0000026922 00000 n 0000066393 00000 n 0000046510 00000 n 0000039870 00000 n 0000052064 00000 n >> << /S /GoTo /D (subsection.4.2) >> 0000048682 00000 n << /S /GoTo /D (subsection.3.3) >> << /S /GoTo /D (subsubsection.2.1.2) >> 0000067736 00000 n 57 0 obj 36 0 obj endobj 0000053206 00000 n 0000068106 00000 n endobj 84 0 obj xڕWMs�6��W�-�:@�d'���I�C�f���LC"b�d ȱ���V$eَDe�4h������PF*!M�����DBB�Q$��DJ��$r�xDA2�#H�j!%,S|!�J��T�a�R@ 2MKLPZ��R�V�i�D`i��$��,#]�j�1��m���,����S)0Tb[-���`Eˆp9���$��%F���(%�K•qI,��J0W�y���S�R0F�e` ��P���ӈ��F�%�4�#�,B�"�B$�b�_ ���'0f5hP����u��)�_�$_X�$�d�Y6П�G�0/������NB)�ɑ� ��[A����E�gl,S�T 9��ȔɄ2�P���$� 9c72� &Ro �� �xO'�׃���/ 0000002111 00000 n 0000070523 00000 n 92 0 obj 0000065027 00000 n 0000068524 00000 n 24 0 obj 0000068050 00000 n 0000025145 00000 n 0000034037 00000 n 60 0 obj 0000069534 00000 n stream endobj << /S /GoTo /D (subsection.5.2) >> 108 0 obj 41 0 obj 1 0 obj As we shall see, this is the case for any sequence of vertex-transitive graphs of polynomial growth. Mixing Times of Markov Chains: Techniques and Examples A Crossroad between Probability, Analysis and Geometry Nathana el Berestycki University of Cambridge N.Berestycki@statslab.cam.ac.uk November 22, 2016 The purpose of these notes is to showcase various methods which have been developed over the last 30 years to study mixing times of Markov chains and in particular the cuto … (Markovian coupling and other metrics) 0000040632 00000 n << 0000048209 00000 n /Filter /FlateDecode /Type /ObjStm /d=Iєp�i��-�;~P��_-Kg/��A���W`4�O���Lˣ�����A���� �$eE�����Ƣ�#���G�&��{�`�M�Z�^�q��2H��\��@N���;�d�Z���v⌗`��בg��vယx6����V��f�z^,4�p��#�Q`׬4fH ��e��3�#��. /N 100 (Path coupling) endobj (Mixing times) endobj 0000012055 00000 n 0000070393 00000 n endobj endobj 32 0 obj 44 0 obj (Background) endobj 0000010691 00000 n 8 0 obj endobj << /S /GoTo /D (section.2) >> endobj 0000030617 00000 n 28 0 obj endobj endobj 0000025757 00000 n >> 0000017767 00000 n (Algorithm) 16 0 obj 0000071394 00000 n 0000051991 00000 n /Length 1229 << /S /GoTo /D (section.1) >> stream 0000017714 00000 n 0000067994 00000 n 100 0 obj << /S /GoTo /D (section.5) >> endobj /Length 1388 17 0 obj The theorem above says that the Markov chain run long enough will converge to equilibrium, but it does not give information on the rate of convergence. 0000025704 00000 n endobj 101 0 obj 113 0 obj Markov Chains, Mixing Times and Coupling Methods with an Application in Social Learning Senior Thesis submitted by Jinming Zhang June 4, 2020 Advisor: Ursula Porod Northwestern University. trailer << /Size 864 /Info 749 0 R /Root 777 0 R /Prev 710079 /ID[<4b68ec8e98c3a020314dd4f4afb2882f><697dcc76edc4c537a22dbff4032ce220>] >> startxref 0 %%EOF 777 0 obj << /Type /Catalog /Pages 762 0 R /Metadata 750 0 R /JT 775 0 R /PageLabels 748 0 R >> endobj 862 0 obj << /S 11356 /T 11740 /L 11931 /Filter /FlateDecode /Length 863 0 R >> stream 29 0 obj endobj 2 0 obj 48 0 obj << /S /GoTo /D (subsection.2.3) >> 97 0 obj (Comparison technique) The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the 0000030640 00000 n endobj endobj 0000067512 00000 n << /S /GoTo /D (subsection.5.3) >> 0000047599 00000 n Math, physics, history, economics, all of these appeared plausible choices which I had some interest in. 0000040148 00000 n 0000065050 00000 n 105 0 obj 0000017543 00000 n endobj 0000018727 00000 n << /S /GoTo /D (subsection.5.1) >> 0000026174 00000 n << 0000055249 00000 n Academia.edu is a platform for academics to share research papers. 77 0 obj 0000068923 00000 n << /S /GoTo /D (subsection.1.2) >> (Spectral decomposition and relaxation time) (Dirichlet form and the bottleneck ratio) (Ising model) (Lp distance) 0000052087 00000 n %PDF-1.5 endobj 0000068315 00000 n 73 0 obj (Total variation distance and coupling) endobj endobj << /S /GoTo /D (subsection.1.3) >> /Filter /FlateDecode This book is an introduction to the modern approach to the theory of Markov chains. (Coupling) 5 0 obj 0000050777 00000 n 104 0 obj 89 0 obj What is the order mixing time? 21 0 obj (Canonical paths) endobj 4 0 obj 0000008740 00000 n 0000051750 00000 n << /S /GoTo /D (subsection.3.2) >> 1 Acknowledgements When I rst arrived at Northwestern, I had no idea what I wanted to major in. endobj endobj << /S /GoTo /D (section.3) >> endobj 93 0 obj (Random walk on the d-ary tree of depth ) << /S /GoTo /D [114 0 R /Fit] >> 0000067795 00000 n 68 0 obj 0000067935 00000 n (Coupling from the past) 69 0 obj 33 0 obj endobj (Expander graphs) << /S /GoTo /D (subsection.2.4) >> 80 0 obj 0000029505 00000 n endobj endobj 0000026899 00000 n 88 0 obj << /S /GoTo /D (subsubsection.2.1.1) >> 0000011013 00000 n 0000040438 00000 n << /S /GoTo /D (subsection.2.1) >> endobj 0000018258 00000 n 0000053681 00000 n 0000011720 00000 n 0000018311 00000 n /Length 2509 �5��nW�z�U6��� �]ϟ�T����y4����9�f6M�1��{�9���V���j*���!vv8��p�h{x��m�m����ls;�-�B[��/#'v�P}-Ň�ST1ׁM��>�W63�˅/2��kS���pc�׼��_7�r ���ڏf�BOm7(��۫��=����aP��۽f�h?qW���xU=��8&�������+�ٵW "�z��m�TyǗ��$ɧ$9������C�����3���=��'�G���r�B��>��;�S�*@�p&�?&��n0��xŔ�/B���u���g�� 0000047209 00000 n (Transportation metric) /First 812 0000032864 00000 n endobj 0000008717 00000 n endobj %���� 0000034934 00000 n 0000031772 00000 n Answer: By the (local) CLT it is n2 (diameter)2. endobj 255 0 obj 76 0 obj 13 0 obj endobj endobj