Abstract
We prove that the half plane version of the uniform infinite planar triangulation (UIPT) is recurrent. The key ingredients of the proof are a construction of a new full plane extension of the half plane UIPT, based on a natural decomposition of the half plane UIPT into independent layers, and an extension of previous methods for proving recurrence of weak local limits (while still using circle packings).
Highlights
The half plane uniform infinite planar triangulation, abbreviated as the HUIPT below, is a random planar triangulation, closely related to the well-known and extensively studied uniform infinite planar triangulations (UIPT), but with the topology of the half plane
The HUIPT is an interesting object in its own right, and in some ways is nicer than the UIPT
Recurrence of the HUIPT does not follow from their work, as it is not known if it is possible to realize the HUIPT as a subgraph of the UIPT
Summary
The half plane uniform infinite planar triangulation, abbreviated as the HUIPT below, is a random planar triangulation, closely related to the well-known and extensively studied uniform infinite planar triangulations (UIPT), but with the topology of the half plane. The HUIPT is an interesting object in its own right, and in some ways is nicer than the UIPT It possesses a simpler form of the domain Markov property Theorem 1 The simple random walk on the half plane uniform infinite planar triangulation is almost-surely recurrent. The methods of [14,20] do not apply directly, since the HUIPT is not a weak local limit of finite planar graphs. The domain Markov property of HUIPT gives this decomposition a elegant structure. Such a decomposition has great potential for the study of random maps. A continuum version of this decomposition has been introduces in recent work of Miller and Sheffield as part of a characterization of the Brownian map [28]
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