part 2 from vicious walk model to random matrix …...1 part 2 from vicious walk model to random...
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1
Part 2 From Vicious Walk Model
to Random Matrix Theory
中央大学理工学部
香取眞理（かとりまこと）
集中講義：東京大学大学院理学系研究科物理学専攻「物理学特別講義
BXI：非平衡統計力学の最近の展開」
２０１０年１１月２９日-１２月１日，東京大学本郷キャンパス
2
• Let be the N-dimensional Markov chain starting from
x
= (x1 , x2 , …xN ), such that the coordinates S j (t) , j = 1,2,…,N, are independent
simple symmetric random walk on Z.
• Take the starting point x from the set
• With a constant T > 0
we impose the noncolliding
condition
up to time T ;
•
We denote by the conditional probability of under this noncolliding condition.
1. Vicious Walker Model
.,.....,2,1 ),(.....)()( 21 TttStStS N =<<<
)P)}(({ ,,...2,1,0xS =tt
{ }. ...:)2(),...,,( 2121 NN
NN xxxxxx <<<∈==< ZxZ
Michael Fisher calleda vicious walker model
in his Boltzmann
medal
lecture.
(M. E. Fisher, J. Statistical Physics 34
(1984) 667-729.)
( )xS Ttt Q ,)}({ ...3,2,1,0=
xx P QT
3• We will assume that
• Each realization of vicious walk is represented by an N-tuple
of
nonintersecting lattice paths
on the 1+1 spatio-temporal plane, Z×{0,1,2,…,T}.
An example is given by Figure 1 on the case N=4 and T=6.
.1 ),1(2)0( NjjS j ≤≤−=
4
•As a model of Wetting or Melting Transitions(Fisher (J. Statistical Physics 1984))
•As a model of Commensurate-Incommensurate Transitions(Huse
and Fisher (Physical Review B 1984))
Physical Motivations to Study Vicious Walker Models
5
•As a model of Directed Polymer Networks(de Gennes
(J. Chemical Phys. 1968),
Essam
and Guttmann (Phys. Review E 1995))
(a) polymer with star topology
(b) polymer with watermelon topology
62. Young Diagrams, Young Tableaux and Schur
Polynomials
A bijection
between vicious walks
and semistandard
Young tableaux (SSYT). [ Guttmann, Owczarek
and Viennot, J. Physics A (1998),
Krattenthaler, Guttmann and Viennot, J. Physics A (2000) ]
(1) Let
Lj = the number of leftward steps
among T steps of the j-th
walker, for j =1,2,…., N.
Draw a collection of boxes with
N columns,
in which the number of boxes in the j-th
column is Lj .
[
e.g., L = ( L1 , L2 , L3 , L4 )=( 3, 3, 2, 1 ) ](2)
For each walker, we label
each leftward step by the integer 1,2,…,T, which is the time when that leftward step is done.
(3)
Then for the j-th
column of the collections of boxes, fill the boxes
by the labels
of leftward steps
of the j-th
walker, from the top to the bottom,
j =1, 2,…, N.
7Remark 1
L j = the number of leftward steps
among T steps of the j-th
walker, for j=1,2,…., N.
[
e.g., L=(L1 , L2 , L3 , L4 )=(3,3,2,1) ]
◎
The noncolliding
condition
of vicious walkers
guarantees the equalities
◎
Let
Then the equalities (2.1) imply
•The collections of boxes with conditions concerning the numbers of boxes in rows (2.2)(and in columns (2.1)) are called Young diagram (YD).
•The total number of rows in the YD is called the length l of YD.[ e.g., length l = 3. In general, length l =0,1,2,…, T ]
(2.1) . ....21 NLLL ≥≥≥
row.th - in the boxes ofnumber the kk =λ
(2.2) . ...21 lλλλ ≥≥≥
] )2,3,4(),,( [ 321 == λλλλe.g.,
Young diagram
8Remark 2Let
T (j, k)=
the integer in the box located
in the
j-th
row and k-th
column,
◎
The noncolliding
condition
of vicious walkers
guarantees the equalities
strictly
increasing
in each column,
weakly
increasing
in each row.
Young diagram with integers T=(T(j, k))satisfying the above ordering
of T(j, k)’s
are called Semi-Standard Young Tableaux (SSYT).
)1,( ),(
),1( ),(
+≤
+<
kjTkjT
kjTkjTYoung Tableau
9• YD with boxes in the
k-th
row, k = 1,2, …, l , is
said to be the YD of shape
• Let L j = the number of boxes in the
j-th
column.
The YD with the shape L = (L1 , L2 , …
, LN ) is regarded as
the conjugate
of the YD with the shape
and denoted by
• As shown in Figure 3, they are mirror images with
respect to the diagonal line.
kλ. ),...,,( 21 lλλλ=λ
. ),...,,( 21 lλλλ=λ
. ~or ~ LλλL ==
• The sequence of integers in the weakly decreasing order(representing the number of boxes in rows of YD)
is regarded as a partition
of an integer
• The number of parts l is equal to or less than
T .We introduce a set of T variables
z =(z1 , z2 , …, zT ).
TkT λλλλλλλ ≥≥≥∈= ... , , ),...,,( 2121 Nλ
∑=
=T
kkn
1λ
Figure 3
10
• For each SSYT, T=( T(j,k) ) , define a monomial
• e.g., for the SSYS shown in Figure 2 (b),
∏
∏
=
=
=
T
m
mm
kjkjT
z
z
1
Tin occurs integer that the timesof #
),(),(
Tz
365
3432
65
644
6432T
zzzzz
zzzzz
zzzz
=
×××××
×××=z
11
•
Notice that for one YD
with a given shape , there are different ways of filling boxes with integers
to make SSYT.
•
For each YD
with shape , we define a polynomial
of z = (z1 , z2 , …, zT )
by summing the monomials over all SSYTdefined on that YD:
This polynomial is called the Schur
Function
indexed by (the partition/YD with shape)
λ
λ
. ),...,,( shape same with theSSYT all:T
T21 ∑=
λλ zTzzzs
• shape
of YD the final positions
of vicious walkers at t =
T.• variety of SSYT
on a YD variety of different vicious walks
with the same final positions
λ
123. Enumeration of Vicious Walks with Fixed Final Positions
• There is a simple relation between
the number of leftward steps Lj , j =1, 2,…, N,
and the final positions yj , j = 1, 2,…, N, as
Lj = (T -
yj )/2 + ( j - 1), j = 1, 2,…, N,
or equivalently
yj = T - 2 Lj + 2 (j - 1), j = 1, 2,…, N.
bijections• final positions
y
of vicious walkers YD
with the shape
• one realization of vicious walk
one SSYT
T=( T (j, k) )from S(0)=(2(
j-1 )), j=1,..,N , to S(T) = y
with the shape
• set of all vicious walk
a Schur
functionfrom S(0)=(2( j-1 )), j=1,..,N , to
S
(T) = y
λ
λ
),...,,( 21 Tzzzsλ
)(conjugate ~ and Lλ Ly =⇔
13
For L = (L1 , L2 , …, LN ) = ( (T -
yj )/2+(
j –
1) ) , set Then
. ),...,,()(1...21,
21 =====
TzzzTTN zzzsM λy
, ~Lλ =
DefinitionFor y1 < y2 < …
< yN , define the number
MN,T (y) = total number of distinct vicious walksof N walkers from the initial positions
Sj (0) = 2(j-1) ,
j=1,2,…, N to the final positions
Sj (T) = yj , j=1,2,…, N .
The above bijection
between vicious walks and Schur
functions gives the following identity.
14Jacobi-Trudi
Formula ( )
( ) , det
det),...,,(
,1
,121 kT
jTkj
kTjTkj
T zz
zzzsk
−≤≤
−+≤≤=
λ
λ
( )( )
.
11 lim
detdet
lim
),...,,,1( lim)(
1
1
)1(
1
))(1(,1
))(1(,1
1
121,
1
∏
∏
≤<≤
≤<≤−
−+−−
→
−−≤≤
−+−≤≤
→
−→
−
−+−=
−−∑
=
=
=
=
Tkj
kj
Tkjjk
jkm
q
kTjTkj
kTjTkj
q
TqTN
jkjk
qqq
qqqsM
kjT
mm
k
λλ
λλλ
λ
λy
Remark:
This is nothing but the dimension of the irreducible representation
R
with the highest weight of the unitary group U(T) λ
( ) ).(det tdeterminan eVandermond
1,1 ∏
≤<≤
−≤≤ −=
Tkjjk
kTjTkj zzz
15
Jacobi-Trudi
formula. )],...,,([det),...,,( 21)(~,121 TkjNkjT zzzezzzs
k −+≤≤= λλ
By definition of elementary symmetric polynomials
t)coefficien (binomial )1,...,1,1( is, that
, )1( then , 1 all if ),...,,()1(0
2101
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛=+=⇒=+ ∑∑∏
===
kT
e
kT
zzzzez
k
kT
k
Tj
kT
T
kk
T
jj ξξξξ
. 2/))1(2(
det
det
~det )(
, )1(2/)( and ~ Since
,1
,1
,1,
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
=
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=
−+−==
≤≤
≤≤
≤≤
kNkj
kNkj
kNkjTN
kk
yjTT
kjLT
kjT
M
jyTL
λy
Lλ
16
. 2/))1(2(
det )( ,1, ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
= ≤≤k
NkjTN yjTT
M y
Since
this expression of binomial-coefficient determinant
is regarded as a special case of
Karlin-McGregor formula
known in probability theory, and
Lindstrom-Gessel-Viennot
formula
known in combinatorics
theory.
, timefinal at the position theto 0 timeinitial at the )1(2position thefrom
plane temporal-spatio on the possible ofnumber the2/))1(2(
Ttytj
yjTT
k
k
==−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
paths single
174. Enumeration of Vicious Walks with All Possible Final Positions
. 1
1
)(
10...: 1
...:,,
21
21
∏∑ ∏
∑
≤<≤≥≥≥≥≥ ≤<≤
<<<
−+−++
=−
−+−=
=
TkjN Tkj
kj
yyyTNTN
kjkjN
jkjk
MM
T
N
λλλ
λλ
λ
yy
DefinitionMN,T = total number of distinct vicious walks
of N walkers from the initial positions Sj (0) = 2(j-1) ,
j=1,2,…, N
to any possible final positions at time T
Asymptotics
of the Survival Probability of Vicious Walkers
[Fisher (1984), Guttmann et al. (1998), Krattenthaler
(2000)]
).1(41 with
2,
, −=≈≡ − NNTM
P NNTTN
TNN ψψ
18Summation Formula
. 1
110...: 121
∏∑ ∏≤<≤≥≥≥≥≥ ≤<≤ −+
−++=
−
−+−
TkjN Tkj
kj
kjkjN
jkjk
Tλλλ
λλ
λ
( )
( ) )conjecture sKnuth'-(Bender 1
1,...,,,1
)conjecture s(MacMahon'
1
11
1,...,,
,equalities twofollowing theofany oflimit 1by obtained is This
11
1
0...:
12
1 11
1
2/)12(
2/)12(
0...:
2/12/)32(2/)12(
21
21
∏∑
∏ ∏∑
≤<≤−+
−++
≥≥≥≥≥
−
= ≤<≤−+
−++
−
−+
≥≥≥≥≥
−−
−−
=
−−
−−
=
→
Tkjkj
kjN
N
T
T
i Tkjkj
kjN
i
iN
N
TT
qqqqqs
qqqqqs
q
T
T
λλλ
λλλ
λλ
λλ
19
More general summation formula is given in the textbook of Macdonald
. )(det)(det
),...,,( 21,1
21,1
0...:21
21
jTi
jiTji
jTNi
jiTji
TN zz
zzzzzs
T
−−≤≤
−+−≤≤
≥≥≥≥≥ −
−=∑
λλλλλ
Remark{ }
{ }T
T
NTN
TN
N
∈×
⎭⎬⎫
⎩⎨⎧
≤≤
≥≥≥≥≥
λ
λ
λ
:rectangle an in included YD
)( parts ofnumber with theand rowfirst theoflength with theYD
0....:
1
21
c
l
c
λ
λλλ
205. Diffusion Scaling Limit
{ }
. )|,(f y),(Ν
by given is 0 t time toup of
boundary hit thenot does at startedmotion Brownian y that theprobabilit The
. in motion Brownian absorbing theofdensity n transitio the
2)(
exp 2
1det)|,(f
formula,McGregor -Karlin theof By vietue ).A typeofchamber (Weyl ...: Let
,....2,1,0 ),( walk random theofion interpolat thebe toconsidered here is )( where
, )],0([ paths continuous of space on the , 1 , . )(1Q) . (
measuresy probabilitconsider we, ,0For
.,...,2,1,0 , )(...)()( ; time toupcondition ngnoncollidi with the walk vicious thedenotes )Q,)}(({hat Remember t
2
,1
121
2
21
,...2,1,0
2
xyxR
R
Rx
R
xy
RxRSS
RS
Zx
S
xx
x
tdt NN
N
N
N
N
kjNkjN
NNNN
NTLL,T
NN
t
tyx
tt
xxxttt
TCLtLL
T
TttStStSTt
∫=<
>
∈•
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧ −−=
•<<<∈=•
=
→≥⎟⎠⎞
⎜⎝⎛ ∈=
∈>•
=<<<
•
<
<
<
≤≤
−<
<
=
π
μ
21Theorem 1.
. || ),2/(/2 ,(0,0,...0) where,, ,0for
,),(N
),(N)|,(f),;,(g
),,(N)(2
||exp),;,0(g
densityy probabilitn transitiowith the],,0[ , ))(),...,(),(()( processdiffusion ousinhomogeney temporall theof law the
weakly toconverges ) . ( , as ,0 and fixedany For
1
22
1
2/4/)1(2/
,
1
2
,
21
,
2
∑∏
∏
==
−−−<
≤<≤
<
=Γ==∈≤<≤
−−−
=
−−⎭⎬⎫
⎩⎨⎧−×=
∈=
∞→>∈
N
jj
N
j
NNNNN
N
NNTN
NNji
ijTN
N
TLN
yjtTcTts
sTtTstts
tTyyt
ct
TttXtXtXt
LT
y0Ryx
xyxyyx
yyy0
X
Zx xμ
22
Corollary 2.
motions.Brownian standardt independen denote )}({ where
,,...,2,1 ),,0[ , )()(
1)()(
:modelmotion Brownian sDyson' of equations thesolves )( processdiffusion The (ii)
.)(/)2(' where,, ,0for
)()(h with (y)h)|,(f)(h
1),;,(p
,)(h2
||exp '),;,0(p
densityy probabilitn transitiowith the),,0[ , ))(),...,(),(()(
processdiffusion shomogeneouy temporall theof law the weakly toconverges ) . ( , as ,0 and fixedany For
. as with offunction increasingan be Let (i)
1
,1
1
2/2/
1
2
21
,
2
2
Nii
ijNj jiii
N
j
NNN
jNkj
kNNNN
N
NN
N
TLN
LL
tB
NitdttYtY
tdBtdY
t
jtcTts
xxstts
tct
ttYtYtYt
LT
LTLT
=
≠≤≤
=
−−<
≤<≤
<
=∞∈−
+=
Γ=∈≤<≤
−=−=
⎭⎬⎫
⎩⎨⎧−×=
∞∈=
∞→>∈
∞→∞→
∑
∏
∏
Y
Ryx
xxyx
yx
yyy0
Y
Zx x
π
μ
Next we consider the case that the noncolliding
time-period
T=TL goes to infinity.
23
REMARKS:
When all the particles are starting from the origin 0,
The Process Y(t)
= NONCOLLIDING Browniam
Motions
2)(e 2
1),;,0(p 11
2/2
iNji
j
N
k
tkyN yy
tt −×
⎭⎬⎫
⎩⎨⎧∝ ∏∏
≤<≤=
−
πy0
Product of Independent Gaussian Distributions
Strong Repulsive Interactions
. 0),;,0(p ,0|| As →→− y0 tyy Nij
24
REMARKS:
Here we set N = 3 as an example.
The process
Y(t) is identified with Dyson’s Brownian motion model
( ) ( )( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−−−−−−
×=)|,G()|,G()|,G()|,G()|,G()|,G()|,G()|,G()|,G(
det)(h)(h,,,;,,,p
333231
232221
131211
321321
xystxystxystxystxystxystxystxystxyst
yyytxxxsN
NN x
y
h-transform
in the sense of Doob
Karlin-McGregor formulaLindstrom-Gessel-Viennot
formula
. )()(h ,kernel)(heat
)(2
)(exp)(2
1)|,(
1
2
iNji
jN xxx
styx
stxystG
−=
⎭⎬⎫
⎩⎨⎧
−−
−−
=−
∏≤<≤
π
25
Dyson’s Brownian motion model1.
For all t > 0, with Probability 1.
2.
The process is given as a solution of the stochastic differential equations,
where Bi (t) are independent standard Brownian motions i =1, 2 ,…, N .
Nt <∈RY )(
[ )∞∈≤≤−
+= ∑≠≤≤
,0,1)()(
1)()(,1:
tNidttYtY
tdBtdYijNjj ji
ii
• Strong repulsive forces
among any pair of particlesdistance particle1
∝
26Remark.
HERMITIAN MATRIX-VALUED PROCESS
AND DYSON’S BROWNIAN MOTION MODEL
•Let be mutually independent (standard one-dim.) Brownianmotions started from the origin. Define
,,1),(~),( NjitBtB ijij ≤≤
⎪⎪⎩
⎪⎪⎨
⎧
>−
=
<
=
⎪⎪⎩
⎪⎪⎨
⎧
>
=
<
=
)()(~2
1)(0
)()(~2
1
)(
)()(2
1)()(
)()(2
1
)(
jitB
ji
jitB
ta
jitB
jitB
jitB
ts
ij
ij
ij
ij
ii
ij
ij
•Consider the Hermitian
Matrix-Valued
stochastic processNN ×( ) ( )
NjiijijNjiij tatstt≤≤≤≤
−+==Ξ,1,1
)(1)()()( ξThat is,
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
−−−−−−
−+−−−−−+−+−−−+−+−+
=Ξ
)(...)(1)()(1)()(1)(..............................
)(1)(...)()(1)()(1)()(1)(...)(1)()()(1)()(1)(...)(1)()(1)()(
)(
332211
333323231313
222323221212
111313121211
tstatstatstats
tatststatstatstatstatststatstatstatstatsts
t
NNNNNNNN
NN
NN
NN
27
•Consider the variation of the matrix, It is clear that
And by the previous observation, we find that
They are summarized as
•
Since is a Hermitian
matrix-valued process,at each time t there is a Unitary Matrix
, such that
where the eigenvalues
are in the increasing order
•
We can regard
as an N-particle stochastic process in one dimension.
)(tΞ
( )Njiij tdtd
≤≤=Ξ
,1)()( ξ
),1(0)( Njitd ij ≤≤=ξ
( )
( ) )1()(
)1()()()()(
)1(0)(
2
*
2
Nidttd
Njidttdtdtdtd
Njitd
ii
ijijjiij
ij
≤≤=
≤≠≤==
≤≠≤=
ξ
ξξξξ
ξ
),,,1()()( Nnkjidttdtd jkinknij ≤≤= δδξξ
Njiij tutU ≤≤= ,1))(()({ })(),....,(),(diag)()()()( 21 tttttUttU Nλλλ=Λ=Ξ+
[ )∞∈∀≤≤≤ ,0),(....)()( 21 tttt Nλλλ
( ) NN tttt Rλ ∈≡ )(),....,(),()( 21 λλλ
28
QUESTIONBy the diagonalization
of the matrix, what kind of interactions emerge
among the N particles in the process ?)(tλ
•From now on we assume that
•And we consider the following conditional configuration-spaceof one-dim. N particles,
(This is called the Weyl
chamber of type . )
)0(....)0()0( 21 Nλλλ <<<
{ }NNA
N xxx <<<∈= ....: 21RxW
1−NA
29
ANSWER 1 (by Dyson 1962)1.
For all t > 0, with Probability 1.
2.
The process is given as a solution of the stochastic differential equations,
where are independent standard one-dim. Brownian motions .
ANt Wλ ∈)(
[ )∞∈≤≤∑−
+=≠≤≤
,0,1)()(
1)()(,1:
tNidttt
tdBtdijNjj ji
ii λλλ
)(tBi )1( Ni ≤≤
• This process is called Dyson’s Brownian motion model.
• Strong repulsive forces
emerge among any pair of particles distance particle1
∝
30
• Let
•Consider
It solves the Fokker-Planck (FP) equation in the form
( )⎪⎩
⎪⎨
⎧
=≤≤∂∂
=∑−
=
−∏=
≠≤≤
≤<≤
)(),....,(),()( , )1( )h( ln1)(
)sdifference ofproduct ( )()h(
21,1:
1
xxxxbxx
x
Ni
ijNjjji
i
iNji
j
bbbNixxx
b
xx
[ ]. )...., , ,( ), ...., , ,( where
,)( to)( fromdensity y probabilittransition),;,p(
2121 NN yyyxxxtsts
=====
yxyλxλyx
),;,p()(),;,p(21),;,p( yxxbyxyx xx tststs
t∇⋅+Δ=
∂∂
ANSWER 2Introduce a determinant
Then the solution of the FP equation is given by
•
If (all particles starting from the origin)
[ ] tj
yi
x
ijijNji txytxytt
2/2)(
,1e
21)|,G( with )|,G(det)|,f(
−−
≤≤==
πxy
. , ,0for )h()|,f()h(
1),;,p( ANtsstts Wyxyxy
xyx ∈∞<<<−=
,0at )0,....,0,0( ==→ s0x
∏Γ=∑=⎭⎬⎫
⎩⎨⎧−=
==
− N
i
NN
ii
NiCy
tCtt
1
2/1
1
2222
1
2/2
. )()2( , || where)h(2
||exp),;,0p( πyyyy0
31
Noncolliding
Diffusion Processes
Random Matrix TheoryMatrix Models in Statistical Phys.
andString Theory, etc…..
Infinite Systems of Noncolliding
Diffusion Particles
Theory of Entire Functions
Representation Theory
ofLie Algebra/Lie Groups
I hope …..Statistical PhysicsProbability Theory
Other Fields of
(Infinite) Particle Systems
Noncolliding Processes Mathematics
Remaks
limit functions ncorrelatio of
sexpression taldeterminan
∞→N
Enumerative CombinatoricsYoung TableauxSymmetric FunctionsVicious Walker Model
Diffusion Scaling limit
32
6. Temporally Inhomogeneous Noncolliding
Brownian Motions
. || ),2/(/2
,(0,0,...0) where,, ,0for
,),(N
),(N)|,(f),;,(g
),,(N)(2
||exp),;,0(g
densityy probabilitn transitiowith the],,0[ , ))(),...,(),(()(
The
1
22
1
2/4/)1(2/
,
1
2
,
21
2
∑∏
∏
==
−−−
<
≤<≤
=Γ=
=∈≤<≤
−−−
=
−−⎭⎬⎫
⎩⎨⎧−×=
∈=
N
jj
N
j
NNNN
NN
NNTN
NNji
ijTN
N
yjtTc
Tts
sTtTstts
tTyyt
ct
TttXtXtXt
y
0Ryx
xyxyyx
yyy0
X process diffusion ousinhomogene temporally
. ),0[ , )( lim)( processdiffusion shomogeneouy temporallThe∞∈= ∞→ ttt T XY
33Case t = T
Case ∞→T
34
Transition in Time of Particle Distribution• This observation implies that there occurs a transition.
For a finite but large T
Ttxxyt
tg jNkj
k
N
iTN i <<<−
⎭⎬⎫
⎩⎨⎧−∝ ∏∑
≤<≤=
0for )(21exp),;,0( 2
11,
2y0
As the time
t goes on from 0
to T
TtxxyT
Tg jNkj
k
N
iTN i =−
⎭⎬⎫
⎩⎨⎧−∝ ∏∑
≤<≤=
at )(21exp),;,0(
11,
2y0
35
Three Standard (Wigner-Dyson) Random Matrix Ensembles
[1]
The distribution of Eigenvalues
of Hermitian
Matricesin the Gaussian Unitary Ensemble (GUE)
is given in the form
[2]
The distribution of Eigenvalues
of Real Symmetric Matricesin the Gaussian Orthogonal Ensemble (GOE)
is given in the form
[3]
The distribution of Eigenvalues
of Quternion
Self-Dual HermitianMatrices
in the Gaussian Symplectic
Ensemble (GSE)
is given in the form
2)(2
1exp11
22 j
Nkjk
N
ii λλλ
σ−
⎭⎬⎫
⎩⎨⎧− ∏∑
≤<≤=
NN ×
NN ×
NN ×
)(2
1exp11
22 j
Nkjk
N
ii λλλ
σ−
⎭⎬⎫
⎩⎨⎧− ∏∑
≤<≤=
4)(2
1exp11
22 j
Nkjk
N
ii λλλ
σ−
⎭⎬⎫
⎩⎨⎧− ∏∑
≤<≤=
36•Let be mutually independent standard Brownian motions
started from the origin. Define,,1),(~),( NjitBtB ijij ≤≤
bridges)(Brownian ].,0[ ,1 , )(
)(~)( where
,
)()(2
1)(0
)()(2
1
)(
)()(2
1)()(
)()(2
1
)(
0
TtNjidssTs
tBt
jit
ji
jit
ta
jitB
jitB
jitB
ts
tij
ijij
ij
ij
ij
ij
ii
ij
ij
∈≤<≤−
−=
⎪⎪
⎩
⎪⎪
⎨
⎧
>−
=
<
=
⎪⎪
⎩
⎪⎪
⎨
⎧
>
=
<
=
∫β
β
β
β
•Consider the Hermitian
Matrix-Valued
stochastic processNN ×
( ) ( )NjiijijNjiijTN tatstt
≤≤≤≤−+==Ξ
,1,1, )(1)()()( ξThat is,
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
−−−−−−
−+−−−−−+−+−−−+−+−+
=Ξ
)(...)(1)()(1)()(1)(..............................
)(1)(...)()(1)()(1)()(1)(...)(1)()()(1)()(1)(...)(1)()(1)()(
)(
332211
333323231313
222323221212
111313121211
,
tstatstatstats
tatststatstatstatstatststatstatstatstatsts
t
NNNNNNNN
NN
NN
NN
TN
37
•
Since is a Hermitian
matrix-valued process,at each time t there is a Unitary Matrix
, such that
where the eigenvalues
are in the increasing order
•
We can regard
as an N-particle stochastic process in one dimension.
We have proved the following theorem.
)(tΞ
Njiij tutU ≤≤= ,1))(()({ })(),....,(),(diag)()()()( 21 tttttUttU Nλλλ=Λ=Ξ+
[ )∞∈∀≤≤≤ ,0),(....)()( 21 tttt Nλλλ
( ) NN tttt Rλ ∈≡ )(),....,(),()( 21 λλλ
TheoremThe eigenvalue
process
is equivalent
in distribution
with
the system of
noncolliding
Brownian motions
X(t)with the initial condition X(0)=0 (i.e. all particles start from
the origin),
which are obtained as the scaling limit of vicious walker model.
( ))(),....,(),()( 21 tttt Nλλλ≡λ
387. PATTERNS of NONCOLLIDING PATHS
AND RANDOM MATRIX THEORIES7.1 STAR CONFIGURATIONS• There occurs a transition in distribution from GUE to GOE.
•
This temporal transition can be decribed
by the Two-Matrix Model
of Pandey
and Mehta, in which a Hermitian
random matrix is coupled with a real symmetric random matrix.See Katori and Tanemura, PRE 66
(2002) 011105/1-12.•
Techniques developed for multi-matrix models
can be used to evaluate the dynamical correlation functions. Quaternion determinantal
expressions
are derived.See Nagao, Katori and Tanemura, Phys. Lett. A307 (2003) 29-35.
• Using the exact correlation functions, we can discuss the scaling limits
ofinfinite particles and the infinite time-period .
See Katori, Nagao and Tanemura, Adv.Stud.Pure
Math. 39 (2004) 283-306.∞→N ∞→T
39
7.2 Watermelon Configurations• Consider a finite time-period [0,T] and set
y=0
at the initial time t=0 and the final time t=T.• The transition probability density is given as
• The distribution is kept in the form of GUE.• Only the variance
changes as a function of t as .
( )2)h(
/12||exp11),;,0(q2
1
2/2
watermelon yy0⎭⎬⎫
⎩⎨⎧
−−
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ −=
−
Ttty
Ttt
Ct
N
⎟⎠⎞
⎜⎝⎛ −=
Ttt 12σ
40
7.3 Banana Configurations• Consider 2N particle system. Set y=0
at the initial time
t=0.
At the final time
t=T, we assume the following Pairing of Particle Positions.
• The transition probability density is given by
• As , there occurs a transition from the GUE distribution to the GSE distribution.
. .... with ,.... , , 12312124321 −− <<<=== NNN yyyyyyyyy
. )|,G()|,G(det),(N where
, , ,0for ),(N
),(N)|,f(),;,0(q
1,21
2
banana
banana
bananabanana
⎥⎦⎤
⎢⎣⎡
∫=
∈≤<<−
=
≤≤≤≤ iji
ijNjNi
AN
xyttxxytxt
TtsT
tTtt
ANW
Wyxx
yxyyx
Tt →= 0
41
7.4 Star Configurations with Absorbing Wall• Put an Absorbing Wall
at the origin. Consider the N Brownian particles started from 0
conditioned never to collide with each other nor to collide with the wall.• This is identified with the h-transform of the N-dim. Absorbing Brownian motion in
• For , we can obtain a process showing a transition fromthe class C
distribution
of Altland
and Zirnbauer
(1996);
to the class CI distribution
(studied for a theory of quantum dots)
{ } ). typeofchamber (Weyl ....0: 21 NNNC
N Cxxxx <<<<∈= RW
∞<T
Ttyyyt
tN
kki
Njij
C <<<−⎭⎬⎫
⎩⎨⎧−∝ ∏∏
=≤<≤0for )(
2||exp),;,0(q
1
222
1
22yyx
.at )(2
||exp),;,0(q1
2
1
22
TtyyyT
TN
kki
Njij
C =−⎭⎬⎫
⎩⎨⎧−∝ ∏∏
=≤<≤
yyx
42
7.5 Star Configurations with Reflection Wall• Put a reflection wall
at the origin. Consider the N Brownian particles started form 0
conditioned never to collide with each other.• This is identified with the h-transform of the N-dim. Absorbing Brownian motion in
{ } ). typeofchamber (Weyl ....|:| 21 NNND
N Dxxxx <<<∈= RW
For , we can obtain a process showing a transition from the class D distribution
of Altland
and Zirnbauer
(1996);
to the ``real”
class D distribution
∞<T
Ttyyt
t iNji
jD <<<−
⎭⎬⎫
⎩⎨⎧−∝ ∏
≤<≤0for )(
2||exp),;,0(q 22
1
22yyx
.at )(2
||exp),;,0(q 2
1
22
TtyyT
T iNji
jD =−
⎭⎬⎫
⎩⎨⎧−∝ ∏
≤<≤
yyx
43
7.6 Banana Configurations with Reflection Wall• Put a reflection wall
at the origin.
• Consider the 2N Brownian particles started from 0 in Banana configurations.
• For , we can obtain a process showing a transition from the class D distribution
of Altland
and Zirnbauer
To the class DIII
distribution.Ttyy
tt i
Njij
D <<<−⎭⎬⎫
⎩⎨⎧−∝ ∏
≤<≤0for )(
2||exp),;,0(q 22
1
22
, banana yyx
∞<T
.at )(||exp),;,0(q1
122
121
212
24
oddbanana , Ttyyy
TT
N
kki
Njij
D =∏−∏⎭⎬⎫
⎩⎨⎧−∝
=−−
≤<≤−
yyx
447.7 CONCLUDING REMARKS
• There are 10 CLASSES
of Gaussian Random Matrix Theories.
Standard (Wigner-Dyson)GUE
Star configurations
GOEGSE
Banana configurations
Nonstandard (chiral
random matrices)
Particle Physics of QCDchGUEchGOE
Realized by Noncolliding
Systems of
chGSE
2D Bessel processes
and Generalized MeandersNonstandard (Altland-Zirnbauer)
Mesoscopic
Physics with Superconductivity
class Cclass CI
Star config. with Absorbing Wall
class Dclass DIII
Banana config. With Reflection Wall
All of the 10 eigenvalue-distributions can be realized by the Noncolliding
Diffusion Particle Systems (Vicious Walks).
See Katori and Tanemura, J.Math.Phys.(2004)
45
Noncolliding
Diffusion Processes
Random Matrix TheoryMatrix Models in Statistical Phys.
andString Theory, etc…..
Infinite Systems of Noncolliding
Diffusion Particles
Theory of Entire Functions
Representation Theory
ofLie Algebra/Lie Groups
I hope …..Statistical PhysicsProbability Theory
Other Fields of
(Infinite) Particle Systems
Noncolliding Processes Mathematics
8. Remaks
(again)
limit functions ncorrelatio of
sexpression taldeterminan
∞→N
Enumerative CombinatoricsYoung TableauxSymmetric FunctionsVicious Walker Model
Diffusion Scaling limit
46
小林奈央樹 et. al. : Phys. Rev. E 78 (2008) 051102