Today : Reminders about reversibility ,finishing with mixing times �1�

Default setting : Markov chain X=CXn,

no ),

State space V. Trans.matrix P

,Stat

.

dist. it

Reversibility

Chain reversible if Tupou= I pun for all a. VEV

.

Def : Random walk on weighted graph G= ( V. E. w ),

w( e) e [ a - ) is MC.

with

puu:

- w(uD/§ WC e).

Can always assume E={ { Uv } :u,VeV } by using some 0 - weightsUEE

Pap

Therandom walk on G- ( U

,E. w ) has Stat

.

distribution tu= E wluv )Vi UVEE

Proof :Let µyµ,ueV ) be µu=u§⇐w/uv)

.2£Ew=Then (µP)x = §yµupux= E §w⇐w( uvlpux = E § wluvl . wld: UEV UEV WEE E W (e)

Us ( c. E UKEE n' CEE e :ueE

= §e.

WM " ) =µ " . Soµ

is a stationary measure .

To make it a distribution divide through by u§µ=§§u4e1=2§ew(e)( each edge has 2 endpoints

.

).

�2�

In particular : for an unsighted graph G=CU , E),

Stat.

dist.

ofsrw is to ) =d{gC¥, .

Corollary Random walk on G = ( V. E. w ) is reversible .

Proof : Tupuv =(§a⇐w(e) )'w(uv)/§⇐[w( e) = wluv ) = want Trpvu .

Conversely , suppose X is reversible . Setting duh =

tupuv then X is RW on

G= CV, Ec ) where Et{ { uv } : a ,ueV }

. 0

so Reversible M.C. s

= RWS on graphs .

NB : This connection requires the existence of a stationary distribution.

Example: Consider the graph 12 with edge weights wli,

IH )=2i for all i.

TqheuRW on Z with weights w has pine , =z?z÷

. , =§ .

( Drift tonight)

No reasonable def. of reversible should include this Me

.

Warning : Random walks on weighted graphs are different in discrete and continuous time

-30Natural continuous generalization : for a graph G= ( V

,E

,w ) ,

w ( uv ) is rate of jumps from u to v on v to U .

Consider, e.g. ,

star with n leaves.

÷¥4,

;→ All weights 1

.

CTRW-

leaves hub at rate n,

returns at rate 1.

Stationary dist.

uniform on V

Discrete RW → itlhubkt,

I ( i ) =

zlg for all it, .

→n

.

In general CTRW on a weighted finite graph always has it = unit

.distribution

But there are obviously others. E.

g.

.

§¥O with @bT¥0 for all i .

More closely"

mimics" discrete chain

.

Total Variation . Recap .

( All this works in either discrete on continuous time.)

�4�

Total variation distance is Hn -

Nowmax { IMAI - VA ) 1 : Ac V }

=L,§v1µm - Hal

= inf { PCXH ' ) :( X. Y ) coupling ofµ

and v }.

DHK IF

HEPI- it Htv IHK

,;yqgHER - of Pttkv

.

(distribution of X , given that X. =x

.

Fact : DH ) EDTH for all t.

Seen last week .

Atso : d- H ) =

,;gqg11 dcpt - of Ptlkv €, ;gqgHEPt- till

,+ Hit . of Ptlkv ) = Zdtt )

.

Prop : For all s,t > o ,

at ( stt ) e d- ( s ) . at ( t ).

proof : Fix States y,zeV

.

Let (Ys,

Zs) be an optimal coupling ofdy Ps anddzps

.

In other words,

P(Ys=x) - (

ofPs )x

,

PCZs=x) = (

ofPs )

. ,PC Ystz ) = 11

ofPs -

ftp.ltr.

= Py( Xs=x ) =P.

IXs=x )

Note : For all uv ,n

,

Pu ( Xn=v ) = Phu = @up"

)v.

�5�

Next. Psgttw - § Ps

,.

B.tw = § PlYs=x ) Pxtw = ElPIs ,w ]Likewise ,

PIHW = E [ Pts,w ]

so Psjtw - PIHW = El PI, ] - E (Pta,if = IE (PIs; Ptzsw )

Take absolute values,

sum over w to gett.E.lpj.tw- PIII = 's EIIE(BB ; Pta,w ) l '

IEIEIBBIPtzswl

n=Et El RBIPtzswl

ygypstt.gzpstty2

ev

= IEHokspioieptyFinally ,IE Hok

,PicksPty = § IPCH, Zskcy ,

⇒tlldjpt- ofPtyEV

£ § ,

Pllktskcy,

⇒ ) . d- H ) = Post Zs ) - d- H ) =d- Gidtt )

.

o

�6�

Mixingtime is In :X =I* (f) = inf ( t :dltl ' t )

NB : If t i. Tmix then

a ( t , Idtt) =ak¥÷¥I"±@ t.DK e C zdH*Dt= It

So have exponential rate of convergence to station anity on this scale.

In preceding proof we used a coupling ( Ys,

E) of GPs and AP ? This coupled

the locations of two copies of the ME.

at a fixed time.

But can also couple copies

of the entire ME. ( we did this in proving the convergence theorem for a pained .

MCs )

Def : A cguplidgof MC

.s with transition matrix P is a process ( ( Ys

,

Zs),

530 ) st.

( Ys) s soand ( Zs ) s >

o are each MC. s with transition matrix P

.

Given any coupling ( ( Ys,

Zs),

530 ) ,the corresponding stickyopting is (01 it's )

,s > o )

where £s= Z V. s,

Is :{ If,

5 , t E- min ( t : YEZD

This is again a coupling ( exercise )

�7�

theorem : For any coupling ( ( Ys,

Zs),

530 ) of MC.

s with transition matrixB any

t 'o,

andany y ,zeV,

11 ofPt -

ofPtlkue Plt >

TKYO, E) =y,z )

Poot "

Note PC Fey1 ( Y. , E) = ( y ,⇒ ) =

BLXEU) =

B.tnVa

and PLEEV1 1 Yo, E) = c y ,⇒ ) = PdXev ) =BY V.v

So ( It,Et) is a coupling of cryptand of Pt .

Thus

11 orypt - Jeptlltv =P ( It ±

£+10. ,ZHy ,

#=P ( t > t 1 Ho

, -201=4,7)).

o