Peeling Random Planar Maps
École d’Été de Probabilités de Saint-Flour XLIX – 2019
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Beschreibung
These Lecture Notes provide an introduction to the study of those discrete surfaces which are obtained by randomly gluing polygons along their sides in a plane. The focus is on the geometry of such random planar maps (diameter, volume growth, scaling and local limits...) as well as the behavior of statistical mechanics models on them (percolation, simple random walks, self-avoiding random walks...).
A “Markovian” approach is adopted to explore these random discrete surfaces, which is then related to the analogous one-dimensional random walk processes. This technique, known as "peeling exploration" in the literature, can be seen as a generalization of the well-known coding processes for random trees (e.g. breadth first or depth first search). It is revealed that different types of Markovian explorations can yield different types of information about a surface.
Based on an École d'Été de Probabilités de Saint-Flour course delivered by the author in 2019, the book is aimed at PhDstudents and researchers interested in graph theory, combinatorial probability and geometry. Featuring open problems and a wealth of interesting figures, it is the first book to be published on the theory of random planar maps.
These Lecture Notes provide an introduction to the study of those discrete surfaces which are obtained by randomly gluing polygons along their sides in a plane. The focus is on the geometry of such random planar maps (diameter, volume growth, scaling and local limits...) as well as the behavior of statistical mechanics models on them (percolation, simple random walks, self-avoiding random walks...).
A “Markovian” approach is adopted to explore these random discrete surfaces, which is then related to the analogous one-dimensional random walk processes. This technique, known as "peeling exploration" in the literature, can be seen as a generalization of the well-known coding processes for random trees (e.g. breadth first or depth first search). It is revealed that different types of Markovian explorations can yield different types of information about a surface.
Based on an École d'Été de Probabilités de Saint-Flour course delivered by the author in 2019, the book is aimed at PhD students and researchers interested in graph theory, combinatorial probability and geometry. Featuring open problems and a wealth of interesting figures, it is the first book to be published on the theory of random planar maps.The first book on probabilistic aspects of planar maps Provides comprehensive coverage of the theory and includes open problems Illustrated with numerous attractive figures
Autor*in
Nicolas Curien
Themen in »Peeling Random Planar Maps«
Graph Theory Planar Maps Combinatorics Markov Property Scaling Limits Stable Processes Combinatorial Geometry Combinatorial Probability Peeling Exploration in the Random Planar Map Coding Processes of Random Trees Random Geometry
Stimmen zu »Peeling Random Planar Maps«
“This lengthy monograph is an excellent addition to the long-running École d'Été de Probabilités de Saint-Flour series of extended lecture notes, continuing their tradition of reader-friendly (for an active researcher in mathematical probability) authoritative accounts of an active technical topic. It has the traditional underlying definition/ theorem/proof format … and numerous well thought out figures, which (to your reviewer) are essential for any work on graph theory. … this monograph will long remain a key account of its topics.” (David J. Aldous, Mathematical Reviews, December, 2024)
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