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2022 First-passage percolation in random planar maps and Tutte’s bijection
Thomas Lehéricy
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Electron. J. Probab. 27: 1-50 (2022). DOI: 10.1214/21-EJP662


We consider large random planar maps and study the first-passage percolation distance obtained by assigning independent identically distributed lengths to the edges. We consider the cases of quadrangulations and of general planar maps. In both cases, the first-passage percolation distance is shown to behave in large scales like a constant times the usual graph distance. We apply our method to the metric properties of the classical Tutte bijection between quadrangulations with n faces and general planar maps with n edges. We prove that the respective graph distances on the quadrangulation and on the associated general planar map are in large scales equivalent when n.


We thank two anonymous referees for their helpful feedback.


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Thomas Lehéricy. "First-passage percolation in random planar maps and Tutte’s bijection." Electron. J. Probab. 27 1 - 50, 2022.


Received: 25 June 2019; Accepted: 7 June 2021; Published: 2022
First available in Project Euclid: 2 March 2022

arXiv: 1906.10079
Digital Object Identifier: 10.1214/21-EJP662

Primary: 05C80 , 60D05

Keywords: first passage percolation , Probability , Random maps


Vol.27 • 2022
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