The Annals of Probability

Classification of half-planar maps

Omer Angel and Gourab Ray

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Abstract

We characterize all translation invariant half-planar maps satisfying a certain natural domain Markov property. For $p$-angulations with $p\ge3$ where all faces are simple, we show that these form a one-parameter family of measures $\mathbb{H}^{(p)}_{\alpha}$. For triangulations, we also establish existence of a phase transition which affects many properties of these maps. The critical maps are the well-known half-plane uniform infinite planar maps. The subcritical maps are identified as all possible limits of uniform measures on finite maps with given boundary and area.

Article information

Source
Ann. Probab., Volume 43, Number 3 (2015), 1315-1349.

Dates
First available in Project Euclid: 5 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1430830283

Digital Object Identifier
doi:10.1214/13-AOP891

Mathematical Reviews number (MathSciNet)
MR3342664

Zentralblatt MATH identifier
1354.60010

Subjects
Primary: 60B99: None of the above, but in this section 51F99: None of the above, but in this section

Keywords
Random maps domain Markov property half planar maps

Citation

Angel, Omer; Ray, Gourab. Classification of half-planar maps. Ann. Probab. 43 (2015), no. 3, 1315--1349. doi:10.1214/13-AOP891. https://projecteuclid.org/euclid.aop/1430830283


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