3 x for Define 6,1 E a E minft P TA hitale t t e max X A - - PDF document

SLIDE 1

Refined mixing and hitting relations

Basu

Hermon Peres

f g S

IR

Cf g

it

Eeg fGglx Cx If

P

is

reversible

wrt

it

invariant distr

then there exist

j

eigenvalues

with

corresponding eigenvectors fjYj

and

1

1 and fell 1

Then

Pffggy

I t 22Ajfjcxfg.ly

Proof

Idea

A

Cx g

Fix

PK y

symmetric

D

Tty

If

P

is

reversible

we

write

the

eigenvalues in decreasing order

I

An

Az 3

3An L

Define his

max 121

A is

an eigenvalueof P

with 21 1

and

j

L Aa

absolute spectral gap

g

I

Iz

spectral gap For

lazy chains

for J

Def

tree

1

relaxation

time

fr f

S

IR

E

f fan Cx

and

Var If

Eat Cf E HD2 Poincare

Var Cptf

t

e 24thVar H

P

reversible lazy

f f

f t

• Pf

x

3

PGy7 f g

Define

for

E a E

6,1

hitale minft

max

P TA

t

t

e

X A ICA x

Theorem BHP

infinite

Let X be

a

reversible and lazy Markov chain

• n 5

with P

• Then

tee

z

truix 2e

t hit.de

12

tree Logfes

t

SLIDE 2

and

twixt

e

E hit eh 2e

t 12 treelog

Exercise

Remark

Mixing happens

in 2 stages

in the first

• ne Chitteces

we

have

to wait

to escape

some

small set

with highprob

and

in

the

second

• ne

we

wait

for

tree

steps

Proof of 1 1 Set

t kiteece

and

s 12

treelog

Want to

show

that

t

x

A

Iptts

Al

A t 2 E

Idea

Want

to define

an intermediate

set

G

s t we

hit

it whp

before

time

t

and

conditional

• n hitting it by

time t

we want

to

be

close

to ICA

at time tts by

at

most

E i e

I DX Xttse At TG

ft

I

A I t e

If

DX TG

t

c

e

then

we

will be

done because I PxCXtts EA

ICA

lpxcxetseAITGC.tlPxCTGEt

1tlPxCXttseAl7G7t

Pxtca7t

it A I

t 1Px Xtts c Aka Et

stall

Px Ta

t

E

qq.mg PyCXreA7

stfAIltPxCTa

t

Define G

y

gyp IPyCXreA

ICA

t

E

Then

I PxCXtts EA

A I t

e t R tea

t

So suffices

to show

that

Px Ta

t

t

e

It

suffices

to prove

that

TCG

I

e

since

t hit ele

NTS G

2 E

f

S

IR define

f Cx

qq.pe P2kf x I

SLIDE 3

ft G

Pth A

ICA

Pt ICA

ICA

x

ft Cx

say.pe P2kfeCx7

sq

poIP2kttCx A ICA Pfe Cx

say.gg

p2kpftCx

qgyoplp2httt1Cx A

A

G

y

fitly

ly

t

e

Gc

E

y

fatty

e

t

st

y Pfstly

E

t

E 1 42

Et

FELT

Markov's iueq.com

We

write

for pea ad

hfltp E.tl

HIP

HPfsH

E k

1122

Var PS ICAN

e

m Var

ICA

T P is

a contraction

Poincare

substitutingvalue of s

EI

ICA

1 HAD E

I

8

Theorem

Starr's maximal ineq

P revers wrt

I

PE la

n

t f i s

R Il f Hp ftp.y

lfllplKPfsYHzt4

llPfsHz.g3

and

HfsH

f 4 Hfsll

E EI

Plug into

as stCGC

L Eg t Eg

E

D

Proof

• f

Starr f

x

qq.pe P2kfCx

et

X

be

a

MC

with Xo rest

Puf Xo

Elf Xen 1Xo

E

E

fkn

XnXo Xo

tower property

E

ElfCXzn Xn IXo

Markov property

Set

Rn

EffCXzn lXn

Goat

Show

that R

is

a backwards martingale

SLIDE 4

in

• ther

words

we

will show

that

if

N is fixed

then

RN n

• enEN

is

a

martingale

Since

Horst and

X

is

reversible

it follows that XmXun

Xan

n

Xn Xm Xo

So

Ru

EfflXzn7lXn

EffHollin Eff

lXn Xun Set

In

Xm Xm

Markov property

and

fix

N

Then

Rn nloenen

is

a

martingale

wrt Fn

• meany.fm tp

lomgxnRnnllp ftp.F

HRollp

pl zkfCXolHp PzzlfHp

Doob's Lpineg

t.mg

2nfCxoi

• ygxe.nElRn1XoI

eEl

no.axm.t2n lxo

Conditional Jensen

implies

that

Hou.iq 2nfCXo1llptllmo.qxenRnllpfpP

HfHp

Letting

N

so

and using

MONOTONE org

completes the proof D