The Annals of Probability

Classification of half-planar maps

Omer Angel and Gourab Ray

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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.

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

First available in Project Euclid: 5 May 2015

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

Random maps domain Markov property half planar maps


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

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