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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'