markov chains and mixing times pdf

endobj 0000008974 00000 n 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 … 101 0 obj endobj 0000025638 00000 n 0000068726 00000 n 0000040632 00000 n 36 0 obj (Background) (Distance to stationarity) << /S /GoTo /D (subsection.3.3) >> endobj 93 0 obj (Lp distance) 0000070393 00000 n endobj 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���$� 9c72� &Ro �� This book is an introduction to the modern approach to the theory of Markov chains. 0000012448 00000 n << /S /GoTo /D (subsection.4.2) >> 57 0 obj endstream endobj 0000053681 00000 n Request PDF | On Oct 31, 2017, David Levin and others published Markov Chains and Mixing Times | Find, read and cite all the research you need on ResearchGate endobj 0000026922 00000 n endobj endobj endobj 0000052064 00000 n endobj Answer: By the (local) CLT it is n2 (diameter)2. 73 0 obj (Path coupling) << /S /GoTo /D (subsection.5.3) >> 0000030617 00000 n 0000002111 00000 n endobj %PDF-1.3 %���� (Spectral techniques) (Markovian coupling and other metrics) 96 0 obj 44 0 obj endobj endobj 0000034037 00000 n 37 0 obj 0000010691 00000 n endobj 24 0 obj 0000068050 00000 n 88 0 obj 53 0 obj endobj 69 0 obj 0000025757 00000 n 25 0 obj 4 0 obj 0000047599 00000 n (Spectral decomposition and relaxation time) 9 0 obj (Algorithm) (Bottleneck ratio) endobj << endobj 0000011013 00000 n endobj (Path metric) 60 0 obj 0000071394 00000 n /Length 1388 (Dirichlet form and the bottleneck ratio) 13 0 obj (Coupling) 21 0 obj << /S /GoTo /D (subsection.2.1) >> << /S /GoTo /D (subsection.1.2) >> (Random walk on the d-ary tree of depth ) Math, physics, history, economics, all of these appeared plausible choices which I had some interest in. 64 0 obj endobj endobj << 0000067935 00000 n Mathematical Aspects of Mixing Times in Markov Chains Ravi Montenegro1 and Prasad Tetali2 1 University of Massachusetts Lowell, Lowell, Massachusetts 01854, USA, ravi montenegro@uml.edu 2 Georgia Institute of Technology, Atlanta, Georgia 30332, USA, tetali@math.gatech.edu Abstract In the past few years we have seen a surge in the theory of finite Markov chains, by way of new techniques … �5��nW�z�U6��� �]ϟ�T����y4����9�f6M�1��{�9���V���j*���!vv8��p�h{x��m�m����ls;�-�B[��/#'v�P}-Ň�ST1ׁM��>�W63�˅/2��kS���pc�׼��_7�r ���ڏf�BOm7(��۫��=����aP��۽f�h?qW���xU=��8&�������+�ٵW "�z��m�TyǗ��$ɧ$9������C�����3���=��'�G���r�B��>��;�S�*@�p&�?&��n0��xŔ�/B���u���g�� 108 0 obj 0000012055 00000 n It … Exercise 1.12. 255 0 obj 0000033650 00000 n 49 0 obj 32 0 obj %PDF-1.5 Check that d(t) is a non-increasing function of t. We de ne the mixing time to be the rst time the total variation distance from stationarity drops below ", i.e. (Monotone chains) �xO'�׃���/ << /S /GoTo /D (subsection.3.1) >> 0000009420 00000 n endobj endobj 0000053206 00000 n endobj (Expander graphs) 0000013379 00000 n endobj << /S /GoTo /D (subsection.1.3) >> 0000008717 00000 n 0000018727 00000 n << /S /GoTo /D (subsection.4.4) >> What is the order mixing time? 17 0 obj << /S /GoTo /D (section.4) >> 0000068315 00000 n endobj 0000067512 00000 n 52 0 obj Jonathan Hermon Mixing times of Markov chains January 7, 202013/31. 0000067994 00000 n 0000014564 00000 n 0000068923 00000 n stream As we shall see, this is the case for any sequence of vertex-transitive graphs of polynomial growth. 138 0 obj << /S /GoTo /D (section.6) >> << /S /GoTo /D (subsection.5.2) >> endobj << /S /GoTo /D (subsection.2.4) >> endobj ;M�Y��diܦ ���y��Y�UH#/����� oy]gp��O���fB�i���=�U��3f����N�NZ�R��,3d�N�.V�v��)3���jTx���E�G�B��^;��V��� �զug�M��m!B�{4/���7$� IT�d߫�O�t{͌��1���\���X������~R���S�UC����|K����xˊ8o3!c�jfo@iPנ�u94���,I���i�I�si�� ��̢��D�z���E�F�A�h'4�'ˣ?Xd�9ܪp�e]V�r�΂� 0000046510 00000 n 80 0 obj (Strong stationary times) endobj 0000040438 00000 n endobj 0000032864 00000 n 2 0 obj /Length 1229 << /S /GoTo /D (subsection.6.1) >> endobj 0000030640 00000 n endobj endobj endobj 41 0 obj 100 0 obj Random walks on graphs Simple random walk on a sequence of graphs. /Filter /FlateDecode 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 /d=Iєp�i��-�;~P��_-Kg/��A���W`4�O���Lˣ�����A���� �$eE�����Ƣ�#���G�&��{�`�M�Z�^�q��2H��\��@N���;�d�Z���v⌗`��בg��vယx6����V��f�z^,4�p��#�Q`׬4fH ��e��3�#��. 77 0 obj 81 0 obj endobj 65 0 obj 0000051750 00000 n endobj endobj 0000046806 00000 n 8 0 obj endobj 0000031772 00000 n 0000028347 00000 n >> (Total variation distance and coupling) 0000047209 00000 n << /S /GoTo /D (subsection.1.1) >> 0000055874 00000 n 0000067795 00000 n 97 0 obj 113 0 obj endobj Academia.edu is a platform for academics to share research papers. (Hitting time bound) endobj >> stream 0000068106 00000 n 20 0 obj 0000018258 00000 n endobj 0000018781 00000 n << /S /GoTo /D (section.3) >> (Canonical paths) H��VK�ww�4�k'�{�@E���ʲl_@��*s (U���k/JD�#�c���]?�Z�T�@b�����Y?W*j��"�H�RrC=��Z��of�DE)��3������5&����������o�����y���`�ܗ�����3'"�;���''Il�����E��S��~�������3�k�!�u�e��c��!�Q+��S�����[�� endobj endobj 0000039870 00000 n endobj << /S /GoTo /D [114 0 R /Fit] >> 0000029505 00000 n %���� 112 0 obj 0000067736 00000 n endobj 0000048209 00000 n endobj << /S /GoTo /D (subsubsection.2.1.2) >> 0000069534 00000 n 0000065050 00000 n (Comparison technique) 85 0 obj 0000069128 00000 n 76 0 obj 0000018311 00000 n 0000065027 00000 n << /S /GoTo /D (subsection.5.1) >> 0000068524 00000 n 89 0 obj endobj endobj E.g., the n-cycle. 0000066370 00000 n /First 812 0000028324 00000 n 56 0 obj 0000052087 00000 n 776 0 obj << /Linearized 1 /O 778 /H [ 2221 6519 ] /L 725729 /E 71696 /N 119 /T 710090 >> endobj xref 776 88 0000000016 00000 n << /S /GoTo /D (subsection.4.3) >> << /S /GoTo /D (subsection.6.2) >> 0000031749 00000 n (Applications) endobj 0000026174 00000 n << /S /GoTo /D (subsection.3.2) >> (Mixing times) (Hitting times: submiltiplicativity of tails and a mixing time lower bound) 92 0 obj 1 0 obj 0000048682 00000 n /E� W$s��2O�- �,ԒE�l#v_e�S��Ŕ�Y�䒝�p*Q@.8+Y^H�j��"�$���d�M!��)��$ݒ ��p��R.T���/��9�x�}���zl]מ�g������R�m�Ƶ+Z�Xw���l.vf��o���q�E�k��W�5U�qm�����]4 ]�OХѴ���V��{w�NG�����ȱlb�\R_۶[�ig��돑���ٰ��Y����>b�nM��OO��,��r6����i��IC�mo}���sH��I�@��O�c�;(������e9��2C>�zۘ�A&f��_rJ��ne�k�l,]ڪ[ON�X˻��/+��D�9_ԏE���f��U3C��r�l��0ɣ;�,�3zWM,�\M��q0�Y�� ͮ�@�ϛ*Z��@P�����.#�:�V�Q� ע�M���̠�\���:��9��9���C����g�˩�Us�ݜ�$瀢��$$5�Tu����`�Ixc��n'b軋�6�����i��z٭���?���ڕ�O��3����ck�i�M_k$���^����$�^�n���G[��U�˚HWַ�����H�f�ݵ�90�(�h��u��Mv �P��v��)� (Ising model) 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 << /S /GoTo /D (subsection.5.4) >> 0000011720 00000 n 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. 0000050777 00000 n << /S /GoTo /D (subsubsection.2.1.1) >> 68 0 obj 45 0 obj 0000025092 00000 n 0000034934 00000 n 0000066393 00000 n 0000011531 00000 n << /S /GoTo /D (subsection.2.3) >> endobj 48 0 obj endobj endobj 0000008740 00000 n 0000014884 00000 n 0000017543 00000 n 0000026899 00000 n << /S /GoTo /D (subsection.2.2) >> x�u˒��>_�[����#�d�Yo&�ǖ=�l0"$��$� 9�����$jV{� 4��F?����d��.������w?��*��,-���n�q�4�*k��W�����'�Ō��&O�������x�I��uVE�^��������I=��&��uVG��|�5v�>���U�UV! %���� x��YYs�6~ׯ�#9S���>�I��43m�>9� ,���r��R��#��M*��4�r���v�$��˄. 5 0 obj 0000055249 00000 n 0000011238 00000 n 0000051991 00000 n 40 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. 61 0 obj (Examples) endobj 84 0 obj /Filter /FlateDecode endobj 0000025145 00000 n endobj << /Length 2509 endobj 0000070523 00000 n 0000034516 00000 n /Type /ObjStm stream (Coupling from the past) << /S /GoTo /D (section.5) >> /N 100 72 0 obj << /S /GoTo /D (subsection.4.1) >> %PDF-1.5 �'�E�B>1�����]�_��~7��,��}{n�x��S؛��|�. endobj 12 0 obj 0000002221 00000 n

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