The parking problem

The question is to develop a microscopic model for the reported random-matrix behavior of car-parking spacings. The reported contrast is GUE statistics for cars parked on the right side of a two-way street and GOE statistics for left-side parking on a one-way street. The challenge is proposed by analogy with microscopic explanations for other transportation problems; no particular parking rule, probability law, or formal criterion for such an explanation is specified. Deift (2007)

Background

In continuous random sequential adsorption, equal-length cars independently attempt randomly chosen positions and park if they do not overlap cars already present. This irreversible process reaches a jamming limit, but its predicted small-gap density diverges as the gap tends to zero. Measurements of parked cars in London instead found a density tending to zero at small gaps. An equilibrium logarithmically repelling gas can reproduce random-matrix spacing laws, but it assumes a particular interaction and temperature rather than deriving them from parking behavior. Šeba (2007)

Šeba’s model instead describes a fully occupied street in equilibrium: when a car leaves, its two neighboring clearances merge, and an arriving driver splits the resulting vacant space. The model assumes symmetric parking-volatility density, so the split proportions have the same distribution when reflected about one half; it also neglects events in which multiple cars leave in a single time step. This split-and-merge mechanism was proposed as a way to connect driver behavior to the observed spacing distributions. Šeba (2007)

Known results

Under those assumptions, the spacing distribution is described by a distributional fixed-point relation. If $X$ denotes a generic clearance, $X_1,X_2$ are independent copies of it, and $A$ is an independent split variable with density $P$, stationarity gives the scalar marginal relation

$$X\overset{d}{=}A(X_1+X_2).$$

The model also gives the corresponding marginal relation with an independent split variable having the same density for the other new clearance. These are marginal laws; they do not assert a joint law for the two new clearances. Iterative solutions for $P(a)=6a(1-a)$ are reported as practically indistinguishable from the GOE level-spacing distribution, while the symmetric beta density with parameters $(3.3,3.3)$ gives close agreement with the GUE prediction. These are numerical comparisons, not proofs of exact GOE or GUE statistics. Šeba also compared model predictions with car-spacing measurements from two streets in Hradec Králové. Šeba (2007)

A related finite-segment model, presented as a preprint, considers limited resource sharing. With equal split parameters, it obtains a Dirichlet joint distribution for the spacings. The authors motivate the choice of parameter $g=3$ using a proposed distance-perception argument; in the limit of a large number of cars, the normalized one-spacing distribution tends to a gamma distribution with shape $3$. This is a distinct model and does not establish the GOE/GUE comparisons in Šeba’s street model. Parking in the city: an example of limited resource sharing (preprint)