and hitting times for Markov chains Mixing between mixing times and Overview 1 up to constants Equivalence of large sets times hitting different sizes of sets comparison for times 2 flitting Refined mixing and hitting equivalence 3 in a finite state space S X chain Let irreducible Markov be an the P X Let be transition of matrix ti je S Phi jl Picktaj invariant distr STP I Pth y if X aperiodic then x y also t so t step as is prob distr S and Let be 2 u on µ Hp Uktv VIA Anyang 1 µ A ITH w m ax Il Ptc DH minft 70 twixt dit E e t e e 0,1 e turix t twin called reversible X V x.ystcxlpcx.yl utyply.in is minft 0 Xt c A Ex TA where TH 7A x riff lazy version of X R PII Oliveira Peres S Theorem 2 2012 t act for all ca and 7 positive constants reversible s.t ca Markov chains f twixt Catala ca tach lazy twin Katalin
Proof of lower bound twix 3 Catala L a 8 twix Zg t Let E 3 twixt PYX 17 A TCA go Pth A then A ICA t x Take with Iz fo So Geo Zz TA E 16 t t E Za D my reversible lazy Aldous 82 all MC's for tueixkyyqx.CAT fTaI Remark essential Reversibility is Ct laziness biased Exercise 1 RW Eu Consider on a R PII 3 c P 2 Zn ITI show twixt m2 and H x TH a kn the then If theorem is false Remark 12 a Exercise 2 2 cliques on u vertices joined an edge by kn kn Show that and tula twixt m2 kn if a of Theorem 1 Proof Clipperbound Definition at geometric time Mixing a Let Ze be taking values in of parameter 1 a geometric v.v and X of indep myx 11 Px X7t Ilku Define DG Ctl doit e I and ta t 0 geometric mixing ruin
instead of geometric we take Remark If Ut to be uniform on 7 t the Cesaro mixing time this to then rise gives Show that dolt t Exercise 3 is in decreasing Aldous lazy For all reversible chains taktaix Ideas Theorem 2 was and For all Winkler chains ta Katield Theorem 3 t a c 12 Immediate from Thur's Pf of Thou 1 2 and 3 D 2 tula Pf of Thur 3 to easy Ip to constants to ha tuk we I a prove 8 a CBI 3 Iz t c ta find want to a set B with We s t Let 3 Ot for max Ex 2B positive constant O some t c ta Pz Htt C A C ICA A F s t t Ig ICA Pg Htt c A B 3 ICA f y 3 f Claim ICB t ftp.cotlyl PylXZtyA P ICA A yButyPg ft CA Z EI EI c TCB TIBI t ICA CA E stCBI f f D e Dt for Eaters that a suitable constant 0 we will prove assuming leads contradiction to a Markov's ineq Rz By 20Mt 2B 7 E 1 2M ME IN Pz Xzt c At Zt Pz XZt e Al 2B L 20Mt Pz 7 3213,213 28 Mt 2B memoryless property of Zt ymeifg Py XZt EA and strong Markov at EB
Pz 201ft CBC 20Mt It HCA Ig Patt 20Mt Pz TBC 20Mt 20Mt L gu ICA I Lg 1 20Mt L Fm fun g I 20M D qq.rs PzCXztC A 74TH f 2 I Choosing shows Pz Xzt EA ICA M large I enough Taking which contradiction is I a Idea of geometric mixing due to Oded Schramm Ex Ax Ax is randomised stopping time tstop m ax ruin a s.tl CXn 7 stl I Baxter and Chacon 76 filling rule Lou'asz Winkler Aldous reversible Thou 2 tstop k twixt stop E 8 twin easy The hard to 1 direction stop 2 twix show is Prove that for reversible chains tstop E 8twix Exercise 4 flint to define Use separation distance appropriate stopping time an u
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