VICIOUS%20WALK%20and%20RANDOM%20MATRICES

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VICIOUS WALKandRANDOM MATRICES

• Makoto KATORI
• (Chuo University, Tokyo, Japan)
• Joint Work with
• Hideki 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
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1. INTRODUCTION

• Consider a Standard Brownian Motion B(t) in One
  Dimension.
• Stochastic Properties of the Variation

B(t) ?
B(s)x

• Transition Probability Density from x at time s
  to y at time t (gts) is given by

• This solves the Heat Equation (Diffusion
  Equation)

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• Next we consider a pair of independent Brownian
  Motions,

We define a complex-conjugate pair of
complex-valued stochastic variables,

• By definition

which give
We have a correlation between the
complex-conjugate pairs.
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2. HERMITIAN MATRIX-VALUED PROCESS AND
DYSONS BROWNIAN MOTION MODEL

• Let be
  mutually independent (standard one-dim.) Brownian
• motions started from the origin. Define

• Consider the Hermitian Matrix-Valued
  stochastic process

That is,
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• 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.

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QUESTION By the diagonalization of the matrix,
what kind of interactions emerge among the N
particles in the process ?

• From now on we assume that
• And we consider the following conditional
  configuration-space
• of one-dim. N particles,
• (This is called the Weyl chamber of type
  . )

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• ANSWER 1 (by Dyson 1962)
• For all t gt 0, with
  Probability 1.
• The process is given as a solution of the
  stochastic
• differential equations,
• where are independent standard one-dim.
  Brownian motions .

• This process is called Dysons Brownian motion
  model.
• Strong repulsive forces emerge among any pair
  of particles
  

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• Let
• 
  
• Consider
• It solves the Fokker-Planck (FP) equation in the
  form

• ANSWER 2
• Introduce a determinant
• Then the solution of the FP equation is given by
• If
  (all particles starting from the origin)

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REMARK 1 When all the particles are starting
from the origin 0,
Strong Repulsive Interactions
Product of Independent Gaussian Distributions
Dysons Brownian motion model NONCOLLIDING
Diffusion Particle Systems
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REMARK 2 Here we set N 3 as an example.
h-transform in the sense of Doob (Probab.Theory)
a stochasic version of Slater determinant (Karlin-
McGregor formula in Probab.Theory) (Lindstrom-Gess
el-Viennot formula
in Combinatorics )
Dysons Brownian motion model a Free Fermion
System
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3. VICIOUS WALKS As 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.
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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.

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Estimations 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.

SCHUR FUNCTION EXPANSION
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3 The limit of the transition
probability density is well defined as follows.

• Two Limit Cases
• Case tT Since
• we have
• (2) Case

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Case t T
Case
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• Transition in Time of Particle Distribution
• This observation implies that there occurs a
  transition.
• For a finite but large T

As the time t goes on from 0 to T
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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
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4. 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 of
• infinite particles and the
  infinite time-period .
• See Katori, Nagao and Tanemura, Adv.Stud.Pure
  Math. 39 (2004) 283-306.

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4.2 Watermelon Configurations

• Consider a finite time-period 0,T and set
• y0 at the initial time
  t0 and the final time tT.
• 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
  .

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4.3 Banana Configurations

• Consider 2N particle system. Set y0 at the
  initial time t0.
• At the final time tT, 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.

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4.4 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)

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4.5 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

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
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4.6 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.

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5. CONCLUDING REMARKS

• There are 10 CLASSES of Gaussian Random Matrix
  Theories.

Standard (Wigner-Dyson) GUE
Star configurations
GOE GSE
Banana configurations Nonstanda
rd (chiral random matrices) Particle
Physics of QCD chGUE
chGOE Realized by
Noncolliding Systems of chGSE
2D Bessel processes and
Generalized Meanders Nonstandard
(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)
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REMARK 3. Relations between Random Matrices and
Vicious Walks are very useful to analize other
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 matrices
• and 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).

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References
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) 369-390arXiv.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.
arXivmath-ph/0402061.