Abstract
We introduce the Incipient Infinite Cluster ($\mathsf{IIC} $) in the critical Bernoulli site percolation model on the Uniform Infinite Half-Planar Triangulation ($\mathsf{UIHPT} $), which is the local limit of large random triangulations with a boundary. The $\mathsf{IIC} $ is defined from the $\mathsf{UIHPT} $ by conditioning the open percolation cluster of the origin to be infinite. We prove that the $\mathsf{IIC} $ can be obtained by adding within the $\mathsf{UIHPT} $ an infinite triangulation with a boundary whose distribution is explicit.
Highlights
The purpose of this work is to describe the geometry of a large critical percolation cluster in the Uniform Infinite Half-Planar Triangulation (UIHPT for short), which is the local limit of random triangulations with a boundary, upon letting first the volume and the perimeter tend to infinity
The study of local limits of large planar maps goes back to Angel & Schramm, who introduced in [5] the Uniform Infinite Planar Triangulation (UIPT), while the half-plane model was defined later on by Angel in [3]
Local limits of large planar maps equipped with a percolation model have been studied extensively
Summary
The purpose of this work is to describe the geometry of a large critical percolation cluster in the (type 2) Uniform Infinite Half-Planar Triangulation (UIHPT for short), which is the local limit of random triangulations with a boundary, upon letting first the volume and the perimeter tend to infinity. The probability measure PIIC is called (the law of) the Incipient Infinite Cluster of the UIHPT (IIC for short) and is supported on triangulations of the half-plane. In Theorem 2.1, we decompose the UIHPT into two infinite sub-maps distributed as the closed percolation hulls of the IIC, and glued along a uniform necklace The idea of such a decomposition goes back to [18]. The boundary of an infinite planar map is the embedding of edges and vertices of its root face. The probability measure P∞,k is called (the law of) the UIPT of the k-gon, while P∞,∞ is (the law of) the Uniform Infinite Half-Planar Triangulation (UIHPT). The hull H of C is the coloured map obtained by filling in the finite holes of C
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