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The half plane UIPT is recurrent.

Omer Angel1, Gourab Ray2

  • 11Department of Mathematics, University of British Columbia, Vancouver, Canada.

Probability Theory and Related Fields
|July 2, 2019
PubMed
Summary
This summary is machine-generated.

The study proves that the half plane version of the uniform infinite planar triangulation is recurrent. This finding is based on a novel full plane extension and adapted recurrence methods.

Keywords:
60B9960G50

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Area of Science:

  • Probability theory
  • Geometric probability
  • Random graph theory

Background:

  • The uniform infinite planar triangulation (UIPT) is a fundamental object in random graph theory.
  • Understanding the behavior of UIPT in different domains, such as the half-plane, is crucial for advancing probabilistic models.
  • Previous research has explored recurrence properties of related structures, but the half-plane case presented unique challenges.

Purpose of the Study:

  • To establish the recurrence property of the half-plane version of the uniform infinite planar triangulation (UIPT).
  • To develop novel techniques for analyzing random planar maps in restricted domains.
  • To extend the understanding of random graph behavior in geometric settings.

Main Methods:

  • Construction of a full plane extension for the half-plane UIPT.
  • Decomposition of the half-plane UIPT into independent layers.
  • Adaptation of existing methods for proving recurrence of weak local limits, incorporating circle packing techniques.

Main Results:

  • The half-plane uniform infinite planar triangulation (UIPT) is proven to be recurrent.
  • A new method for constructing full plane extensions of UIPT variants was developed.
  • The study successfully extended techniques for analyzing weak local limits in the context of UIPT.

Conclusions:

  • The recurrence of the half-plane UIPT is definitively established.
  • The employed techniques offer a new framework for studying other random planar structures.
  • This work contributes significantly to the field of random graph theory and geometric probability.