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Planar maps, random walks and circle packing / Asaf Nachmias (2020)  

Public Titre : Planar maps, random walks and circle packing : École d'été de probabilités de Saint-Flour XLVIII - 2018 Type de document : Guide/Manuel Auteurs : Asaf Nachmias, Éditeur scientifique Editeur : Berlin, Heidelberg, Vienne, New York, ... : Springer Année de publication : 2020 Collection : Lecture notes in Mathematics num. 2243 Importance : 120 p. ISBN/ISSN/EAN : 978-3-030-27968-4 Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Statistiques [Termes IGN] arbre aléatoire [Termes IGN] fonction harmonique [Termes IGN] graphe planaire [Termes IGN] modèle de Markov Index. décimale : 23.60 Statistiques et probabilités Résumé : (Editeur) This open access book focuses on the interplay between random walks on planar maps and Koebe’s circle packing theorem. Further topics covered include electric networks, the He–Schramm theorem on infinite circle packings, uniform spanning trees of planar maps, local limits of finite planar maps and the almost sure recurrence of simple random walks on these limits. One of its main goals is to present a self-contained proof that the uniform infinite planar triangulation (UIPT) is almost surely recurrent. Full proofs of all statements are provided. A planar map is a graph that can be drawn in the plane without crossing edges, together with a specification of the cyclic ordering of the edges incident to each vertex. One widely applicable method of drawing planar graphs is given by Koebe’s circle packing theorem (1936). Various geometric properties of these drawings, such as existence of accumulation points and bounds on the radii, encode important probabilistic information, such as the recurrence/transience of simple random walks and connectivity of the uniform spanning forest. This deep connection is especially fruitful to the study of random planar maps. The book is aimed at researchers and graduate students in mathematics and is suitable for a single-semester course; only a basic knowledge of graduate level probability theory is assumed. Note de contenu : 1. Introduction 1.1 The Circle Packing Theorem 1.2 Probabilistic Applications 2. Random Walks and Electric Networks 2.1 Harmonic Functions and Voltages 2.2 Flows and Currents 2.3 The Effective Resistance of a Network 2.4 Energy 2.5 Infinite Graphs 2.6 Random Paths 2.7 Exercises 3. The CirclePacking Theorem 3.1 Planar Graphs, Maps and Embeddings 3.2 Proof of the Circle Packing Theorem 4. Parabolic and Hyperbolic Packings 4.1 Infinite Planar Maps 4.2 The Ring Lemma and Infinite Circle Packings 4.3 Statement of the He–Schramm Theorem 4.4 Proof of the He–Schramm Theorem 4.5 Exercises 5. Planar Local Graph Limits 5.1 Local Convergenceof Graphs and Maps 5.2 The Magic Lemma 5.3 Recurrence of Bounded Degree Planar Graph Limits 5.4 Exercises 6. Recurrence of Random Planar Maps 6.1 Star-Tree Transform 6.2 Stationary Random Graphs and Markings 6.3 Proof of Theorem 7. Uniform Spanning Trees of Planar Graphs 7.1 Introduction 7.2 Basic Properties of the UST 7.3 Limits over Exhaustions:The Free and Wired USF 7.4 Planar Duality 7.5 Connectivity of the Free Forest 7.6 Exercises 8. Related Topics Numéro de notice : 26541 Affiliation des auteurs : non IGN Thématique : MATHEMATIQUE Nature : Manuel de cours DOI : 10.1007/978-3-030-27968-4 En ligne : http://doi.org/10.1007/978-3-030-27968-4 Format de la ressource électronique : URL Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=97764