Asymptotics of Parking Search in Hyperfractal Networks

2026-04-27

AI Summary

This study investigates the asymptotic behavior of the expected distance to the first available parking spot in a recursive Manhattan street network endowed with a hyperfractal intensity structure, as the total intensity grows. By integrating a Poissonian parking-release process, self-similar harmonic sums, and Mellin-transform-based asymptotic analysis, the work reveals that the expected search distance decays as a power law, with an exponent equal to the reciprocal of the hyperfractal dimension—an exponent dictated solely by the large-scale geometry of the network. This scaling law proves robust against multiplicative random perturbations of street intensities and extends consistently to higher moments such as variance, the expected number of turns, and alternative jump-based search strategies, where stochastic heterogeneity influences only the proportionality constant.

Abstract

We study the asymptotic behaviour of the distance to the first available parking slot in a recursive Manhattan street network endowed with a hyperfractal intensity structure, where slot-release events occur according to Poisson processes along the streets. We establish, by analysing the associated self-similar harmonic sums via Mellin-transform asymptotics, a power-law decay of the expected distance as the total intensity grows, with exponent equal to the inverse of the hyperfractal dimension. In particular, the scaling exponent depends only on the large-scale geometry of the network. We further prove that this exponent is robust under random multiplicative modulations of the street intensities: mild stochastic heterogeneity affects only the multiplicative constant. Similar scaling behaviour holds for the variance, the number of turns before parking, and for a jump-over variant of the search strategy.

Problem

• parking search
• hyperfractal networks
• asymptotic behaviour
• Manhattan street network
• Poisson processes

Innovation

• hyperfractal networks
• parking search
• Mellin-transform asymptotics
• power-law scaling
• self-similar harmonic sums