Refined mixing and hitting relations Hermon Peres Basu S f g Cf g Eeg f G glx Cx IR it invariant distr there exist P reversible then If wrt it is j eigenvectors fjYj 1 and fell and 1 with 1 eigenvalues corresponding t 22 Aj fjcx fg.ly Then I Pffggy Fix A PK y Proof Cx g Idea symmetric D Tty eigenvalues in decreasing order If the P reversible write is we Az 3 3 An An I L with 21 1 an eigenvalue of P Define his A is max 121 absolute spectral gap L Aa and j Iz I spectral gap g For lazy chains for J Def relaxation tree 1 time fr Eat Cf E HD2 Var If f and IR f S E fan Cx e 24thVar H reversible lazy Var Cptf P Poincare t f f f t o PG y7 f g Pf 3 x for Define 6,1 E a E minft P TA hitale t t e max X A ICA x Theorem BHP infinite reversible and lazy on 5 Let X Markov chain be with P o a t hit.de Then tee truix 2e t tree Logfes 12 z
E hit eh 2e t 12 twixt and tree log e Exercise in the first in 2 stages Remark one Chitteces Mixing happens small set have with high prob to wait to we escape some and for the second wait tree in one we steps t kite ece Proof of 1 1 s 12 Set and tree log that t A Want to show x Iptts Al t 2 E A it whp hit to define G Idea an intermediate Want set s t we on hitting it by and t conditional we want before time t time tts by be close to ICA to at at time most E i e A I t e I DX Xttse At TG ft I will be DX TG then done because If t c e we lpxcxetseAITGC.tl PxCTGEt I PxCXtts EA ICA 1tlPxCXttseAl7G7t it A I Pxtca7t t 1 Px Xtts c Aka Et stall Px Ta t stfAIltPxCTa E qq.mg PyCXreA7 t G ICA gyp I PyCXreA Define t E y Then A I t t R tea I PxCXtts EA t e that Px Ta So suffices show t to t e t hit ele that It TCG suffices to I e since prove NTS G 2 E x I qq.pe P2kf f IR define S f Cx
Pth A Pt ICA ICA ft G ICA x ICA ft Cx say.pe P2kfeCx7 poIP2kttCx A sq qgyoplp2httt1Cx A p2kpftCx A Pfe Cx say.gg G fitly t ly e y y Pfstly E Gc fatty E st e t y E 1 42 Markov's iueq.com Et t FELT hfltp E.tl for pea ad HIP We write m Var E k PS ICAN ICA Var HPfsH 1122 e T a contraction Poincare substituting value of s P is 1 HAD E I ICA EI 8 Starr's maximal ineq PE la P revers wrt Theorem n I t f i s R Il f Hp ftp.y lfllplKPfsYHzt4 E EI and Hfs H f 4 Hfsll llPfsHz.g3 Plug into as stCGC L Eg t Eg E D qq.pe P2kfCx Proof of f Starr x X with be MC Xo rest et a Elf Xen 1 Xo XnXo Xo Puf Xo fkn E E tower property El fCXzn Xn I Xo E Markov property Rn EffCXzn lXn Set that a backwards martingale R Goat Show is
that will show other words N is fixed then if in we RN n is oenEN martingale a it follows that Horst and X Since reversible is Xn Xm Xo Xm Xun Xan n Eff Eff Hollin EfflXzn7lXn lXn Xun So Ru Markov property Set In Xm Xm wrt Fn Rn nloenen fix and Then N 0 is a martingale lomgxnRn nllp ftp.F omeany.fm tp HRollp zkfCXolHp PzzlfHp pl Doob's Lp ineg eEl no.axm.t2n lxo 2nfCxoi t.mg oygxe.nElRn1XoI Conditional Jensen that implies Hou.iq 2nfCXo1llptllmo.qxenRnllpfpP HfHp MONOTONE org completes the proof and using N Letting so D
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