Translator Disclaimer
October 2021 The fractal cylinder process: Existence and connectivity phase transitions
Erik I. Broman, Olof Elias, Filipe Mussini, Johan Tykesson
Author Affiliations +
Ann. Appl. Probab. 31(5): 2192-2243 (October 2021). DOI: 10.1214/20-AAP1644

Abstract

We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension d2, and a connectivity phase transition whenever d4. We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point.

A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this restriction results in a fractal ellipsoid model which we describe in detail, as it is key to obtaining our main results.

In addition we also determine the almost sure Hausdorff dimension of the fractal set.

Funding Statement

The first and fourth authors were supported by the Swedish Research Council.

Acknowledgments

The authors would like to thank the anonymous referees for providing comments and suggestions improving the quality of the paper.

Citation

Download Citation

Erik I. Broman. Olof Elias. Filipe Mussini. Johan Tykesson. "The fractal cylinder process: Existence and connectivity phase transitions." Ann. Appl. Probab. 31 (5) 2192 - 2243, October 2021. https://doi.org/10.1214/20-AAP1644

Information

Received: 1 August 2019; Revised: 1 September 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

Digital Object Identifier: 10.1214/20-AAP1644

Subjects:
Primary: 28A80 , 60K35 , 82B43

Keywords: fractal percolation , Poisson cylinder model , Random fractals

Rights: Copyright © 2021 Institute of Mathematical Statistics

JOURNAL ARTICLE
52 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE