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GMGHGH G H e K
G≡a 1 JG H
¡JG G MGFM MGH¥F H JG GGHG H FG MGM J MGHG JG HFG
HGM LM L JG n MJ GGGMH HHG J JG n ¬¥HGGGMG GGF¥GHK
MG MGHGF G M JG HG HG MGH
¨©
V
= {i ∈ V | B ≈ b for a = 1, . . . , n}
¨HHF FG HG HG ¥H © FH HGM JG MHGMG GMHH
¨©
V
V
up to
V
=∪ V
=∪
¡JGHG GHG MH MG G JG H JG LFF HGHG
A"?#" D C- 2?# % "$
. %A" #" ?$
)#" <+)$ !)# )% %A" %#) "4%?# ?
) #)7? ¢) £"# #" ) $ t%A"¿ #"τ ? ¢A"#" % V$%)#%$ "%A" 7" a A"#" !)# "$ {1, . . . , n}# ) ")#$ )$ %A" 4A)#)4%"# $% 4 % " ? %A" %#)$ % ?$ #? ) #" ? V
%? )?%A"# ?"
τ ®®
¢ %A ?" )7 %A" $) " V
2?# τ ¿ t ¿ τ . %A" ® #? ) % 7 $%# +% ?
V
A)$ #")4A"7 ) <+)$ ,$%)% ?)# $%)%" ¢ %A ")4A V . ? ?¢ #? %A" #" ) )% ? %? 0 ? %A"
?7"$ ¢ %A a > a $ 4" %A" e ¿ 1 ¡JG HGMF FHH L J¥H MGH JFG JGMMJF ¥MG L JG HG HG H
GFM JG MRF GGMG R K ¡ F¥G § ¥GMFG GGMF HJ LFF
LM JH H¥ JG GG L JG MMGF GH JG MGF GH JG JMGMH
GH L MH ¨GMHG MH MGH GGG GHFG MGH© MGGM FF MGF
MGFG JG GGF¥GH L R K
a=0
(a)
a=0
(a)
a
(0)
i
N
i=1
(n)
b1 ,...,bn
a
(a) (a0 )
i
i
(a)
i
aa0
(stat)
−t/τa
a
(n)
bn b1 ,...,bn
(n−1)
b1 ,...,bn−1
(0)
(1)
b1 ,...,bn−1
(n)
b2 ,...,bn b1 ,...,bn
(a)
b1 ,...,ba
a
(a)
b1 ,...,ba−1 ,ba
a
(a)
b1 ,...,ba−1 ,b0a
(a−1)
b1 ,...,ba−1
0
a+1
(a)
b1 ,...,ba
−t/τa0
a
G HMGG HG L GFGGH GF M
MGF HG )7$4) " H G¤G H
L¥ Φ V J MGFV HG F¥GHK § JH G JG GGH (i, j) ∈ E L ®MJ (V, E)
G HGG H HG L GFGGM GH JG FHGK H GFG 7 K7 JG MJ G¤GH
M F V JHG MH M R H G JM¥J JG MGM MF« L JG
MJ ¯G MK MGHM JG HG L H j J G MGJG LM i JG G¤
L E FFH ¥ LM M¥H HMH ¨H¥FGG JG GGMG HMH HHG
J JG FHG© LM HG HGM FH HGM JMG M FF MHK H
GFG GF FH FFH M JG GFM GKK GLM FMG HGH M MGG
00 3 ( 303 ( 030 226( ' 3) ' ≈
('0 . / 0 00 )1 (
4
=
>n
τa = O(1)
¿ τ
n
'0 ) 4
62001 )' . 0 2 ' '()( * a0
'
& )4 04 ( 4 0 4 0 2 ∞.
K § HJFF JGMG MGHG J H¥J MM GFM L V MGH HJH ¥ JG
FHG LG¥MGHK
¡JG 4?##" )% ? % " t (Φ, R) H G¤G H JG JMGMH MG L JG MMGFH GGG
JG F¥GH G Φ F
M¯GM L JG M F GGMG R ¨ MM ©
H M¥JF JG ¥GM L HGH J JG M FGM GGH MGJ MG
JHG F¥G H
FGM MMGFG J JG HM F¥GK § H J¥H ¯ JMGMH L JG FHG
JG MJ (V, E) ¨M G¬¥FGF R©K ¡ ¥G JH G LM JG FGG L MGMG
Φ
M H¥FG M¯GM
J T HGH G GFH GM M¥G FF
(i ) G¤
GHG HHF GMGH
J GMF GMGH F F MGF«
L JG M F
¨GGG L t©
C (θ) ≡ hΦ(i )Φ(i )i − hΦi
#
"
# "
X
¨ ©
1 X
1
Φ(i ) ∼ e
Φ(i )Φ(i ) −
≈
corr
t t=1...T
Φ,R
t+θ
2
t
T −θ
2
T
T −θ
t
t+θ
T
−θ/tcorr (Φ,R)
t
¨H¥J G¤ G G JG F H T → ∞ MG R H MMG¥FG©K ¡JG
H JG G¤G H JG GH¬¥MG HG MGFG ¥M
)7$4)® " 4?##" )% ? "%A
GGH
HGH
K °JGMGH l (Φ)
R l (Φ) H MH JMGMH L JG F
tHG(Φ) GH GHG
t
L
JG
H«G
L H GFFH K
¡JGΦ HG E L GFGGM HGH HHG J MJ (V, E) J G GFG H FMM
L %"$%, ?!"$ ¨ MM © G
MF H¥FK § H¥J H¥F JG FHG
H¥GMHGH ""#"% 4 )44" %)4" 4# %"# ? LM HG JG GMFH MGM L Φ ≤ Φ
JG GHG i → j H GG® J MF ¦ L Φ ≥ Φ H GG J MF
GH¥M J V H HFG J GJ MMF e JGMG β H ¥FG
eMGGM ¨JG GMHG GGM¥MG
HHF GJH©K ¡JG FHH L JG MH
M R L JG M MGH F¥GH JG GH MGMGH LM HG H GM L R H
MMG¥FGK
§ JH G G JG MHG ) ?!" (?%" *)# ? ) ?# %A %? ® #?!" %A" )7$4)® "
"® ?#)% ? H¥FH GFM M HF V K §GG HGH L H«G LM FMGM J
MGH¥F MM JG M¯GM LM GFF JGM G MGG MG
l
(Φ) MG
FF ¥ L JG FHG JH H F¥FG MGM JG F LM JG
M¥ HG M LM HF JG JFG HG V K °JG F LM HF¥G ¥ L Φ H
G¥J H GJ GFF ¦ H FH GGHHM GFMG JG GFF G¥J MGHGF GGMG
JG F¥G L Φ K FM GJ GFF H FH MG¬¥MG LM HF ¥MHGHK § HG¬¥GG
J H GGG H ) %"#)% ? ? ?4) $%"® $. $ 7" %A" ¢"$. )7 $%"® $ ? ? #)" l > l (Φ)
JG MH LM GFF L JGMK ¡JG GJ MG G JG ¥F F JG
FHG JFF J¥H H¥GMHGH Φ ¥FFG LHJ JG H GJ L JG GH
GH ¨GMHGF MMF JG GMGG L JG HG JG G LM©K ¡JG GH¥ GFMM
GJH H HFF M J JG JGMMJF HGG HM¥¥MG L JG FHG
MG GH JG HGM¥ L JG HHG MH M ¨ M¥ L J JG MH
M HHG J JG GGMG GG MGM© H GHG 7 KKR¥J GHG G
MF FMJ H HGFF GFFH¥G H¥H GJ HM GGM HMH ¨GKK
HGM JMG H M¥FM G FGM HHGH M FF MH H J
GFH MH M JGM MH FH FG M ¤FGH© 7 K
t=1
t=1
corr
corr
corr
corr
j
j
−β(Φj −Φi )
i
i
−βΦ
corr
corr
0 3 4'0 ) 3 3 4 )1 1 )' (stat) .
R
301 1 1( 0 ( V 2 2 4 p
0(
V 4 0 2 (3 ('0V.
201 0 0 5' '01 23 ( 20 ' * &
0 12001 (('(
02 .
lcorr (Φ)
Φ
G HG L H JG HFGH GF L M MJ GGFG M H G
N F J MF GGG H L GGGF LM
¥H VGHFHJ
p
JH J¥H M GV GH HHFF
FF JG GMG MH L HK ¡JG ¯G M
GGG G¬¥F J MF p ¦ JGMA MH JG MG 7"® "7"% #)7? >??,
") !)# ) "$K GMG MJ JGM GGH M M JGM K ¡JG F H L¥FF MGHMG
JG HG L GGH E %A"#" $ ? ?"# ) +7"# #") $® )4" JG GJMJ L HG
H HG L FF JG HGH MGF GG K ¡JG HG GGG HGH H G
iJG ¥GM L HGH L JG F J G JGK
¡JG GM GGM H JG GMG
HG GGG M L HGHK
M L JH M MJ GF H MJGM MGM«G JG ¥GM M L FH FG
M ¦ JGM MH ¥H JHG M MH JG N (N − 1)/2
HHFG GHK
¡JH M GH J JG ¤MH GF JG F H N → ∞ )% ; "7 p = 2M/(N − 1) K §
J HGH JG GMG GMGG hki = 2M/N = N p MMGHH JG F GMGG F¥GK
M GM JMGMH H JG 7"#"" 7 $%# +% ? GHM JG MF
LM G G MGF GG k JGM GHK § JG F N →P (k)∞ ?¢ )% ; "7 hki = N p JG
M H G GF H HJ GJ HH GMGG HM¥ GG M¥ JG GMG
GMGG hki GMGH FMG k )$%"# %A) ) "® ?"% ) ¨ª¥MG ©
¨ ©
λ e
with λ = hki
P (k) =
k −λ
k!
P(k)
P(k)
k
<k>
k
−γ
k
!" #$ % & ' ()
# *(
$ ( ! + # −k log k )
hki
e
# ! ,P (k)
- (
(!() + * " #"
% %
) # #"# *(
#
!(
- 2<γ<3
−γ
k=1
γ>1
!+ #$)
! ' , P (k)
*( ∼k ' ,!
%
../012 3045678 9130 :564 ;0721<=> :531?=315@2 30=@ A6=;0 3045678 64<4@3>7 B4?4>5;4B :=1@>7 1@ 304 <5@34C3 5D
=@3 F : :4<0=@1<2 =@B 1@ ;=631< F >=6 5D EF =@3 F : <0=528 B4=>1@A 9130 304 EF =@3 F : G40=?156 5D 27234:2 90524
EF
<>=221<=>
=@=>5A2 =64 <0=531< HIJKL
.MN4<=>> 30=3 304 O51225@ B12361G F 315@
=;;65=<042 = P= F 221=@ D56 >=6A4 :564 ;64<124>78 304 ?=61=G>4
Pλ (k)
√ 34@B2 1@ >=9 35 = @56:=> B12361G
315@ R<4@3464B 5D ?=61=@<4 IS 904@ λ Q L
F
(k − λ)/ λ
λ→∞
T
GKK H¬¥MG M© M HF ¨d = 3 GKK ¥ GHJ© FGH
G GG FG ¨
2
GH MG GGG MGF HGK § H LM HG
HGM HM MJHd =JHG
JG HG L *) " %#"" ¨M >"%A" )%% 4" LM JG JHHH© M LM GMF LM
JJ MG z GGH JG ¥F MG¥MHGF GJ G H MGFG z − 1 G GH M
GJ JGM M¥FM GGK ¡JG GMGG L GJ G H J¥H z ¨FG M
JG GF G GH L GMGG © FF GH MG G¬¥FG JG ¤GH«G F N → ∞ K
¡JH HGGF M¤F GF FH M MFG ¥G JG HGG L ¥F GGG
GH MGF MGFG F JG MGG FGH HGFL G FF MJGH L HGGMF
HH¥GH ¨LM HG JG GGM L GMF JMGHJF HGG GF K©K § MJH J
FHG JH GMG FGH JGHG MJGH G MGFG F JG MG L "),; " 7
)®® #? )% ? J HHH MGHGF M JG MGHGG L FGH J¥H MG¥ JG MF
MJ
FG MGGK ¡JG FGM GMH H + !"#$) ?7" M FF MJH JGMG
G GGH J MMGFH LFF LM FGH F GFFG MFG ¨L ¥MHG G
JGG ¬¥GF FGH HGMM©K
G´FG G H HGM
FG MGG H FG L G z ; %"
7 "$ ? GGMF JGMGF M¥GH LFF LM JG JGM L MF JGG
MGMF« M¥ G JH ¤G F¥G d = ∞ L JG GH H¥M L JG G
¤GF MJ F 7 K
GMF GMH GMG M¥G H HFG ¨¥FF JG HFGH© GFH L 7 $?#7"#"7
"7 )K ¡JG FG MG¥FM FGH ¨GKK H¬¥MG JGF M M¥FM MH d = 2 ¥
MH d = 3©K JFGH MG M¥G MG JG H L JG MG¥FM MH J
MF
¨M MGF JGM GH J JFGH J MF p HG L HG
GMF©K H1 −p JG M H G GF JG MGHGG M HGG L F 1H −GGG
L JG
H¥H L JG JGM FHK ¡JH L HFF HMGG GF J GGG FG FF HG
MFGH H HGFF GFFH¥G LM ¥GMF H¥GHK G JG HGMGH MGMFG HHF
MGMGH GF "#4? )% ? %#)$ % ? JG M JG MF MGGM MMGH
JG GMG L ® GG G H JG JFG GMK § JG pF N → ∞
LM p ≥ p JH H F¥HGM H ¤G ¥GM L GH H ¥MMGF G
JG ¤G F¥HGM K J JG "#4? )% ? %A#"$A? 7 p ¨ L ¥MHG GF LM p < p ©
H HJ LM P (p® ) = 0 L FH ¦ JHp =LM
GGH HMF HG
(p)
LM p > p M MGFG MGM MGGM G JGP GMF
MH J
¥¥H $"4?7,?#7"#, £" %#)$ % ? ¥MGK MF JG MH H GHMG HF
FH GKK
¨©
P (p) ∼ (p − p )
(p ≥ p )
JG ¤G F¥HGM GJH LMF ¥MG JG GMF JMGHJFK GGMF FF
MJGH JG GG GGFG ´GF GHMG JG GMF MH F H
JMGHJF p JG HF GGH JGGFF HF JGMGH G¤GF M
MGMF« GJH 7 K ¡JG GGFL JG LM¥L¥FGHH L FH GH GJH
L HHF GJH G JGM MF F FG¥FM HHGH J JG LMG
M GHFHJG F« HK
c
c
c
c
c
c
β
c
c
( '0 0 31 6) 10 0 '
d
z 4 10 ( 2
0
0
'0 0
&z = 6 *. ( &z'= 4 *3 2 (
d → ∞ .4 ) ' 01 0(d 1 2 ( 0
00 ( 4 .
pc (N )
¡JG M H G M MJ GF G HGG H ¤GGHF GGH L
GMF GFH FF HGH MG MM GHHFG MG F LM G G ¨
F JG z = 2d GMGH GJMH H JGM¥ FG GH d© ¥ F LM
L FH MG ¥FF MGHGK G HJH J JG GMF JMGHJF H MGJG JG JG
pGMG GMGG H
K ¡JG MGF hki = N p JG GH JG ¤GH«G GMF JMGHJF
hki =H 1 G GF HJ
M
JG ¤GH«G F JG ¤G F¥HGM
L
JG
pMG¥GH
= 1/N
H MGGK ¡JG M MJ GMF JMGHJF G MGGMG J
¬¥FG M¥G GL JG GMF JMGHJF L N HG M MJ J J
G¬¥F p = 1/(z − 1) L
FG MGG JGMG JG ¥GM L GJMH
L G HG JG FF
H MG MGHG H z = N 7 K
c
c
M H G GF GMG JG ¤MH GF L M MJ
JJ JG MGF HG JGMG
¡JG
JG HGH FG ¨L © FH MFGK §H F H GF¥HGF MGHMG JG G GMK
H H¥J H G H M J FFG GGF GH ¨GMGG HM¥
GGM© HGGMF GJH GHG M GMH K § JH MG G H¥GHHL¥F
HGGMF FH LM HG ¨JG FH H LM F HFG© GH «
FMJH LM ¥H M MGH¥MGH FF GM MG F¥HGM
FHH¤ GJH M MFH GJH ¥HG MG MJH MGMGH FG " JG
F ¥GM L GG MHHH GM FF MH L HFG MJ JG FGK
GGMJGFGHH JG MGFGG L JG M H G GF H LMF GF MGMGHG L #") FG GMH JH GG GGF ´GHG LM HJM GG 7 K MGF GM L N GH
FH G LMG M MJ J JG GMG JG HG ¥GM L FH
KGK MM MJ J N GH G¤G JG MF p = 2M/N (N − 1) M MG MGF
JG M GF G¤G H ¥GM M L FHK G JG HGGH J JG #)7? #)® A ?7"
7?"$ ?% )44?+%. )#. ?# %A" ® #?® "#% "$ ? ?$% #") 4? ® " "%¢?#£$ LM HG GF
GMH L GH HF GMH
M JG §GMGK § M¥FM MGF GMH JG GMGG
HM¥ P (k) GMGH LM MG HFF FMG k F¥GH FF M ® ?¢"# )¢ ¨HGG
ª¥MG ©
¨©
¡JH GMF GGGG MG GH JG HGG L F F¥G LM JG GMGG H¥J G
MH MG ¥MMGF GMG $4) ", #"" "%¢?#£$ ¦ JGM MGMGH FF G MGHGG K
JH JH JG M H G GF MGH MGLGMGG J JM¥JF
HHF FF MGMGHK HGM GMG MGMGH MGF GM GGGH
MH J GHHGFF GMH LM M MJ J JGM MGGH HJ¥F G G
¥ H GFK
P (k) ∼ k −γ
(with γ > 1)
¡JG GGM L MH FFGG GJM L FMG ¥GM L MH GFG
GH FGH JG GM MG L HHF GJHK § JG HHGH ¥H¥FF HGMG HHF
GJH GMH GGG JGHG GFGGH MG FH HFG JGG¥H GG MF
JGGK G ¥H ¬¥G LM HG LGMMG GMH GGG HH GF¥GF¥G
GMH GGG JG GMH L FGM J HF¥ HJMMG ¨ ¥HG ¥F
JGMG GJ GFGG H ¥FG H GMGH GJMH© M FMG ¨FF ¥F JGMG GJ
GFGG H ¥FG JGM G© GMH GGG HFFMH ¥FG FG GFHK
0 4 2
10
2 4 )' /
hk i ≥ 2hki
hkic = 1
.
§GM GMH MG JG HFG MG¥FM G GJGM H¬¥MG M ¥ FGH ¨M JG MG
HM M¥FM M JGF MH© GJGM ¤GGHF MJH JGMG GJ G H
GG GJ JGMK LG JG HF MGH¥F LM HMG« L JG MGF HG JGMG
JG HGMG JGG ¥MHK
§GMH MG FH H HFG MGF HHGH JG LM GMH J MMG¥FM
FG MJG¥MGK § F LM HG GMH G MGFG M¥H GH
HHF G FF MGG F GFH M GG J GMG HF MGH JG HG
HHG K
¥MMG GFGH MG GG GMH GF GMH 77 G¥MF GMH 7 L
GH ¥ GMH H JG °MF °G °G M JG §GMG 7 HF GMH
K
HF GJH L GMH H L H¥ H¥J FG HHGH L¥H JG
G GM H FF HHF MGMGHK § GGH JG HM L GMF
GFH MG FG FGH LG HGFLM«G JJ MGF HG HHG
¥MF HG J G HG M L FFG MGFGGK ¤MH GG¤ L JH LMF« H
MG GH FF ¬¥L JG GM FGK ¥ JG GMGH H HM MGF
GMH FHHGH G¤G HG L F MGMGH M JG MGFG $%)% $% 4) "$" "$K § JH G HHF GHGFG JGMH FF MJH HJM HG MGHMG
MGGH GKK JG GMGG HM¥ M MJ GJH 7 K ¡JG JGMGF HH¥G H JG
GGMG JG MGMGH GH¥ LM JGHG H MGGH GJGM LM MG ¥MHGH
H¥H JGMG JG MGFGG L JG HHF GHGFG H GFF HHGHHG GJGM FG H
MGFGG LM JG MGH JG FFG GGMGF ¦ HMG ¥F
GGG JG GG L F¥ F LG¥MGH JG G¤ L JG HHF GHGFG
G G¬¥G GF L JG MGF GM ¥GM GHK
ªFF LM GGHG GGMGF FHH MGGF GMG J JG MGFG
M GHMG MGF GMH H MJGM J L $4) ", #"" "%¢?#£$K § H¥J GMH JG
GMG ¥GM L FH GM G GH FF MGMGHG JG F GMGG ¥FF JGMG H ?
% 4) ?7" A"4" ? % ® 4) 7"#""K H LG JG GMF GMGG HM¥ H HHG J
LM GF GMF GGGG P (k) ∼ k J γ > 1 K
JG® HFGH GGM FF
JJF GG
¡JG GMF G L P (k) GH GFFG MF HGMG
GH FH JJGM MF HGMG GF GG GH ¨k = 1 M k = 2© J M MJ J hki À 1 K
¨MF J FH GGM
°JG FH MG MGG GMG HM¥H
G G© P (k) ¨MF J k FH G ¥P (k)G G© MG GkH¥HJG
HHG J GGH K ªM HG H¥J ¥F GM F H HGMG GF GMH J γ =γ2.4 γγ = 2.0 77 K § H FH JG HG L JG §GMG
JG °MF °G °G J JGMG FH GMG GGH γ γ L HGGMF FF
GMH § JGM H¥H LM HG GG GMH ¨JGMG JG GH MG JG GGH
G (i, j) MG GH J JG MG G GG i MG¥FGH JG GG j © MG GGM
¤G J JM GJM GMF ¥GMGG HM¥ P ¨ MG G
G GG MG¥FG MMM FMG ¥GM L JGM GGH© GGFF G
GMGG HM¥ P J ¤G JMGMH HFG HG JHF MGHH LM
HG HGM JMG MGG MMMF FMG ¥GM L MGH G FG JG
−γ
in
out
in
out
in
out
in
out
out
in
30'
' 3 3 (00 30' ) 0' 30'
γ >20'
2
2
(0 23 )' γ ' 0 30'
*
.
γ<
P (k)
P (k) ' 2( k 3
1& ' '( 00 2
3 303
. . .*( 1 &
i '0 62
3 '01 2 4 2))01 c . * 0 ( '22 620
P (k) ' i
3 '0 k 0 P &(k) ∼ ck i
)' 4 k . /= 1/ log(1/c) ('0 ( ) 3 41
/
3 120 ' c 4 4 . 120
2 2201 .
c
kc
GJMJ L G GG MGM K GGMJGFGHH GGMGF MG HMG
LJL¥FF GGMG JG ¥F ¥MG L JG HM¥K § H HGH HGGMF GFH G
H¥MG JG HG GGMGF K
§ H GG G ¥GMFG J H JJF GFG GM HHGHH J HG GM
JG GMF GGGG L GGMGF GMGG HM¥ ¨HGG ª¥MG 7 ©K ªMH H MG
GG GHG JG 4+ + )% !" 7"#"" 7 $%# +% ? +4% ? = P P (k ) MJGM J
JG HM¥ P (k) HGFL GFH HJGM FH ¨HHFF (k)
¥GM GMMGMGHG
J J MHG JG HFG ¥G ¤GH«G ¥¥H GMG ¥© H JG MMM
JG MG¬¥MG M JG JHM JG GMGG F¥GH JG G G G
H¥ G HHHK G JG GMF GMGGH (k ) G GHGH
¨©
1
1 − Fb (k) =
Card{i, k ≥ k}
(Fb ≈ F when N → ∞)
N
§ HG JG P (k) ∼ k FMG k JG Fb (k) GJGH H k K ¥ GG J JH MG¥MG
GM HGGMF GGH MG MG´MG J MMGHH JGMG GMGGH M GGG
FGHK § GGMGF H¥H H¥J GMGG F¥GH MG HF G¥GMG
JG H¥M L JG GMF HM¥ H LM MMGMK ¡JG F GH JG GMGG
HM¥ J G LM JG GGMGF HG ¥ FH ¨H H LMG¬¥GF JG HG F© LM JG HHG HGFLK § JH HG JG GM HH¥G L JG HM¥ ¥MG ¨
HH F
!$ GM F© H HGJ FFHG GFGHH JG HM¥ MG¥GH JG MGMG
LG G ¤ GJGM L JG L¥F LMH ¦ F M¥GH HJ¥F G G
¯¥HL JG GF JGK ¡ H¥M«G GMF HJG GMH GH H JG HFGH
¤ HHG J JG G MG ¥MG MG LJL¥F J GGF ¤K § H L JG $ ® "$% ""# 4 7"® "7"4" ¢A" %A"#" $ ? 4A)#)4%"# $% 4 $4) "K
MGGM HGM GMF GMGG HM¥ GG J ¥M GM GH
MGMGHG HGFL GF L H M ¨HGG LM HG JG MGG GHH K ª
GFFGM M JHMF GMHGG MF HHGHHG L JH JJF LHJFG
L HFGLMGG GM ©K ¡JG F ¥¬¥GHFG F¥H J G M H JG JJ
JGGMGG JG GH MG G JG HGG L JMGMH GMGG F FH JG
GMGG HM¥ FF JG MG HGMHG J γ H HFFK GMF GMGG HM¥ J
M¥H MJG¥MGH GJGM F¥HGMG M GJGM ¥FM M GJGM JGMMJF
M LM HG L¥MJGM HG¤ L JG HHF GHGFG J G GGHHMK F
GF MJ HG F FGG L JG HHG H¥GH JG MGFG GHGFG
MG MGF ¥GMH L JG GM MGMGH GJHH M JG HGMG
GJMHK
k0 <k
0
i i=1...N
N
i
N
−γ
1−γ
N
log Y
log Y
log X
log X
+ # #
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G GHGK ¥FF MGH G ¤GF L GHH GHMG JG GJM L
FG GMH ¥ J JG HGM HFG GGMG J H JG MGFG G¨H© L
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GGG GF JGM HGFLHFM L 7 K
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FG GMH GGMG JG ¥ M HG M« MFGH
H¥G M¥H HGFG GJHH K °J MGFH GJH J ¥ L JG
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GG MJ G MJGM ¦ JH GH MFG M¥GH GMF GMGG HM¥ J
K
γ=3
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L FHK °G JG GHG LM HG JG H¥ JGMG GJ G i GH M
LM GJ F
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p MFG J
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G hk i = n p K GF HHG MG¬¥MGH MG JG FH H¥HJ FH MM
G FH ¥K ¥GMGG HM¥ MG GH JG H«G HM¥ (n )
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J G LMG GMH JG GMGG HM¥ H FH HH F
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= hk i = hki
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¥ LM JG GMF G L FG
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FGMG GFH ¥ LM JG GMG L GM F ¨HGG LM HG 7
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JH HG G JG G MHH JG GM GMGG HM¥K ¡ JGM M
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FF MH L GH L JG ¥GM L HGH L JG F J G JG© JG 4 +$%"#
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MGMH JGHG JMGG LG¥MGH MG¥FM MH M H G M MJH HFFMFH G LGM
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.
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JG MGLGMGF JG MFG K ¡FG H¥M«GH JGM ¨©F ¥ LM JG
MGF FG GM LG¥MGHK
6 possible triangles
R U em_^XdZcq b\X bZXc^o hX cXh
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small
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M¥G© H JG GG L ¥GH KGK MG GHGF GG MGHK
¤MH MJ H HMG LM FHH¤ GJH GGFG LM GGMF HGH L H
HG ¬¥G GHG L JGM HFM ¥MMGF GH L HGK G ¥H ¬¥G
LM HG JG K , ")$ "%A?7 JG A "#)#4A 4) 4+$%"# K § JG ¤MH GJ JG ¥GM
L FHHGH H JHG MM ¨JH GJ H J¥H GMG H¥GMHG GJ © ¦ JG MG¥MG
K
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JG FHGH GGM x K G JH HM JH GG G LM FF H JG GGM L M L
GJ H¥M¥ H ¥G JG MGH¥F H MG JG ¥HG ¥G JG H L JG
GGMH x K K Kx K ¡JG MG¥MG H GMG ¥H H MGLGMGG H JGHG G GGMH
H LMJ FF
JG M GH GFG MGK ¡JG HG GJ HHH GGM
MGG M¥ JG FHGH H GJGM H¥GHHG HGHK G FHGGHH H ¬¥¤G
GH L HFM HG G G¤G GJ HG¤ HGK ¡JH JGMMJF GJ
GFH LF L MH MG M FGHH MG M JG HGMG FGGF JG MGG
M LM JG GFGGM FGGF JGMG GJ M¥ MG¥GH HFG ¥ JG MGG M
JGMG FF H LM HFG FHHK
¡JG HG MJ H FMG MGHGG 7 K7 K § FH MGH JGMMJF ¥MG
¥ M LM GMG MG¥MG HG JG #)® A "® ?#)% ? ) #)7? ¢) £ ¢ %A
%#)$ % ? )%# R = A /k ¦ ¥GH MG JG G¤G J ¬¥HM MGH JJ JG F MGH
MG LM FK ¡JH GJ MGGMH JGMMJF F¥HGM MG
G M¥GH HG GGG JG HGH MG G JG G MG¬¥MG LM JG F
G JG K
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JGMGF FMH ¥GFGH GMG MGLGMGG GFH J GFF MFFG
GJMK ¥J ¥MG H FFG LM HG LM F HHGH ¨MM GMGF G¬¥
H L F GH© LFF LM JG GGM L FF JG HHFG H GJMH
JG GHM L JGM L¥MH JG GGG L MF LMH J H HFG ¥GMHF
GF¥ FH JJ HJ GGM H MG¥G ¯¥K ¡JH H JG HM JJ GH GHH L #)7? >?? ") "%¢?#£$ JG GG ¥GK ¡JGHG GFH
FG GMH (V, E) JGMG JG HG L GJ HG i ∈ V H GHMG FG MFG x
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GJ HG i M HH GJ L JG 2 HHFG ¤¥MH JG k GJMH L
G F¥G M LM x G t + 1 K ¡JGHG 2 JGH G¤ FMM HG i MG
i GGMFtGMLMG M
GGGF GJ LM JG JGMK
¡JG GMGH L H¥J M GF J H HG GF GJH L GM
GGG HG H GJMH H )44?+% ) " "4% !" ¢) ?# %A" ® #"$"4" ? ?%A A %?# )7 " 4 %)%?# 4?"4% ?$. )7 ? ")# %"#)4% ?$ J ¥F G ¥F L HHFG
GHMG GFK ¡JG ¤ GGG JG GF JG MGMGH L JG ¥F H H GMLMG
7 #"4% )% %A" "!" ? $%)% $% 4) ")%+#"$ L JG GMH LM HG H MH
L MG ¤¥MH GFH x = 1 J x = 0 K § H JG HHFG JGG HHG
GH L JG MMH JGM H L M MHGH ¥M GGMG
JG GGGG L JGHG LG¥MGH J MGHG JG GM H«GK GGMF H
MGGH J GMG FGGFH L FG G HGMG K ¡JH L GFH JG GG
M¥G JG G L GG GMH 7 JGMG JG MG HFF GGFG K ¡JG MG G
LM¥L¥F ¥GM¥H JGM GH MG JG GH JMG GM GFHK
i
ki
i
ki
i
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§ FF JGMG G HJM GMG L JG L HH¥GH J HHF GF L GMH
FFH ¥GMG J L¥H JG M¥F GGG JGM FF FF LG¥MGHK
MGMFG LG¥MG L HFGLMGG GMH H JG MGHGG L J¥H GF JJF GG
GHK H G L JGM G GH H MG¥G JG GM GGM JG F MG JG G G
L ¥H GM MGMGH FF G ¥J GG JG MGF L H¥J GHK
FGLMGG GMH ¨ M¥FM MGF GMH GJ FF GMGG HM¥© MG
J¥H JJF HGHG H GF MGG GHM¥ L JG GH LFF GMGH
GMGGK § MH JGHG GMH MG M¥H J MGHG JG M MGF L GH FMG
LM L GH G MGG J¥ F JGM ¥H J JG GM
GM JGMG MGF J M MJHK G JGMG MGGMH JG HGMG M¥HGHH L MGF
FG GMHK ¡JH H¥HH GH JGM FF¥HM L JG GGG GMGGH GGG
JG G J MGF FG GMH JG JGM J GMF GMH M MJH
¥HG LM F GF JG GHG FGMG GFHK
HHF GF L GMH L GMGH LM H HG HM MGF GMH ¥
GMHF FHHGH GGG MGFG HGMFGH GHHGF MGGMH H FH MGFM
HG MH H¥GHK ¡JH GMHGG JG GG GGFG GGHGF LM GMF
GMHK ¡JG GMF GF GH H HG H ¨FF HHF© MGMGH
L JG MGF GM ¦ JG FFH <+)% %)% !" H¥ L F JGG F JG G
MK ¥GM¥H H¥GH G MG MF M «G JG GJM
L JG HHG JG GG ¥G JH HM GKK LM LMGH ¤MG M M F¬¥
HG M¥H G¥ 7 K § H L HGM M JGG J JG GHGG
L GMF JMGHJF GMH JG H ¥HF H JMGHJF HGM H¥H JGMG
M HH LM JHG JGMG GGFH MHHGH JG JFG GM JG ¤MG GJGM HH
M MGJGH JG JGM HG L JG LMGH JG GG HGGH JG ¥ M K ¡JG HGF F
L MGF FG GMH MG¬¥MGH ¥GMG GHH L M JGG MG
MMG GM GFHK FH MG ¥GM¥H M LM JG M L LM
M M¥HGH JG §GMG JG HMG L ¥H HGHG ¥F HM¥¥MG
HF MGFH M F GM L MFGH M JG LG L M¥H JG GF
GMK ¡JG MGMFG MGH¥F H JG ) $"4" ? ® "#4? )% ? %A#"$A? 7 $4) ", #"" "%¢?#£$K ¬¥G LMGHGGFG F¥H H J JG FGG L JG GM F H GHHGF MG
GHMG ¬¥GF ¨ FMG ¬¥FGF© M JGG ¥MM JG
GM 7 K
§ JG MG¥H H¥HG K7 G JG GG HG MGH¥FH ¥ JG H ? JG
GM ¦ K JG HH¥G GMG HG MGMGH ? JG GM HGFLK °G JGMG ¥M
JM FHH L ¬¥GHH GM %A" 7 ) 4$ ? %A" "%¢?#£ ?7"$ KGK JG GF¥ L JGM
¥F HGH s (t) JG JG GGH GG GMH GGG JG GH JGM HGHK
G HG L ¥FG GFGGH JG ¥GG L JG GM GM F JGM H
i
%% ) 4 4 4 3 ((0 24 04 )'
m ) (3 4' ) 4 62
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g ≤ gc = 1 − [m(γ
− 2)/(γ
− 3)
− 1]−1
4 γ ('0
0
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≤ 3
gc (N
)
2 4 N. γ3
0
30'
4
2( = 0.9
3 ( 62(0 (' gc .
G GHG J JG LMG MG GFF¥FM ¥ K G FG GMH
HG MFGH MG HMGG ¥ HHF MG J GMG F¥GHK ¡JG GMGG
FGH JG M¥FGH JGMG JG HMGGG ¥ M¥FGH FG JG HG L GJM
HGH ¨ GGMF JGM GMGHH GJGM GGMH M HJH H HFF JGG¥H©K
¡JG GF¥ L JG HG x (t) L HG i H J¥H GGMG JG HGH x (t) L FF GJMH j L
LM HG JM¥J JMGHJF P x (t) ≥ θ JGMG θ H ¤GK
i
i
j
j
E
Q
j
R
Q
R
R
E
E
Q
R
E
t+1
t
Q
t+2
t+3
R U gcY[ZbYe [\^Z] Zc Y b\mfeXh X`bZ^YieX hgcY[Zb_k yYbW c\hX bYc iX Zc XZ^WXd \] ^WX ^WdXX
_^Y^X_ E pX`bZ^YieXro Q p}mZX_bXc^r \d R pdX]dYb^\dgrk ~^ _ Xx\em^Z\co Zc hZ_bdX^X ^Z[Xo ]\ee\a_ E → R → Q Ych
Z] Y cXZq Wi\d Z_ X`bZ^Xho Xe_ X ^WX c\hX dX[YZc_ }mZX_bXc^k VWZ_ ^dZYcqeX [\^Z] WY_ ^WX dX[YdjYieX fd\fXd^g
Q→E
^\ iX Zc_Xc_Z^ ZxX ^\ ^WX _^Y^X Ych hgcY[Zb_ \] ^WX _mdd\mchZcq cX^a\djo jXXfZcq Z^_ fXdZ\hZb hgcY[Zb_k mbW Y
[\^Z] Yb^_ Y_ Y ufYbXv[YjXdwo qXcXdZbYeeg Z[f\_Zcq Z^ _ fXdZ\h ^\ ^WX aW\eX cX^a\dj \c ^WX e\ cq ^Xd[k
§ MGG M G H¥G JG ¥GG L GM F MG HG¤FF H GMGG
HM¥ ¨GJGM HH F M GM F© GFG H G GF
F J¥J MGFH G¥MF GM GJ HG G GJGM L JMGG HGH
¨GFG© ¨¬¥GHG© M R ¨MGLMM©K §H HMGG G GF¥ MGH E → R → Q
E
H GG JGMHG MGH ¬¥GHGK ¥G H¥MMHF
Q → E L QG L H GJMH
MG JG MGH¥FH HG K K7 JG ¥GG L F JGH G MJGM
G HG F ¬¥G JG GGM GJM H F MG M¥H MG JGMG
GMF GMK F L JH F HGH GFG ª¥MG FGH JG MGHGG
L LH H JGM MH H JG JFG GM JGGM H F
HK ¡JH MGH¥F FGH ¥H H¥HH MG GF JG MFG L FF MGMGH LH JG GMFF
GM GJMK
H MG¥HF ¥GMFG K7 GGMF G MG¥ JG F JG GMGG
HM¥ J GFF G GMHF¤ JGM LG¥MGH ¥F G MG HM
LM JG MGH¥F GJM LM HG JG F¥HGM G G M JG HM¥ L
FG H«G ¨FGJ L JG F MF J G G HGFL L ©K MGFM
H¥GH JG J¥H G ¥G GGMG J MG JG MGFG HHF GHGFGHKªM
HG JG GFG MGHGG G G L JG MGH¥FH L ¥M H¥ H MGHGF H¥GH
J G ¥GHHG ¥G M¥H GF J JG HHH L MFGH H FGF
G G MGG HJ¥F G GFF G ¥ JG GH GFG
H GMK
?% H GMMGMGHGG M ¥GMMGMGHGG ¨ L ©
H¥MJ JG HG
MG GM MG JG MGH L G HHF GFK ¡JH MHGH J¥G
GMGH JG G L FF MG¥FM GMH LM HG GF GMH GG
GMHK LM LH H HGMG G MFGG MJ GMF
% 6(201 ' (001 ' 1( ' ) 0 4 00'0 '( 22'0 30'1 1(
* & *
*.
&
&
HMJLMMF GL¥F ¥MGF FF L¥H JGM MH ¨MGH M G«
MGH LM HG© K
MH L LH HHH MG¥FM GMH L GMG
HGGH JH GG MHG H J MH JG GF¥M JHM L JGHG GMHK
JGM MGMFG FF HM¥¥MGH G GHG LM HG ¥GH ¨MG¥HF H
¥HHG 7 K7 K © MMGH GHGF GG MGH GMMGG H L¥F
¥FGHK ¥ JGHG MJGH HJ¥F G LMG L¥GF HH¥GH MGM JG GJM
L GM JG MGFG L ¥HF GGMG MGMGH HGMG JG FGGF L
JG JFG GM ¥ HHFG LMGHGG HF F JG GFG MGMGH L JG
GFGGM MGGH ¨GH GGH FF HM¥¥MGH©K G GH GH¥G HG ¥ H
MG´MG JG FHH GMGMMG L JG LH JGM MFG JG GMFF
GJM L JG GM H MG¥FK
L H FF MGM H J¥H GGHHM HGM H GG JG JFG
GM GHG JG GGMG HG¬¥GGH L JH FF HHF LG¥MG K ¡JG
MG L L HJ¥F G J¥H GF¥G GMH L H ¥GG HG FF MGM
L JG GM LM HG M M H MGMGHK JGMHG JG MGHGG L LH
HF MGH¥F H ¥FM L¥F L JG MGHG GM L JG MJ
MGHH M FF G MG G HG MG L H GF¥M JHM JGM
MH GMG HHF H MG G HG HG¤ L JG GM GGHHK § HFM JG
¥ HM¥¥MG H HGFL H¤ GM MGM ¥FGHH FH JG MH
G FG GH HF GMK § JG GGMF HG JGMG GH F GG
GMH M G JG HHF L HG M ¥GH GGHHMF MG G HG
GHHGF M«F M L¥F MGMK °G J¥H ¥GMFG JG MG ¥ JG L
GMMG G¤G G JGJGM H GMG FF MGM MGM
LFF LM ¥ FHH M LG¥MG H ª¥MG ¦ JG MGHGG L GMMGMGHGG
LH FF JG JG HG M JG HG G GJ HGK
¥¬¥GHFG GMGH LM L FHH ¥ GG MGF GM H
LM JGHG HHF MGMGH J JHG L HHFG GFH GHGFF JG HGGMF ¥M
MG GH MG FFGK ¡JH G JH GG FMGF GGFG JG G L GMF
GMH JGMG JG F¥HGM HHH MGH ¬¥G MGM MGG JG G¬¥
L GF JG MGF HHG GLMG GF JH GF GKK MG GJM L
JG HHG K JGM GMGH LM L GGM H ¥ HMH FF H¥GH
LM HG JG MG¬¥MGG G ¥ FF MMGFH ¨MG G JG GHGG L
¥GM M GMMGMGHGG LH©K ¡JG L L H J¥H MGFG JG H F¥G
JG JGM FF HHF LG¥MGH ¨GMGG HM¥ F¥HGM G G LM HG© G
¥ JG GM FHH J H¥HG JG JH GGHHM H¥K
¡ H¥M«G GG L H MGF GL¥F MGH¥FK § HFM HM H
MGM GL MG GHGF GG MGH J ¥FGH JG HGHG L L¥FF
GGG FHK ¡JH L GMMG MJGM MG G JG HJ GHG JG GM
HG¤ H MGMH ¥HF MGFHJH ¥M F L FH MH JG H JFG
¥F G HFGM GHG GMH L JG ¥F G MG¥G MM L GGG
¥F FH J GFFG¤G GLMGG MGMGH ¨GKK M MGHHG L¥H©K
§ H MMGF JG HG GM FG MGHGF MGH JG L J JG MGMGH
GG JG GFGH L JG GFGGH MG HMF GG JG GM JGG ¥MM
K ªM HG JG ¥G L JG G GGG HGGMF HHFG JH H JJF
GGG L JG H¥MM¥ HG GF¥K G FF LGHH KGK GGH HGMG
FFF ¨G L G G LM HG© LFF LM JG FFGG H L JG GMK
GMH J¥H MHG J¥G JFFGG JG JGMGH G GGF HHG GJH
GHMG MG JG HG¬¥GGH L ¥GM¥H GMG M GG GMH
LGGH GGG FF FF GM MGMGHK
§ JG MGHGG JGMG F JG HFGH H¥ L GMH J HGM ¥GJG
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JGM GMHGGH G ¥ F GM MJ JGM GMG JG ¥MHG L F¥
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G ¤H L JGM FF HM¥¥MGK § H JFG MF J GMGH FH FF HG¤
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NT QP U § MGL¥F FG J FGHHM ªMG LM G
M¥G JH HGF HH¥GK § MF FGG K GGG K GM¥ K K
MG KK J¥F K GMHHF LM LM¥L¥F H¥HHHK
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