Planar Maps Random Walks And Circle Packing

Author : Asaf Nachmias
Publisher : Springer Nature
Page : 120 pages
File Size : 46,9 Mb
Release : 2019-10-04
Category : Mathematics
ISBN : 9783030279684

Get Book

Planar Maps, Random Walks and Circle Packing by Asaf Nachmias Pdf

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.

Author : Asaf Nachmias
Publisher : Unknown
Page : 122 pages
File Size : 48,7 Mb
Release : 2020-10-08
Category : Mathematics
ISBN : 1013271130

Get Book

Planar Maps, Random Walks and Circle Packing by Asaf Nachmias Pdf

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.

Author : Nicolas Curien
Publisher : Springer Nature
Page : 293 pages
File Size : 51,9 Mb
Release : 2023-11-20
Category : Mathematics
ISBN : 9783031368547

Get Book

Peeling Random Planar Maps by Nicolas Curien Pdf

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.

Author : Itai Benjamini
Publisher : Springer
Page : 133 pages
File Size : 53,5 Mb
Release : 2013-12-02
Category : Mathematics
ISBN : 9783319025766

Get Book

Coarse Geometry and Randomness by Itai Benjamini Pdf

These lecture notes study the interplay between randomness and geometry of graphs. The first part of the notes reviews several basic geometric concepts, before moving on to examine the manifestation of the underlying geometry in the behavior of random processes, mostly percolation and random walk. The study of the geometry of infinite vertex transitive graphs, and of Cayley graphs in particular, is fairly well developed. One goal of these notes is to point to some random metric spaces modeled by graphs that turn out to be somewhat exotic, that is, they admit a combination of properties not encountered in the vertex transitive world. These include percolation clusters on vertex transitive graphs, critical clusters, local and scaling limits of graphs, long range percolation, CCCP graphs obtained by contracting percolation clusters on graphs, and stationary random graphs, including the uniform infinite planar triangulation (UIPT) and the stochastic hyperbolic planar quadrangulation (SHIQ).

Author : Kenneth Stephenson
Publisher : Cambridge University Press
Page : 380 pages
File Size : 41,7 Mb
Release : 2005-04-18
Category : Mathematics
ISBN : 0521823560

Get Book

Introduction to Circle Packing by Kenneth Stephenson Pdf

Publisher Description

Author : Vladas Sidoravicius
Publisher : Springer Nature
Page : 341 pages
File Size : 55,8 Mb
Release : 2019-10-17
Category : Mathematics
ISBN : 9789811503023

Get Book

Sojourns in Probability Theory and Statistical Physics - III by Vladas Sidoravicius Pdf

Charles M. (Chuck) Newman has been a leader in Probability Theory and Statistical Physics for nearly half a century. This three-volume set is a celebration of the far-reaching scientific impact of his work. It consists of articles by Chuck’s collaborators and colleagues across a number of the fields to which he has made contributions of fundamental significance. This publication was conceived during a conference in 2016 at NYU Shanghai that coincided with Chuck's 70th birthday. The sub-titles of the three volumes are: I. Spin Glasses and Statistical Mechanics II. Brownian Web and Percolation III. Interacting Particle Systems and Random Walks The articles in these volumes, which cover a wide spectrum of topics, will be especially useful for graduate students and researchers who seek initiation and inspiration in Probability Theory and Statistical Physics.

Author : Wolfgang Woess
Publisher : Cambridge University Press
Page : 350 pages
File Size : 47,8 Mb
Release : 2000-02-13
Category : Mathematics
ISBN : 9780521552929

Get Book

Random Walks on Infinite Graphs and Groups by Wolfgang Woess Pdf

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.

Author : Itai Benjamini,Olle Häggström
Publisher : Springer Science & Business Media
Page : 1199 pages
File Size : 44,6 Mb
Release : 2011-08-12
Category : Mathematics
ISBN : 9781441996756

Get Book

Selected Works of Oded Schramm by Itai Benjamini,Olle Häggström Pdf

This volume is dedicated to the memory of the late Oded Schramm (1961-2008), distinguished mathematician. Throughout his career, Schramm made profound and beautiful contributions to mathematics that will have a lasting influence. In these two volumes, Editors Itai Benjamini and Olle Häggström have collected some of his papers, supplemented with three survey papers by Steffen Rohde, Häggström and Cristophe Garban that further elucidate his work. The papers within are a representative collection that shows the breadth, depth, enthusiasm and clarity of his work, with sections on Geometry, Noise Sensitivity, Random Walks and Graph Limits, Percolation, and finally Schramm-Loewner Evolution. An introduction by the Editors and a comprehensive bibliography of Schramm's publications complete the volume. The book will be of especial interest to researchers in probability and geometry, and in the history of these subjects.

Author : Christopher J. Bishop,Yuval Peres
Publisher : Cambridge University Press
Page : 415 pages
File Size : 44,6 Mb
Release : 2017
Category : Mathematics
ISBN : 9781107134119

Get Book

Fractals in Probability and Analysis by Christopher J. Bishop,Yuval Peres Pdf

A mathematically rigorous introduction to fractals, emphasizing examples and fundamental ideas while minimizing technicalities.

Author : Geoffrey Grimmett
Publisher : Cambridge University Press
Page : 260 pages
File Size : 48,7 Mb
Release : 2010-06-24
Category : Mathematics
ISBN : 9781139488365

Get Book

Probability on Graphs by Geoffrey Grimmett Pdf

This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. Schramm–Löwner evolutions (SLE) arise in various contexts. The choice of topics is strongly motivated by modern applications and focuses on areas that merit further research. Special features include a simple account of Smirnov's proof of Cardy's formula for critical percolation, and a fairly full account of the theory of influence and sharp-thresholds. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.

Author : Ken’ichi Ohshika,Athanase Papadopoulos
Publisher : Springer Nature
Page : 724 pages
File Size : 41,7 Mb
Release : 2020-12-07
Category : Mathematics
ISBN : 9783030559281

Get Book

In the Tradition of Thurston by Ken’ichi Ohshika,Athanase Papadopoulos Pdf

This book consists of 16 surveys on Thurston's work and its later development. The authors are mathematicians who were strongly influenced by Thurston's publications and ideas. The subjects discussed include, among others, knot theory, the topology of 3-manifolds, circle packings, complex projective structures, hyperbolic geometry, Kleinian groups, foliations, mapping class groups, Teichmüller theory, anti-de Sitter geometry, and co-Minkowski geometry. The book is addressed to researchers and students who want to learn about Thurston’s wide-ranging mathematical ideas and their impact. At the same time, it is a tribute to Thurston, one of the greatest geometers of all time, whose work extended over many fields in mathematics and who had a unique way of perceiving forms and patterns, and of communicating and writing mathematics.

Author : Philip L. Bowers,Kenneth Stephenson
Publisher : American Mathematical Soc.
Page : 118 pages
File Size : 49,5 Mb
Release : 2004
Category : Circle packing
ISBN : 9780821835234

Get Book

Uniformizing Dessins and BelyiMaps via Circle Packing by Philip L. Bowers,Kenneth Stephenson Pdf

Introduction Dessins d'enfants Discrete Dessins via circle packing Uniformizing Dessins A menagerie of Dessins d'enfants Computational issues Additional constructions Non-equilateral triangulations The discrete option Appendix: Implementation Bibliography.

Author : Jorgen Bang-Jensen,Gregory Z. Gutin
Publisher : Springer Science & Business Media
Page : 769 pages
File Size : 44,5 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781447138860

Get Book

Digraphs by Jorgen Bang-Jensen,Gregory Z. Gutin Pdf

The study of directed graphs (digraphs) has developed enormously over recent decades, yet the results are rather scattered across the journal literature. This is the first book to present a unified and comprehensive survey of the subject. In addition to covering the theoretical aspects, the authors discuss a large number of applications and their generalizations to topics such as the traveling salesman problem, project scheduling, genetics, network connectivity, and sparse matrices. Numerous exercises are included. For all graduate students, researchers and professionals interested in graph theory and its applications, this book will be essential reading.

Author : Bhatia Rajendra,Pal Arup,Rangarajan G
Publisher : World Scientific
Page : 4144 pages
File Size : 53,9 Mb
Release : 2011-06-06
Category : Mathematics
ISBN : 9789814462938

Get Book

Proceedings Of The International Congress Of Mathematicians 2010 (Icm 2010) (In 4 Volumes) - Vol. I: Plenary Lectures And Ceremonies, Vols. Ii-iv: Invited Lectures by Bhatia Rajendra,Pal Arup,Rangarajan G Pdf

ICM 2010 proceedings comprises a four-volume set containing articles based on plenary lectures and invited section lectures, the Abel and Noether lectures, as well as contributions based on lectures delivered by the recipients of the Fields Medal, the Nevanlinna, and Chern Prizes. The first volume will also contain the speeches at the opening and closing ceremonies and other highlights of the Congress.

Author : Anonim
Publisher : World Scientific
Page : 814 pages
File Size : 51,7 Mb
Release : 2011
Category : Electronic
ISBN : 9789814324359

Get Book

Proceedings of the International Congress of Mathematicians 2010 (icm 2010) (in 4 Volumes) - Vol. I: Plenary Lectures and Ceremonies, Vols. Ii-iv: Invited Lectures by Anonim Pdf