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1
VICIOUS WALKand
RANDOM MATRICES
Makoto KATORI(Chuo University, Tokyo, Japan)
Joint Work withHideki Tanemura (Chiba University) and
Taro Nagao (Nagoya University)
Nonequilibrium Statistical Physics of Complex Systems Satellite Meeting of STATPHYS 22 in Seoul, Korea, KIAS International Conference Room, 29 June-2 July 2004
21. INTRODUCTION
• Consider a Standard Brownian Motion B(t) in One Dimension. Stochastic Properties of the Variation .))((,0)( 2 dttdBtdB
•Transition Probability Density from x at time s to yy at time t (>s) is given by
.)(2
)(exp
)(2
1),;,(
2
st
yx
stytxsp
•This solves the Heat Equation (Diffusion Equation);
).(),;,(lim),,;,(2
1),;,( 2
2
yxytxspytxspx
ytxspt st
B(s)=x
B(t)= ?
3•Next we consider a pair of independent Brownian Motions, )(~),( tBtB
.00)(~)()(~)( ttBdtdBtBdtdB
We define a complex-conjugate pair of complex-valued stochastic variables,
)(~
1)(2
1)()(
~1)(
2
1)( * tBtBtxtBtBtx
•By definition
)(~
1)(2
1)()(
~1)(
2
1)( * tBdtdBtdxtBdtdBtdx
which give
dttBdtdB
tBdtdBtBdtdBtdxtdx
tBdtdBtBdtdBtdx
tBdtdBtBdtdBtdx
22
222
2
222
2
)(~)( 2
1
)(~1)(2
1)(~1)(
2
1)()(
0)(~)( 2
1)(~1)(
2
1)(
0)(~)( 2
1)(~1)(
2
1)(
*
We have a correlation between the complex-conjugate pairs.
42. HERMITIAN MATRIX-VALUED PROCESS AND DYSON’S BROWNIAN MOTION MODEL
•Let be mutually independent (standard one-dim.) Brownian motions 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
,1,1
)(1)()()( That is,
)(...)(1)()(1)()(1)(
...............
...............
)(1)(...)()(1)()(1)(
)(1)(...)(1)()()(1)(
)(1)(...)(1)()(1)()(
)(
332211
333323231313
222323221212
111313121211
tstatstatstats
tatststatstats
tatstatststats
tatstatstatsts
t
NNNNNNNN
NN
NN
NN
5
•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.
)(t
Njiij tdtd
,1
)()(
),1(0)( Njitd ij
)1()(
)1()()()()(
)1(0)(
2
*
2
Nidttd
Njidttdtdtdtd
Njitd
ii
ijijjiij
ij
),,,1()()( Nnkjidttdtd jkinknij
Njiij tutU ,1))(()(
)(),....,(),(diag)()()()( 21 tttttUttU N
,0),(....)()( 21 tttt N
N
N tttt Rλ )(),....,(),()( 21
6
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 N
N
NA
N xxx ....: 21RxW
1NA
7ANSWER 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 .
A
Nt Wλ )(
,0,1)()(
1)()(
,1:tNidt
tttdBtd
ijNjj jiii
)(tBi )1( Ni
• This process is called Dyson’s Brownian motion model.
• Strong repulsive forces emerge among any pair of particles
distance particle
1
8
• Let
•Consider
It solves the Fokker-Planck (FP) equation in the form
)(),....,(),()( , )1( )h( ln1
)(
)sdifference ofproduct ( )()h(
21,1:
1
xxxxbxx
x
N
iijNjj
ji
i
iNji
j
bbbNixxx
b
xx
. )...., , ,( ), ...., , ,( where
, )( to)( fromdensity y probabilit transition),;,p(
2121 NN yyyxxx
tsts
yx
yλxλyx
),;,p()(),;,p(2
1),;,p( yxxbyxyx xx tststs
t
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
2
1)|,G( with )|,G(det)|,f(
xy
. , ,0for )h()|,f()h(
1),;,p( A
Ntsstts Wyxyxyx
yx
,0at )0,....,0,0( s0x
N
i
NN
ii
N
iCytC
tt
1
2/1
1
2222
1
2/2
. )()2( , || where)h(2
||exp),;,0p( yy
yy0
9REMARK 1: When all the particles are starting from the origin 0,
Dyson’s Brownian motion model = NONCOLLIDING Diffusion Particle Systems
2)(e 2
1),;,0p(
11
2/2
iNji
j
N
k
tk
yyy
tt
y0
Product of Independent Gaussian Distributions
Strong Repulsive Interactions
. 0),;,0p( ,0|| As y0 tyy ij
10REMARK 2: Here we set N = 3 as an example.
Dyson’s Brownian motion model = a Free Fermion System
)|,G()|,G()|,G(
)|,G()|,G()|,G(
)|,G()|,G()|,G(
det)(
)(
,,,;,,,p
333231
232221
131211
321321
xystxystxyst
xystxystxyst
xystxystxyst
xx
yy
yyytxxxs
jiij
jiij
h-transform in the sense of Doob (Probab.Theory)
a stochasic version of Slater determinant(Karlin-McGregor formula in Probab.Theory)(Lindstrom-Gessel-Viennot formula in Combinatorics )
113. VICIOUS WALKSAs Temporally Inhomogeneous Noncolliding Particle Systems
Physical Motivations to Study Vicious Walker Models
•As a model of Wetting or Melting Transitions(Fisher (JSP 1984))
•As a model of Commensurate-Incommensurate Transitions(Huse and Fisher (PRB 1984))
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•As a model of Directed Polymer Networks(de Gennes (J.Chem.Phys. 1968), Essam and Guttmann (PRE 1995))
(a) polymer with star topology (b) polymer with watermelon topology
From the viewpoint of solid-state physics,we want to treat Large but Finite system with Boundary Effects.
13
t
N
dttAN
time toupother each with collide
from starting particlesBrownian Prob
)|,f( ),N(
not dox
yxyxw
NONCOLLIDING PROBABILITY:We can see that
Noncolliding Condition Imposed for Finite Time-Period (0,T]•We introduce a parameter T, which gives the time period in which the noncolliding condition is imposed.•The transition probability density of the Noncolliding Brownian Motions during time T from the state x at time s to the state y at time t is
•The following are satisfied.
A
N ,T, tssT
tTstts Wyx
x
yxyyx
0for ),N(
),Ν()|,f(),;,q(
. , ,0 , ),;,q(),;,q(),;,q( )3(
, 1),;,q( )2( , )(),;,q( lim )1(
A
N
A
Nst
AN
AN
Tutsusdutts
dtsts
Wzxzxyzyyx
Wxyyxyxyx
W
W
14
A
N ,T, tssT
tTstts Wyx
x
yxyyx
0for ),N(
),N()|,f(),;,q(
15Estimations of Asymptotics
By using the knowledge of symmetric functions called Schur Functions,we can prove the following three facts.[1] Exponent of Power-Law Decay of the Noncolliding Probability: For fixed initial positions
[2] The limit gives the Temporally Homogeneous process.
A
NWx
. in )1(4
1 with )h(~),N( tNNtt xx
T
. 0for ),;,p(),;,q(lim
tststsT
yxyx
. )1(
1
)polynomial(Schur detdet)(with
, )()()h()h(
expdet
1
,1,1
)( :
,1
N
ii
jN
iNji
jN
iNji
N
jiNji
iNa
xxs
ssayx
j
x
yxyx
SCHUR FUNCTION EXPANSION
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[3] The limit of the transition probability density is well defined as follows.0x
N
j
NNNN
jCtTtC
tT
tt
1
2/
2
2
2
2/24/)1(
0||
. )2/(2 with ),N()h(2
||exp
),;,0q( lim),;,0q(
yxy
yxy0x
Two Limit Cases(1) Case t=T Since
we have
(2) Case
, 1)()|,0f(),0N( AN
AN
ddWW
yyzyyzy
. )h(2
||exp),;,0q(
2
2
4/)1(
yy
y0
TC
TT
NN
T
2)h(2
||exp) ; ; ,0 p() , ; ,0 q( lim
2
1
2/2
yy
y0y0
tC
ttt
N
T
17Case t = T
Case T
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Transition in Time of Particle Distribution• This observation implies that there occurs a transition.
For a finite but large T
Ttxxyt
yt jNkj
k
N
ii
0for )(
2
1exp),;,0q( 2
11
20
As the time t goes on from 0 to T
TtxxyT
T jNkj
k
N
ii
at )(
2
1exp),;,0q(
11
2y0
19
Three Standard (Wigner-Dyson) Random Matrix Ensembles
[1] The distribution of Eigenvalues of Hermitian Matrices in the Gaussian Unitary Ensemble (GUE) is given in the form
[2] The distribution of Eigenvalues of Real Symmetric Matrices in the Gaussian Orthogonal Ensemble (GOE) is given in the form
[3] The distribution of Eigenvalues of Quternion Self-Dual Hermitian Matrices in the Gaussian Symplectic Ensemble (GSE) is given in the form
2)(2
1exp
11
22 j
Nkjk
N
ii
NN
NN
NN
)(2
1exp
11
22 j
Nkjk
N
ii
4)(2
1exp
11
22 j
Nkjk
N
ii
204. PATTERNS of NONCOLLIDING PATHS AND RANDOM MATRIX THEORIES
4.1 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
21
4.2 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
||exp1
1),;,0(q
2
1
2/2
watermelon yy0
Ttt
y
T
tt
Ct
N
T
tt 12
224.3 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
A
N
xytt
xxytxt
TtsT
tTtt
AN
W
Wyxx
yxyyx
Tt 0
234.4 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 from the class C distribution of Altland and Zirnbauer (1996);
to the class CI distribution (studied for a theory of quantum dots)
). typeofchamber (Weyl ....0: 21 NN
NC
N Cxxxx RW
T
Ttyyyt
tN
kki
Njij
C
0for )(
2
||exp),;,0(q
1
222
1
22y
yx
.at )(2
||exp),;,0(q
1
2
1
22
TtyyyT
TN
kki
Njij
C
yyx
244.5 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 NN
ND
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
22y
yx
.at )(2
||exp),;,0(q 2
1
22
TtyyT
T iNji
jD
yyx
254.6 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.
Ttyyt
t iNji
jD
0for )(
2
||exp),;,0(q 22
1
22
, banana yyx
T
.at )(||
exp),;,0(q1
12
2
121
2
12
24
oddbanana , Ttyyy
TT
N
kki
Njij
D
yyx
265. CONCLUDING REMARKS• There are 10 CLASSES of Gaussian Random Matrix Theories.
Standard (Wigner-Dyson) GUE Star configurations GOE GSE Banana configurationsNonstandard (chiral random matrices) Particle Physics of QCD chGUE chGOE Realized by Noncolliding Systems of chGSE 2D Bessel processes and Generalized MeandersNonstandard (Altland-Zirnbauer) Mesoscopic Physics with Superconductivity class C class CI Star config. with Absorbing Wall class D class 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, math-ph/0402061, to appear in J.Math.Phys.(2004)
27
REMARK 3.Relations between Random Matrices and Vicious Walks are very useful to analizeother nonequilibrium models, e.g. Polynuclear Growth Models. (See Sasamoto and Imamura, J. Stat. Phys. 115 (2004))
Future Problems• Calculate the dynamical correlation functions and determine the scaling limits of all these (temporally inhomogeneous) noncolliding systems. (some of them are done by Forrester, Nagao and Honner (Nucl.Phys.B553 (1999) )• The 10 classes are related with the diffusion processes on the flat symmetric spaces. Extensions to the noncolliding systems of diffusion particles in the space with positive curvature Ref: Circular Ensembles of random matricesand in the space with negative curvature. Ref. Theory of Quantum Wire: DMPK equations Beenakker and Rejaei, PRB 49 (1994), Caselle, PRL 74 (1995) Ref. Symmetric Spaces: Caselle and Magnea, Phys.Rep. 394 (2004).
TN and
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[1] M.Katori and H. Tanemura, Scaling limit of vicious walkers and two-matrix model, Phys.Rev. E66 (2002) 011105.[2] M.Katori and H. Tanemura, Functional central limit theorems for vicious walkers, Stoch.Stoch.Rep. 75 (2003) 369-390;arXiv.math.PR/0204386.[3] T.Nagao, M.Katori and H.Tanemura, Dynamical correlations among vicious random walkers, Phys.Lett.A307 (2003) 29-35.[4] J.Cardy and M.Katori, Families of vicious walkers, J.Phys.A Math.Gen. 36 (2003) 609-629.[5] M.Katori, T. Nagao and H. Tanemura, Infinite systems of non-colliding Brownian particles, Adv.Stud.in Pure Math. 39 ``Stochastic Analysis on Large Scale Interacting Systems”, pp.283-306, Mathematical Society of Japan, 2004; arXiv.math.PR/0301143.[6] M.Katori and N. Komatsuda, Moments of vicious walkers and Mobius graph expansion, Phys.Rev. E67 (2003) 051110.[7] M.Katori and H. Tanemura, Noncolliding Brownian motions and Harish-Chandra formula, Elect.Comm.in Probab. 8 (2003) 112-121; arXiv.math.PR/0306386.[8] M.Katori, H.Tanemura, T.Nagao and N.Komatsuda, Vicious walk with a wall, noncolliding meanders and chiral and Bogoliubov-deGennes random matrices, Phys.Rev. E68 (2003) 021112.[9] M.Katori and H. Tanemura, Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems, to appear in J.Math.Phys.; arXiv:math-ph/0402061.
References