, including exact Catalan-based counts and the distribution of the number of lucky cars. These results connect parking functions to interacting particle systems and classical combinatorial structures, providing exact, computable characterizations with potential implications for phase transitions and stationary measures in related models.

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.

Figures (4)

• Figure 1: Illustration of the probabilistic parking protocol as a random mapping from $[n]^n$ to $S_n$. The dynamics induce a distribution over outcome permutations, and our main question of interest is to characterize this distribution $\Pr\left[ \operatorname{Out}(\alpha) = \sigma\right]$ where $\alpha$ and $\sigma$ are fixed, and we use $\operatorname{Out}(\alpha)$ to denote the outcome of $\alpha$ under the probabilistic parking protocol.
• Figure 2: The Dyck path UUDUDUUUUDDDDUDD, of length $16$ corresponds to the weakly increasing parking function $(1,1,2,3,3,3,3,7) \in \mathsf{PF}^{\uparrow}_8$.
• Figure 3: The Dyck path corresponding to $(1,1,3,3,4)\in\mathsf{PF}^{\uparrow}_5$ can be thought of as a composition of the two different Dyck paths, each with no returns except the last.
• Figure 4: The color of each cell $(p, y)$ in the image above corresponds to the number of parking functions $\alpha \in \mathrm{PF}_n(\mathrm{id})$ where the probability that $\alpha$ parks with parameter $p$ is at most $y$, where $y$ is calculated according to Theorem \ref{['thm:FormulaswPF']}. For example, when $p = 0$, the probability that $\alpha$ parks is always $0$ and the pixels are dark red for all $y$.

Theorems & Definitions (23)

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