VICIOUS WALK and RANDOM 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

1. INTRODUCTION B(t)= ? B(s)=x 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 (>s) is given by This solves the Heat Equation (Diffusion Equation);

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.

2. HERMITIAN MATRIX-VALUED PROCESS AND DYSON’S 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,

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.

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 . )

ANSWER 1 (by Dyson 1962) For all t > 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 Dyson’s Brownian motion model. Strong repulsive forces emerge among any pair of particles

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)

REMARK 1: When all the particles are starting from the origin 0, Strong Repulsive Interactions Product of Independent Gaussian Distributions Dyson’s Brownian motion model = NONCOLLIDING Diffusion Particle Systems

Dyson’s Brownian motion model = a Free Fermion System 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-Gessel-Viennot formula in Combinatorics ) Dyson’s Brownian motion model = a Free Fermion System

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))

From the viewpoint of solid-state physics, 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.

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.

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

Two Limit Cases Case t=T Since (2) Case [3] The limit of the transition probability density is well defined as follows. Two Limit Cases Case t=T Since we have (2) Case

Case t = T Case

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

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

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.

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 .

4.3 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.

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)

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

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.

5. CONCLUDING REMARKS Standard (Wigner-Dyson) There are 10 CLASSES of Gaussian Random Matrix Theories. Standard (Wigner-Dyson) GUE Star configurations GOE GSE Banana configurations Nonstandard (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)

REMARK 3. Future Problems 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).

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) 369-390;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.