Planar Maps, Random Walks and Circle Packing - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Planar Maps, Random Walks and Circle Packing write by Asaf Nachmias. This book was released on 2019-10-04. Planar Maps, Random Walks and Circle Packing available in PDF, EPUB and Kindle. 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.
Planar Maps, Random Walks and Circle Packing
Planar Maps, Random Walks and Circle Packing - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Planar Maps, Random Walks and Circle Packing write by Asaf Nachmias. This book was released on 2020-10-08. Planar Maps, Random Walks and Circle Packing available in PDF, EPUB and Kindle. 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. This work was published by Saint Philip Street Press pursuant to a Creative Commons license permitting commercial use. All rights not granted by the work's license are retained by the author or authors.
Peeling Random Planar Maps
Peeling Random Planar Maps - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Peeling Random Planar Maps write by Nicolas Curien. This book was released on 2023-11-20. Peeling Random Planar Maps available in PDF, EPUB and Kindle. 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.
Introduction to Circle Packing
Introduction to Circle Packing - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Introduction to Circle Packing write by Kenneth Stephenson. This book was released on 2005-04-18. Introduction to Circle Packing available in PDF, EPUB and Kindle. Publisher Description
Random Walks on Infinite Graphs and Groups
Random Walks on Infinite Graphs and Groups - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Random Walks on Infinite Graphs and Groups write by Wolfgang Woess. This book was released on 2000-02-13. Random Walks on Infinite Graphs and Groups available in PDF, EPUB and Kindle. The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. The author considers Markov chains whose state space is equipped with the structure of an infinite, locally finite graph, or as a particular case, of a finitely generated group. The transition probabilities are assumed to be adapted to the underlying structure in some way that must be specified precisely in each case. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. Links with spectral theory and discrete potential theory are also discussed. This book will be essential reading for all researchers working in stochastic process and related topics.
