LEADER 04034 am 22007093u 450 001 996418262803316 005 20230125212657.0 010 $a3-030-27968-5 024 7 $a10.1007/978-3-030-27968-4 035 $a(CKB)4100000009453382 035 $a(DE-He213)978-3-030-27968-4 035 $a(MiAaPQ)EBC5940845 035 $a(Au-PeEL)EBL5940845 035 $a(OCoLC)1135666257 035 $a(oapen)https://directory.doabooks.org/handle/20.500.12854/37818 035 $a(PPN)245286985 035 $a(EXLCZ)994100000009453382 100 $a20191004d2020 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aPlanar Maps, Random Walks and Circle Packing$b[electronic resource] $eÉcole d'Été de Probabilités de Saint-Flour XLVIII - 2018 /$fby Asaf Nachmias 205 $a1st ed. 2020. 210 $cSpringer Nature$d2020 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2020. 215 $a1 online resource (XII, 120 p. 36 illus., 8 illus. in color.) 225 1 $aÉcole d'Été de Probabilités de Saint-Flour,$x0721-5363 ;$v2243 311 $a3-030-27967-7 330 $aThis open access book focuses on the interplay between random walks on planar maps and Koebe?s circle packing theorem. Further topics covered include electric networks, the He?Schramm theorem on infinite circle packings, uniform spanning trees of planar maps, local limits of finite planar maps and the almost sure recurrence of simple random walks on these limits. One of its main goals is to present a self-contained proof that the uniform infinite planar triangulation (UIPT) is almost surely recurrent. Full proofs of all statements are provided. A planar map is a graph that can be drawn in the plane without crossing edges, together with a specification of the cyclic ordering of the edges incident to each vertex. One widely applicable method of drawing planar graphs is given by Koebe?s circle packing theorem (1936). Various geometric properties of these drawings, such as existence of accumulation points and bounds on the radii, encode important probabilistic information, such as the recurrence/transience of simple random walks and connectivity of the uniform spanning forest. This deep connection is especially fruitful to the study of random planar maps. The book is aimed at researchers and graduate students in mathematics and is suitable for a single-semester course; only a basic knowledge of graduate level probability theory is assumed. 410 0$aÉcole d'Été de Probabilités de Saint-Flour,$x0721-5363 ;$v2243 606 $aProbabilities 606 $aDiscrete mathematics 606 $aGeometry 606 $aMathematical physics 606 $aProbability Theory and Stochastic Processes$3https://scigraph.springernature.com/ontologies/product-market-codes/M27004 606 $aDiscrete Mathematics$3https://scigraph.springernature.com/ontologies/product-market-codes/M29000 606 $aGeometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21006 606 $aMathematical Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/M35000 610 $aMathematics 610 $aProbabilities 610 $aDiscrete mathematics 610 $aGeometry 610 $aMathematical physics 615 0$aProbabilities. 615 0$aDiscrete mathematics. 615 0$aGeometry. 615 0$aMathematical physics. 615 14$aProbability Theory and Stochastic Processes. 615 24$aDiscrete Mathematics. 615 24$aGeometry. 615 24$aMathematical Physics. 676 $a519.2 700 $aNachmias$b Asaf$4aut$4http://id.loc.gov/vocabulary/relators/aut$0791278 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996418262803316 996 $aPlanar Maps, Random Walks and Circle Packing$91768608 997 $aUNISA