Planar Maps, Random Walks and Circle Packing
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 t...
সংরক্ষণ করুন:
| প্রধান লেখক: | |
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| বিন্যাস: | Online |
| ভাষা: | ইংরেজি |
| প্রকাশিত: |
Springer Nature
2021
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| বিষয়গুলি: | |
| অনলাইন ব্যবহার করুন: | 1006832 |
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| _version_ | 1869528595772211200 |
|---|---|
| author | Nachmias, Asaf |
| author_browse | Nachmias, Asaf |
| author_facet | Nachmias, Asaf |
| author_sort | Nachmias, Asaf |
| collection | Directory of Open Access Books |
| description | 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. |
| format | Online |
| id | doab-20.500.12854ir-37818 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2021 |
| publishDateRange | 2021 |
| publishDateSort | 2021 |
| publisher | Springer Nature |
| publisherStr | Springer Nature |
| record_format | ojs |
| spelling | doab-20.500.12854ir-378182025-05-09T10:26:05Z Planar Maps, Random Walks and Circle Packing Nachmias, Asaf Mathematics Probabilities Discrete mathematics Geometry Mathematical physics 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. 2021-02-10T14:49:17Z 2021-02-10T14:49:17Z 2020-03-18 13:36:15 2020-04-01T09:11:36Z 2020 book 1006832 1006832 OCN: 1122742832 http://library.oapen.org/handle/20.500.12657/23323 https://directory.doabooks.org/handle/20.500.12854/37818 eng Lecture Notes in Mathematics open access image/jpeg image/jpeg image/jpeg image/jpeg n/a n/a n/a n/a https://library.oapen.org/bitstream/20.500.12657/23323/1/1006832.pdf https://library.oapen.org/bitstream/20.500.12657/23323/1/1006832.pdf https://library.oapen.org/bitstream/20.500.12657/23323/1/1006832.pdf https://library.oapen.org/bitstream/20.500.12657/23323/1/1006832.pdf Springer Nature 10.1007/978-3-030-27968-4 10.1007/978-3-030-27968-4 9fa3421d-f917-4153-b9ab-fc337c396b5a H2020 European Research Council European Research Council (ERC) EU collection 120 open access |
| spellingShingle | Mathematics Probabilities Discrete mathematics Geometry Mathematical physics Nachmias, Asaf Planar Maps, Random Walks and Circle Packing |
| title | Planar Maps, Random Walks and Circle Packing |
| title_full | Planar Maps, Random Walks and Circle Packing |
| title_fullStr | Planar Maps, Random Walks and Circle Packing |
| title_full_unstemmed | Planar Maps, Random Walks and Circle Packing |
| title_short | Planar Maps, Random Walks and Circle Packing |
| title_sort | planar maps random walks and circle packing |
| topic | Mathematics Probabilities Discrete mathematics Geometry Mathematical physics |
| topic_facet | Mathematics Probabilities Discrete mathematics Geometry Mathematical physics |
| url | 1006832 |
| work_keys_str_mv | AT nachmiasasaf planarmapsrandomwalksandcirclepacking |