Simple peeling of planar maps with application to site percolation
arXiv:1909.10217 · doi:10.4153/S0008414X21000146
Abstract
The peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolation on maps with faces of arbitrary degree are conveniently studied via so-called lazy-peeling explorations. During such explorations distinct vertices on the exploration contour may at later stage be identified, making the process less suited to the study of site percolation. To tackle this situation and to explicitly identify site-percolation thresholds, we come back to the alternative "simple" peeling exploration of Angel and uncover deep relations with the lazy-peeling process. Along the way we define and study the random Boltzmann map of the half-plane with a simple boundary for an arbitrary critical weight sequence. Its construction is nontrivial especially in the "dense regime" where the half-planar random Boltzmann map does not possess an infinite simple core.
arXiv comment32 pages, 16 figures
References in corpus (10)
- Uniform Infinite Planar Triangulations
- Construction of Non-critical String Field Theory by Transfer Matrix Formalism in Dynamical Triangulation
- Invariance principles for random walks conditioned to stay positive
- Percolation on uniform infinite planar maps
- Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop
- Geometry of infinite planar maps with high degrees
- Generalized multicritical one-matrix models
- A Boltzmann approach to percolation on random triangulations
- Infinite random planar maps related to Cauchy processes
- On site percolation in random quadrangulations of the half-plane