coupling argument - faculty.math.illinois.edu

Math 595 - Coupling bounds 09/17/20-

-

Recap .Lower bound : Exhibit a set that separates

the two measures .

upper bound : coupling argument

sup 1mA - VAI = d> ✓( u ,

u ) =int IP ( x # Y) ←x -M

A Y - u

• dt ⇐ It := nyaya dev ( Supt , dgpt)

Couple Xt , Yt where Xt - Supt , Ye - dy Pt

Process level Coupling ( Doellin ' 38 , Aldous '83- Pitman

'

80-1)

modification : if X,= Ys

,Xt -- Ye tf t >s

-

da ( Supt , dy Pt ) s Bay ( recouple > t)

where 7couple

= min { t f Xt -- Yt} .

NOI : . we consider only Martian coupling .

i. e . ( Xt , Ye ) ,

t > o is a MC.

• For any Ergodic MC,

I a coupling s.t .

der ( Supt , dy Pt ) = Pmg ( recouple > t )

But,

this coupling need not be Markarian.

Guiffeath '

78

×Enampk- p 11g clockwise

i ) Rwon : n ,~j. 'II,

# t.YD-fn.us ) → { Itter up . Ys i.

.

• zt "4, a. ,

' tu" J trap

.

L,

i

T•-•! •"

→

( xn .

--net

up . Ic) '

Yet , =LL L Ik k

in c- as

},

= dist ( Xt , Yt ),

{ o," - - -

in } Tien

}µ ,

= },It up . I each

.

k

274

Fouling = min { t 70 f 3£ E fo ,n ) } ,3.

= k =

T - n '

n + n -y

d-ts malign Pm

,( Toping > t)

""

= munn Rd Hitting , ,n,> t)

Ek H{ o ,n }E man -

k tH = M

{ o.tn }

Mt := - t - 3+ ( n - 3T ) E ( Mtt , I Ft))

=- ft -11) - n }t t E ( Eto .TL/Fy)

=- t - n 3ft 35 = Mt

057 ⇒ ten Mfg = Enno

⇒ - IE H = - K ( n - k) ⇒ Eun = kin - k)k

d-ts Mann

kin - k)

e-EIat

⇒ Twin E n'

ii) RW on 2nd ( n . .-- -

, Kd ) ( d , , --

, Td )-

:

I choose a co - and

U .a. P

E ( coalescence time for " t co - one ) I

it nz = Yz , same transition⇐ DI

4 it nz. F Yz ,use I - dim

Z coupling= man ( Coates -ence time

-

couplingtied time for ith co - ord )

d

E Z Co absence time for ith ca - and

i -- I

'I reading E DII ⇒ DI s 9¥⇒ Twin s d- n'

Ese 2mi . E (dbgd) n' .

iii ) RW on hypercube :

-

n = ( n . .-

-

, an ), Y = ( Y .

.

- -

, Tn )- I ~ U

. a. r { I,

-

,n }

- it nz = be , same transition

o.

W,

Nz F by,

" It,

-

- nzie-Tz.io up .

'I

Use, = * Tz 3 up. If

}+ = If i I Xtli) t Yt Cil ) /

3h,

- yBt - 1 up .

'

g."thing

= tho,

} o.w

.

t

I - Mnf I

n E - E- f ← A← c-

#n¥!O l K- l K

& I-

> coupling= I t Creem ( ) t -

- - + Creem (I)-

n Tn

= I wii -_ I

where wi 's indef,

wi - Cream ( in)E Tempting = ÷Z

,

÷ = nlogn

linin £ 4 nlegn / IP ( recouping > nlegnt Cn ) E e-C

{n C 31

[ If Xp - Cream ( p ) ,Piso then

XP =p Xp ⇒d Enya, ]Exits

>coupling

= E receipting t n II,

( wi - Ewi))-

= nlegn + n ÷g÷fmn÷÷)-⇒ Enfold - r

2ceufhy-nEIphg D=, €7,

(Ent")i-

Wy

IP ( Rampling 3 E reading 1- en ) ⇒ ① ( W, > *)

nbgn = e- C .

cnn.Ennkgn.cn#EIhdjy,g.y!ettn

⑦ ( n , y) =

Mls ) PCs, n )

ir) Top 2 Random shuttle / More 2 Front .

-

tem For with uniform stat. dish

,

der ( In Pt , te ) = der ( dutt,m) .

= I § I ptln.si - ¥ ,I = I § Ittfid , -5)

= { § lptlid . yay - ¥,I

- ¥1

- I § / Pt Cid , y ) -¥| Elides )

= F lid , 54

⇐

meat it top @ aids same

,I'-

- IJ = -

not marked in 2⇒mark

, z= a top card is marked in on

→-

. -p - then I'= pas in T

- I -

o.

W. I

'= I

> couplings time for all cards to

get marked.

in o,re

n

E " E NZ E E i

i = ,

v ) Random Transposition-

: Diaconis & Shan Shahani '

90-1

- card I -Twin = In leg n t Olu)

- f - same card--

←

g-

← pas re-

r same Pos .

- J-

--

3t= # disagreements at time t

= ? Hoi Fei

§ c- Og = TgBtu =

§. of = TI'

}t"

←. gj

'

f TI'

E 3+-1 To F neg

RI't ai'

05'= I '

IP ( 3h,

€ St ' ' 13T ) = 4¥) '

⇒ E Zay, E* § = n

Finitebinary-tee.li. . :&:.and mi

. .

A N11111111

' n

Riffleshubbk