Noncolliding Brownian Motion and Determinantal Processes

INTRODUCTION

In the present paper, we discuss noncolliding Brownian motion (BM) in one-dimension with finite number $N$ of particles and its infinite particle limits $N\to\infty$. The condition imposed to $N$ particles in the model, not to collide with each other, causes “entropy forces” between all pairs of particles, which are repulsive long-ranged interactions proportional to the inverse of distances between particles. When we draw sample paths of $N$ particles of the system on the spatio-temporal plane, random sets of nonintersecting $N$ paths are obtained. Viewing them as random patterns of polymers or phase boundaries on a plane, the present system has been used as a model of polymer networks , or a model showing wetting (or melting) transitions in statistical physics; see also . Recently many authors have reported that notion of noncolliding BM and its discrete counterpart called vicious walk is very useful to analyze the polynuclear growth models , time-dependent correlations of quantum spin chains , traffic problems , and the Chern-Simons theory .

In order to demonstrate the important connection between the random matrix theory and the noncolliding BM here we show a couple of observations of the three-dimensional Bessel process. The noncolliding BM can be regarded as a multivariate generalization of the three-dimensional Bessel process given below.

Let $B_{1}(t),B_{2}(t),B_{3}(t)$ be one-dimensional standard BMs (see Section 2.1 for definition). They are assumed to be independent and we consider a $2\times 2$ traceless hermitian matrix

where ${\rm i}=\sqrt{-1}$. Since the four entries $(M^{(1)}_{jk}(t))_{1\leq j,k\leq 2}$ are BMs, $M^{(1)}(t),t\in[0,\infty)$ is regarded as a matrix-valued process, which describe a diffusion process in the space of $2\times 2$ traceless hermitian matrices, which is identified with the three-dimensional real space ${\bf R}^{3}$ (${\bf R}$ denotes the set of all real numbers). At each time $t\in[0,\infty)$, it will be diagonalized by an appropriate unitary matrix and the eigenvalue is given by $\pm X(t)$ with

If we consider a Brownian particle in ${\bf R}^{3}$, \mbox{\boldmathB}(t)=(B_{1}(t),B_{2}(t),B_{3}(t)),t\in[0,\infty), the distance of the particle from the origin (i.e., the radial coordinate of \mbox{\boldmathB}(t) ) is given by (1.2) and thus it equals the eigenvalue process $X(t)$ associated with the matrix-valued process $M^{(1)}(t)$. This is called the three-dimensional Bessel process in probability theory (see, e.g. ), and a simple application of the Itô formula gives its stochastic differential equation (SDE) as

where $\widetilde{B}(t)$ is another one-dimensional standard BM than the above $B_{j}(t),j=1,2,3$. Corresponding to the SDE (1.3), the backward Kolmogorov (Fokker-Planck) equation for the transition probability density $p_{X}(t,y|x)$, starting from $x>0$ at time $t=0$ and arriving at $y>0$ at time $t>0$, is given by

and its solution with $\lim_{t\to 0}p_{X}(t,y|x)=\delta(x-y)$ is obtained as

where $h(x)=x$ and $p(t,y|x)$ is the heat kernel given by (2.1) in Section 2.1. The reflection principle of BM can be used to prove that $p_{\rm abs}(t,y|x)$ is the transition probability density from $x>0$ to $y>0$ during time $t$ of the absorbing BM, in which an absorbing wall is set at the origin and any particle is absorbed if it arrives at the wall. It is a matter of course that $h(x)=x$ is harmonic, $\partial^{2}h(x)/\partial x^{2}=0$, but we dare to say that $p_{X}$ is the harmonic transform ($h$-transform) of $p_{\rm abs}$ looking at (1.5). We introduce another $2\times 2$ matrix

for \mbox{\boldmathx}=(x_{1},x_{2}),\mbox{\boldmathy}=(y_{1},y_{2})\in{\bf R}^{2}, and consider its determinant

In summary the three-dimensional Bessel process $X(t),t\in[0,\infty)$ has two different realizations; (i) the eigenvalue-process of $2\times 2$ hermitian-matrix valued process (1.1), and (ii) the $h$-transform of the absorbing BM with a wall at $x=0$. We also observed that the transition probability density of the absorbing BM has a determinantal expression of a $2\times 2$ matrix (1.6)-(1.8). If we consider the two-dimensional BM, it is represented by motion of a point in the two-dimensional space $(x_{1},x_{2})\in{\bf R}^{2}$. We put an absorbing boundary on a line $x_{2}=x_{1}$ and trace the motion of the point in the region {\bf W}_{2}=\{\mbox{\boldmathx}=(x_{1},x_{2})\in{\bf R}^{2}:x_{1}<x_{2}\}. The transition probability density from \mbox{\boldmathx}=(x_{1},x_{2})\in{\bf W}_{2} to \mbox{\boldmathy}=(y_{1},y_{2})\in{\bf W}_{2} is generally given by the determinant (1.7). As a special case of it (with a time-change $t\to t/2$), (1.8) is given.

2 Dyson’s BM Model, Karlin-McGregor Formula and Noncolliding BM

The noncolliding BM, \mbox{\boldmathX}(t)=(X_{1}(t),X_{2}(t),\cdots,X_{N}(t)),t\in[0,\infty) is a conditional diffusion process. It has the following two kinds of realizations.

(i) In order to generate the random matrix ensembles, Dyson introduced $N\times N$ matrix-valued diffusion processes . For the Gaussian unitary ensemble (GUE), $N^{2}$ independent one-dimensional BMs are used to assign entries of matrix to satisfy the condition that the matrix is hermitian at any time. Eigenvalues are real and define an $N$-particle system in one dimension called Dyson’s BM model (with the parameter $\beta=2$ corresponding to GUE). This process solves the SDE (see for the proof using generalized Bru’s theorem)

where \mbox{\boldmathB}(t)=(B_{1}(t),\cdots,B_{N}(t)) is an $N$-dimensional BM; $B_{j}(t),1\leq j\leq N$, are independent one-dimensional standard BMs. The SDE (1.9) is an $N$-variable generalization of the SDE for the three-dimensional Bessel process (1.3). It was proved that with probability one Dyson’s BM is non-colliding .

(ii) We consider the following subset of ${\bf R}^{N}$,

It is called the Weyl chamber of type $A_{N-1}$ . The absorbing BM is defined by putting absorbing walls at all boundaries of the region ${\bf W}_{N}$, whose transition probability density, when the process starts from \mbox{\boldmathx}\in{\bf W}_{N} at time $t=0$ and arrives at \mbox{\boldmathy}\in{\bf W}_{N} at time $t>0$, is given by

This determinantal expression is known as the Karlin-McGregor formula . (Such a determinantal formula for nonintersecting paths is known as the Lindströn-Gessel-Viennot formula in the enumerative combinatorics, see also .) The noncolliding BM is given by the $h$-transform of the absorbing BM

We note that there appear three different kinds of matrices. The matrices representing Dyson’s matrix-valued process (eq.(1.1) for the simplest case), matrices in the Karlin-McGregor determinants (eqs.(1.7) and (1.11)) and that in the Vandermonde determinant (1.13). Of course, the equivalence of Dyson’s BM model (the eigenvalue process of the first kind of matrices) with the noncolliding BM implies direct connection between the random matrix theory and stochastic processes. In the present paper, however, we will show that the Karlin-McGregor formula is much more important. In Section 3 we will show that the Vandermonde determinant appears in the Schur function expansion of the Karlin-McGregor determinant. With the combination of these two determinants, the orthogonal polynomial method is applicable to study the processes. This method has been developed to analyze multi-matrix models in the random matrix theory .

3 Matrix-Kernels and Determinantal Processes

In Section 2, we will explain that the Hermite orthonormal functions are useful to represent BMs. Precise descriptions of facts briefly mentioned above will be given in Section 3. Staring from Karlin-McGregor’s determinantal expression of transition probability density, we will prove in Section 4 that, if we specify the initial configuration as the GUE-eigenvalue distribution, the generating function of multitime correlation functions is given by a Fredholm determinant for the noncolliding BM, and thus multitime correlation functions are generally given by determinants (Theorem 4.1). The system whose spatial correlations are given by determinants is usually called a determinantal point field (or a determinantal point process) in probability theory . Theorem 4.1 states that the noncolliding BM is not only a determinantal point field at any fixed time $0\leq t<\infty$, but it is also a determinantal point field on the spatio-temporal plane. We say that the noncolliding BM is a (finite) determinantal process to express this situation.

Determinantal processes are generally determined by their matrix-kernels (see, for example, ). The matrix-kernel of our finite noncolliding BM is expressed by the Hermite orthonormal functions $\{\varphi_{k}\}_{k=0}^{\infty}$, which is called the extended Hermite kernel in . In Section 5 we will show that the asymptotic properties of $\{\varphi_{k}\}_{k=0}^{\infty}$ completely determine infinite particle limits of the matrix-kernel and then the determinantal process. Appropriate scaling limits are performed and two infinite particle systems are derived from the noncolliding BM. One of them is the spatially homogeneous infinite determinantal process with matrix-kernel expressed by trigonometric functions (Theorem 5.1) and another is the spatially inhomogeneous one with matrix-kernel expressed by Airy functions (Theorem 5.2). The former kernel is called the extended sine kernel and the latter the extended Airy kernel in . We will claim in Section 6 that these three determinantal processes (one finite and two infinite systems) and others reported in references have a common structure; the matrix-kernels are expressed by spectral projections associated with appropriate self-adjoint operators (effective Hamiltonians) . As explained by Spohn and by Prähofer and Spohn , this common feature is shared also with the 1+1 dimensional Fermi field in quantum mechanics (see also ). It may be due to the similarity between the Karlin-McGregor formula for noncolliding systems and the Slater determinant for free fermion systems with the Fermi exclusion principle . For finite and infinite determinantal processes with matrix-kernels associated with spectral projections, we will prove that the determinantal processes are continuous in time (Lemma 7.1) and discuss the bilinear forms derived from correlation functions (Proposition 7.2). Future problems are given in Section 8.

BROWNIAN MOTION AND HERMITE POLYNOMIALS

Let $(\Omega,{\cal F},P)$ be the probability space. One-dimensional standard BM starting from a point $x_{0}\in{\bf R}$ is defined as a real-valued stochastic process $\{B(t,\omega):t\in[0,\infty)\}$, which satisfies the following conditions (see, for example, ). Here $\omega$ is a label on sample path, $\omega\in\Omega$.

1. $B(0,\omega)=x_{0}$ with probability 1.

2. For any fixed $\omega\in\Omega$, $B(t)$ is a continuous real-function of $t$ with probability 1. (With this property we say that the paths are continuous in time.)

3. For any series of times $t_{0}\equiv 0<t_{1}<\cdots<t_{M},M=1,2,\cdots$, $\{B(t_{m+1})-B(t_{m})\}_{m=0,1,\cdots,M-1}$ are independent and they are normally distributed with mean 0 and variance $t_{m+1}-t_{m}$.

Then if we introduce an integral kernel (Gaussian kernel)

the probability that the BM stays in an interval $[a_{m},b_{m}]$, $-\infty<a_{m}<b_{m}<\infty$, at each time $t_{m}$, $m=1,2,\cdots,M$, is given by

That is, the transition probability density of the BM is given by (2.1). Since it satisfies the one-dimensional diffusion equation (heat equation)

with the initial condition $u(0,x)=\delta(x-x^{\prime})$, it is specially called the heat kernel.

We consider the following transformation of variables, $(t,x)\mapsto(\tau,\zeta)$ ;

Then the diffusion equation (2.2) is transformed to

Note that ${\cal H}_{\rm H}$ is identified with the Hamiltonian in the coordinate($\zeta)$-representation of the one-dimensional harmonic oscillator in quantum mechanics, if we set the mass of the oscillator $m=1$, the circular frequency $\omega=1$, and $\hbar=1$.

Following Dirac’s description , we consider the real-valued Hilbert space with basis $\{|\zeta\rangle:\zeta\in{\bf R}\}$, which is orthonormal $\langle\zeta|\zeta^{\prime}\rangle=\delta(\zeta-\zeta^{\prime})$ and complete

Let $\widehat{\cal H}_{\rm H}$ be the operator such that $\langle\zeta^{\prime}|\widehat{\cal H}_{\rm H}|\zeta\rangle=\delta(\zeta^{\prime}-\zeta){\cal H}_{\rm H}$ with (2.6). We consider a state vector $|\Psi(\tau)\rangle$, which follows the equation

If we multiply $\langle\zeta|$ to (2.8) from the left, we will have the equation (2.5) with $U(\tau,\zeta)=\langle\zeta|\Psi(\tau)\rangle$. Given $|\Psi(\tau^{\prime})\rangle,\tau^{\prime}\in(-\infty,\infty)$, the solution of (2.8) for $\tau\geq\tau^{\prime}$ is obtained as

where (2.7) was used. Insert this result into (2.4), we have

This is the transition probability density previously given as (2.1).

2 Hermite Polynomials and Equalities

Let ${\bf N}=\{1,2,3,\cdots\}$ and ${\bf N}_{0}={\bf N}\cup\{0\}$. The eigenvalues of $\widehat{\cal H}_{\rm H}$ are $n+1/2$ with $n\in{\bf N}_{0}$; $\widehat{\cal H}_{\rm H}|n\rangle=\left(n+1/2\right)|n\rangle.$ Here $\{|n\rangle:n\in{\bf N}_{0}\}$ denotes the set of eigenvectors of $\widehat{\cal H}_{\rm H}$, which is orthonormal $\langle n|n^{\prime}\rangle=\delta_{n,n^{\prime}}$ and complete $\sum_{n=0}^{\infty}|n\rangle\langle n|=1$. Let $\{H_{n}(x):n\in{\bf N}_{0}\}$ be the Hermite polynomials

where $[z]$ denotes the largest number not greater than $z$. They are orthogonal

Then $\{\varphi_{n}(\zeta):n\in{\bf N}_{0}\}$ are orthonormal and

This expression for the heat kernel (2.1) is called Mehler’s formula .

It is easy to see that orthonornality and completeness of $\{|\zeta\rangle:\zeta\in{\bf R}\}$ imply $\langle t,x|t,x^{\prime}\rangle=\delta(x-x^{\prime})$ and

By (2.16), we have the equalities for $0\leq t_{1}\leq t_{2}\leq t_{3}$

Though these seem to be trivial, if we note that (2.9) is rewritten as

they become meaningful; for $0\leq t_{1}\leq t_{2}\leq t_{3}$

The equation (2.18) is called the Chapman-Kolmogorov equation. The equalities (2.19) mean invariance of the functions $\phi_{n},n\in{\bf N}_{0}$, with respect to the heat kernel.

NONCOLLIDING SYSTMES

Here we introduce the noncolliding BM in a finite time-period $(0,T)$ with $T>0$. It is defined as an $N$-particle system of one-dimensional standard BMs conditioned not to collide with each other in $(0,T)$. As mentioned in Section 1.2, the transition probability density of the absorbing BM in the Weyl chamber ${\bf W}_{N}$ of type $A_{N-1}$ is given by (1.11) as an application of Karlin-McGregor formula . The probability that \mbox{\boldmathB}(t) starting from \mbox{\boldmathx}^{\prime}\in{\bf W}_{N} stays in ${\bf W}_{N}$ up to at least time $t>0$ is given by

The transition probability density of noncolliding BM is then given by

for $0<t^{\prime}\leq t<T$, \mbox{\boldmathx},\mbox{\boldmathx}^{\prime}\in{\bf W}_{N} . It should be noted that this process is in general temporally inhomogeneous. In the following, we will consider the $T\to\infty$ limit to make the process be homogeneous in time.

2 Schur Function Expansion

By multilinearity of determinant (1.11) with (2.1),

where |\mbox{\boldmathx}|^{2}\equiv\sum_{j=1}^{N}x_{j}^{2}. Consider F_{N}(\mbox{\boldmathx},\mbox{\boldmathy})=\det_{1\leq j,k\leq N}[e^{x_{j}y_{k}}] for a pair of multivariates \mbox{\boldmathx}=\{x_{j}\}_{j=1}^{N} and \mbox{\boldmathy}=\{y_{j}\}_{j=1}^{N}. By definition of determinant, F(\mbox{\boldmathx},\mbox{\boldmathy}) is skew-symmetric under any exchange of indices of $\{x_{j}\}$ and also it is for $\{y_{j}\}$; Let ${\cal S}_{N}$ be the symmetric group of $N$ variables (the set of all permutations of $N$ variables) and for $\sigma\in{\cal S}_{N}$ write \sigma(\mbox{\boldmathx})=(x_{\sigma(1)},\cdots,x_{\sigma(N)}). Then for any $\sigma\in{\cal S}_{N}$ F_{N}(\sigma(\mbox{\boldmathx}),\mbox{\boldmathy})=F_{N}(\mbox{\boldmathx},\sigma(\mbox{\boldmathy}))={\rm sgn}(\sigma)F_{N}(\mbox{\boldmathx},\mbox{\boldmathy}). A fundamental skew-symmetric polynomials of multivariate $x$ is given by a product of differences (the Vandermonde determinant) (1.13). The quotient of F_{N}(\mbox{\boldmathx},\mbox{\boldmathy}) divided by h_{N}(\mbox{\boldmathx})h_{N}(\mbox{\boldmathy}) is a symmetric function both of $x$ and $y$. The following lemma shows an expansion of the symmetric part using the Schur functions, which are labeled by partitions $\mu=(\mu_{1},\mu_{2},\cdots)$, sets of nonnegative integers in decreasing order $\mu_{1}\geq\mu_{2}\geq\cdots$, and defined by

Proof. By multilinearity of determinant, we have

We can see that for any symmetric function f(\mbox{\boldmathn}) of \mbox{\boldmathn}=(n_{1},\cdots,n_{N})\in{\bf N}_{0}^{N},

Since $\det_{1\leq j,k\leq N}[x_{j}^{n_{k}}]=0$ if $n_{k_{1}}=n_{k_{2}}$ for any pair $1\leq k_{1}\not=k_{2}\leq N$, (3.5) equals

Here we change the variables in summation from $\{n_{j}\}$ to $\{\mu_{j}\}$ by $\mu_{j}=n_{j}-N+j,1\leq j\leq N$. Using (3.3) we obtain the first equation of (3.4). Since $s_{\mu}({\bf 0})=0$ unless $\mu={\bf 0}\equiv(0,0,\cdots,0)\in{\bf N}_{0}^{N}$, and $s_{0}({\bf 0})=1$, the estimation in |\mbox{\boldmathx}|\to 0 is given as shown by the second equation of (3.4). ∎

with $C_{N}=(2\pi)^{N/2}\prod_{j=1}^{N}\Gamma(j)$. The integral formula

is found in (eq.(17.6.7) p.321) as a variation of the Selberg integral , whose proof was given in . If we set $\gamma=1/2,a=1/2t$ and note that the integral over ${\bf R}^{N}$ can be replaced by the integral over ${\bf W}_{N}$ multiplied by $N!$, we have

with $C_{N}^{\prime}=2^{N/2}\prod_{j=1}^{N}\Gamma(j/2)$. Similarly by setting $\gamma=1$ and $a=1/2t$, we have

Using (3.7) with (3.6), we obtain the asymptotics of ${\cal N}_{N}$,

3 Temporally Homogeneous Limit

By the above estimate (3.9), we can take the $T\to\infty$ limit in (3.2) and obtain the transition probability density, which is homogeneous in time, i.e., a function of time difference $t-t^{\prime}$,

From now on, we consider the noncolliding BM, which is defined by this transition probability density. It is a temporally homogeneous process and we denote it by \mbox{\boldmathX}(t)=(X_{1}(t),X_{2}(t),\cdots,X_{N}(t)).

Remark 1. The product of differences (the Vandermonde determinant) h_{N}(\mbox{\boldmathx}) given by (1.13) is a harmonic function in the sense

which has strictly positive values at interior points of ${\bf W}_{N}$ and zero at the boundary. Eq. (3.10) is considered as a transformation from $f_{N}$ to $p_{N}$ associated with the harmonic function, which is called the $h$-transform . That is, the temporally homogeneous noncolliding BM is an $h$-transform by $h_{N}$ of the absorbing BM in the Weyl chamber ${\bf W}_{N}$ . It is easy to confirm that p_{N}(t,\,\cdot\,|\mbox{\boldmathx}) satisfies the following backward Kolmogorov equation

which is a multivariate extension of (1.4). It implies that the process \mbox{\boldmathX}(t) is a diffusion process, which solves the SDE (1.9) of Dyson’s BM model (with $\beta=2$), which describes the time-evolution of eigenvalues of hermitian matrix-valued (GUE) diffusion process . This equivalence between Dyson’s BM model for GUE (the eigenvalue part of the matrix-valued BM) and the present noncolliding BM (BM conditioned not to collide) is a multi-dimensional extension of the equivalence between the three-dimensional Bessel process (the radial coordinate of the three-dimensional BM) and the one-dimensional BM conditioned to stay in the positive region $B(t)>0$, as announced in Section 1. See also .

Let \nu_{0}(\mbox{\boldmathx}) be the probability density at time $t_{0}>0$. Then the probability density of distribution \mbox{\boldmathX}(t),t\geq t_{0} is given by

if the distribution $\nu_{0}$ has finite moment. Note that the integral formula (3.8) guarantees that (3.12) is normalized. It should be noted that the distribution (3.12) is equal to the eigenvalue distribution of hermitian random-matrices in GUE with variance $\sigma^{2}=t$ .

For any initial distribution having finite moment, the asymptote of probability density of distribution of the noncolliding BM, \mbox{\boldmathX}(t), in $t\to\infty$ is expressed by the eigenvalue distribution of GUE with variance $t$.

DETERMINANTAL PROCESS

Consider a sequence of times, $0<t_{0}<t_{1}<\cdots<t_{M},M=1,2,\cdots$ for observations of distribution of \mbox{\boldmathX}(t). Given the initial distribution $\nu_{0}$ at time $t_{0}$, multitime probability density is given using (3.10) as

Now we assume that $\nu_{0}$ is the GUE-eigenvalue distribution with variance $t_{0}$,

Then (4.1) becomes the product of $f_{N}$’s multiplied by a factor

By multilinearity of determinant and the fact that the coefficient of the highest order term of the Hermite polynomial $H_{n}(x)$ is $(2x)^{n}$ (see (2.10)), the following equality is established,

Then if we define the multivariate functions

where $\phi_{n}$ and $\widehat{\phi}_{n}$ are given by (2.21) and (2.22), respectively, the factor (4.3) is readily shown to be equal to \mu_{0}(\mbox{\boldmathx}^{(0)})\mu_{M}(\mbox{\boldmathx}^{(M)}). That is, the multitime probability density (4.1) is written as

For \mbox{\boldmathx}^{(m)}\in{\bf R}^{N}, $0\leq m\leq M$, and $N^{\prime}=1,2,\dots,N$, we put \mbox{\boldmathx}^{(m)}_{N^{\prime}}=\left(x_{1}^{(m)},x_{2}^{(m)},\dots,x_{N^{\prime}}^{(m)}\right). For a sequence $\{N_{m}\}_{m=0}^{M}$ of positive integers less than or equal to $N$, we define the $(N_{0},\dots,N_{M})$-multitime correlation function by

Expectations with respect to the configurations \{\mbox{\boldmathX}(t_{0})\},\{\mbox{\boldmathX}(t_{1})\},\dots,\{\mbox{\boldmathX}(t_{M})\} are denoted by ${\bf E}_{N}$ :

Remark 2. Set t^{\prime}=0,t=t_{0}>0,\mbox{\boldmathx}=\mbox{\boldmathx}^{(0)} in (3.10) and consider the limit \mbox{\boldmathx}^{\prime}\to{\bf 0}. By (3.6), we can see

This fact is essentially the same thing with that stated as Proposition 3.2 for the scaling property of the process. The GUE-eigenvalue distribution (4.2) adopted here as the initial distribution at time $t_{0}>0$ is immediately realized if we start the present noncolliding system at time 0 from $\{{\bf 0}\}$. The state $\{{\bf 0}\}$, which is a boundary of ${\bf W}_{N}$, is entrance .

2 Generating Function

Let $C_{0}({\bf R})$ be the set of all continuous real functions with compact supports. For \mbox{\boldmathf}=(f_{0},f_{1},\cdots,f_{M})\in C_{0}({\bf R})^{M}, and \mbox{\boldmath\theta}=(\theta_{0},\theta_{1},\cdots,\theta_{M})\in{\bf R}^{M}, the generating function for multitime correlation functions is defined for the process \{\mbox{\boldmathX}(t)\},t\in[0,T] as

and write (4.9) as $\Psi_{N}[\chi]$. Then by the definition of multitime correlation function (4.6), we have

By the definition (4.9) with (LABEL:eqn:expect) and (4.5), we have

By definition of determinant, it is easy to prove the following identity for square integrable continuous functions $g_{j},\bar{g}_{j},1\leq j\leq N$,

which is called the Heine identity. By repeated applications of this identity we have

By the Chapman-Kolmogorov equation (2.18) and the invariance (2.20), $F_{jk}=\delta_{jk}$ and then $\Psi_{N}=1$, which implies that (4.5) is indeed normalized. If we use the notations introduced in Section 2, it is written as

Now we introduce an indicator $\hat{1}_{+}$ such that

Then (4.13) can be written of the form of an infinite series

and define an $M\times M$ matrix-kernel $\widehat{\chi}_{+}$, whose $(m,n)$-element is given by the integral kernel (4.15). The $(m,n)$-element of its $q$-th power $(\widehat{\chi}_{+})^{q},q=2,3,\cdots,$ will be the following kernel given by $(q-1)$-multiple integral,

Let $\hat{1}^{m,n}(x,y)\equiv\delta_{m,n}\delta(x-y)$. Since it is easy to see that

3 Fredholm Determinant and Determinantal Process

Let $\widetilde{B}_{j}^{(m)}(x)$ and $\widetilde{C}_{j}^{(m)}(x)$, $0\leq m\leq M,1\leq j\leq N$, be square integrable continuous functions. Then the following formulae can be proved;

The expansion formula of the last expression defines the Fredholm determinant, which is abbreviated as

The generating function (4.12) with (4.18) and (4.19) is then expressed as the Fredholm determinant,

and we have used {\rm Det}\Big{[}[\hat{1}-\widehat{\chi}_{+}]^{m,n}(x,y)\Big{]}=1, which is concluded from the fact that $[\hat{1}-\widehat{\chi}_{+}]^{m,n}(x,y)=0$ for $m<n$ by the definition of $\widehat{\chi}_{+}$, (4.15) with (4.14), and it is $\delta(x-y)$ for $m=n$.

Following the formulae given in Section 2, $S_{N}^{m,n}$ is determined as

Combination of this with (2.14) gives $\widetilde{S}_{N}^{m,n}$ as

where ${\bf 1}_{\{\omega\}}$ is the indicator of a condition $\omega$; ${\bf 1}_{\{\omega\}}=1$ if $\omega$ is satisfied and ${\bf 1}_{\{\omega\}}=0$ otherwise.

As shown above, not $\{\widetilde{S}_{N}^{m,n}(x,y)\}$ themselves, but determinants of matrices made of them are observables. By definition of determinant, factors $\{e^{-x^{2}/4t_{m}+y^{2}/4t_{n}}\}$ of $\{\widetilde{S}_{N}^{m,n}(x,y)\}$ in (4.26) are completely cancelled out, when we calculate determinants. So here we define the following matrix-kernel by omitting these factors in $\{\widetilde{S}_{N}^{m,n}(x,y)\}$,

This Fredholm determinant is by definition expanded as

Comparison of (4.29) and (4.10) determines all of the multitime correlation functions. Now we summarize the above results as a theorem.

The temporally homogeneous noncolliding BM, \Xi^{\bf X}_{N}(t)=\{\mbox{\boldmathX}(t)\}, starting from the GUE-eigenvalue distribution (4.2) at time $t_{0}>0$, is a finite determinantal process in the following sense.

(ii) Any multitime correlation function is given by a determinant; for any $M\geq 0$, any sequence $\{N_{m}\}_{m=0}^{M}$ of positive integers less than or equal to $N$, any time sequence $t_{0}<t_{1}<\cdots<t_{M}<\infty$, the $(N_{0},\dots,N_{M})$-multitime correlation function is given by

The following relations hold for the Hermite polynomials,

From (4.31), the Christoffel-Darboux formula is derived for the Hermite orthonormal functions $\{\varphi_{k}(x)\}_{k\in{\bf N}_{0}}$,

for $x\not=y$. Eq. (4.32) can be used to evaluate the limit $y\to x$ in (4.33) and we find

Then the matrix-kernel have the following simpler expressions if $m=n$,

INFINITE PARTICLE SYSTEMS

The density of $\Xi^{\bf X}_{N}(t)$ is given by

as a special case ($M=0,N_{0}=1$ with setting $t_{0}=t$) of Theorem 4.1 with (4.34). It is easy to confirm that $\int_{-\infty}^{\infty}\rho_{N}(t,x)dx=N$ by the orthonormality of $\varphi_{k}(x)$. The following estimations for asymptote in $N\to\infty$ are established . Let $\varepsilon$ and $\omega$ be the fixed positive numbers. We have

Using them, we will have the asymptote of the density profile in $N\to\infty$,

The distribution of $N$ particles has a finite support, whose interval $\propto\sqrt{N}$, and thus $\rho\sim\sqrt{N}\to\infty$ as $N\to\infty$ for fixed $0<t<\infty$. If we set $x=2\sqrt{Nt}\xi$, we see

which is known as Wigner’s semicircle law . See also . In the following we consider scaling limit, in which long-term limit $t\to\infty$ is taken at the same time with $N\to\infty$.

2 Bulk Scaling Limit and Homogeneous Infinite System

First we consider the central region $x\simeq 0$ in the semicircle-shaped profile of particle density in the scaling limit

In this limit the system becomes homogeneous also in space with a constant density $\rho=1/\pi$. We call this the bulk scaling limit.

For any $M\geq 0$, any sequence $\{N_{m}\}_{m=0}^{M}$ of positive integers, and any strictly increasing sequence $\{s_{m}\}_{m=1}^{M}$ of positive numbers

Proof. For any $u\in{\bf R}$, the formula

for $N\gg 1$ with fixed number $\alpha$. Then (4.27) with $m\leq n$ is evaluated in $N\to\infty$ as

In particular, when $m=n$, i.e., $s_{n}-s_{m}=0$, the integration is readily performed to have $\int_{0}^{1}du\,\cos(u(y-x))=\sin(y-x)/(y-x)$. Similar evaluation in $N\to\infty$ can be done also for (4.27) with $m>n$. ∎

In the bulk scaling limit (5.3), the temporally and spatially homogeneous infinite particle system is obtained, whose multitime correlation functions are given by (5.4). The matrix-kernel (5.5) is called the extended sine kernel in . In the present paper, we will call it simply “sine kernel”. The system with the sine kernel was studied by Spohn , Osada , and Nagao and Forrester , as an infinite particle limit of Dyson’s BM model with $\beta=2$ . See also .

3 Soft-edge Scaling Limit and Spatially Inhomogeneous Infinite System

Since (5.7) gives $x^{2}/2t\simeq 2N$, the vicinity of the right edge of semicircle-shaped profile (5.1) will be closed up, and we will obtain a spatially inhomogeneous infinite particle system in this scaling limit. Following the random matrix theory , we call (5.7) the soft-edge scaling limit.

In order to describe the limit, we introduce the Airy function

In the proof of the following theorem, we will use the formula

and \mbox{\boldmathx}_{N^{\prime}}(s)=(a_{N}(s)+x_{1},a_{N}(s)+x_{2},\cdots,a_{N}(s)+x_{N^{\prime}}).

For any $M\geq 0$, any sequence $\{N_{m}\}_{m=0}^{M}$ of positive integers, and any strictly increasing sequence $\{s_{m}\}_{m=1}^{M}$ of positive numbers

Proof. Putting $k=N-p-1$ in the summation of (4.27) for $m\leq n$, we have

With the same reason mentioned above eq. (4.27), we can omit the factor $(t_{n}/t_{m})^{(N-1)/2}$. Since, when we set $t_{m}=N^{1/3}+2s_{m}$,

Note that the factor $(t_{n}/t_{m})^{(N-1)/2}$, which is irrelevant in calculating determinants, was omitted in the second line in the above equations. Put $u=-\lambda$ to obtain the expression (5.13). Similar evaluation in $N\to\infty$ of (4.27) can be done also for $m>n$. ∎

The infinite system obtained by the soft-edge scaling limit (5.7) is temporally homogeneous, but spatially inhomogeneous as shown by the “Airy kernel” (5.13). Prähofer and Spohn and Johansson studied the right-most path in the present system and called it the Airy process $A(t)$. For a given $t>0$, $A(t)$ has distribution of the celebrated Tracy-Widom distribution, which is governed by the Painlevé II equation . Recently Tracy and Widom derived a system of partial differential equations (PDE), which govern the Airy kernel (5.13) . They also discussed other determinantal processes by PDE . See also .

DETERMINANTAL PROCESSES ASSOCIATED WITH SPECTRAL PROJECTIONS

where $\widehat{\cal H}_{\varphi}$ is given by (2.12), and ${\cal P}_{\varphi}$ is a projection operator defined by

It is called the extended Hermite kernel in .

can be regarded as the generalized eigenfunctions of the Hamiltonian

with spectrum $\lambda>0$. Here $S_{-}$ is an odd function (parity ${\sf p}=-$) and $S_{+}$ is an even function (parity ${\sf p}=+$), respectively. Consider an operator $\widehat{\cal H}_{\rm sin}$ such that $\langle x|\widehat{\cal H}_{\rm sin}|y\rangle=\delta(x-y){\cal H}_{\rm sin}$ with (6.3) and introduce a set of its eigenvectors $\{|\lambda,{\sf p};{\rm sin}\rangle:\lambda>0,{\sf p}=\pm\}$;

Since $\langle x|\lambda,{\sf p};{\rm sin}\rangle=\langle\lambda;{\sf p};{\rm sin}|x\rangle=S_{{\sf p}}(\sqrt{\lambda}x)$, we can confirm the completeness of the set

If we change the variable in the Airy differential equation (5.9) by $x\to u-\lambda$, we have

has ${\bf R}$ as spectrum and the Airy functions of the form ${\rm Ai}(x-\lambda)$ are its generalized eigenfunctions. We can consider the corresponding operator $\widehat{\cal H}_{\rm Ai}$ and its eigenvectors $\{|\lambda;{\rm Ai}\rangle:\lambda\in{\bf R}\}$,

where $\langle x|\lambda;{\rm Ai}\rangle=\langle\lambda;{\rm Ai}|x\rangle={\rm Ai}(x-\lambda)$. We find the completeness

Remark 3. The $2(\nu+1)$-dimensional squared Bessel process (BESQν) $Y^{(\nu)}(t)$ is defined as a solution of the SDE

where $B(t)$ is a one-dimensional standard BM . Its forward and backward Kolmogorov (Fokker-Planck) equations are given as

where $-$ for the forward and $+$ for the backward equations, respectively. We will see that the Laguerre polynomials $\{L_{k}^{\nu}(x)\}$ play for BESQν the similar role to the Hermite polynomials $\{H_{k}(x)\}$ for the BM shown in Section 2. By considering the noncolliding system of BESQνs instead of BMs , we can derive a finite determinantal process whose matrix-kernel is described using the orthonormal Laguerre functions (extended Laguerre kernel ). If we take the so-called hard-edge scaling limit, a spatially inhomogeneous infinite particle system is obtained, which is the determinantal process associated with the matrix-kernel

where $J_{\nu}(x)$ is the Bessel function, $J_{\nu}(x)=\sum_{n=0}^{\infty}(-1)^{n}(x/2)^{2n+\nu}/\{\Gamma(n+1)\Gamma(n+1+\nu)\}$, and $J_{\nu}^{\prime}(x)=dJ_{\nu}(x)/dx$. This kernel was called the extended Bessel kernel in . See also . We can see that $J_{\nu}(2\sqrt{\lambda x})$ is the generalized eigenfunction of the Hamiltonian

with spectrum $\lambda\geq 0$. We introduce the corresponding operator $\widehat{\cal H}_{J}$ and its eigenvectors $\{|\lambda;J\rangle:\lambda\geq 0\}$, $\widehat{\cal H}_{J}|\lambda;J\rangle=\lambda|\lambda;J\rangle$, where $\langle x|\widehat{\cal H}_{J}|y\rangle=\delta(x-y){\cal H}_{J}$, $\langle x|\lambda;J\rangle=\langle\lambda;J|x\rangle=J_{\nu}(2\sqrt{\lambda x})$ with the completeness

if we assign the Hamiltonian $\delta(x-y){\cal H}=\langle x|\widehat{\cal H}|y\rangle$ and projection operator as follows instead of $\widehat{\cal H}_{\varphi}$ and (6.2);

(i) for the sine kernel, set ${\cal H}={\cal H}_{\rm sin}$ with (6.3) and $\displaystyle{{\cal P}_{\rm sin}=\sum_{{\sf p}=\pm}\int_{0}^{1}d\lambda\,|\lambda,{\sf p};{\rm sin}\rangle\langle\lambda,{\sf p};{\rm sin}|,}$

(ii) for the Airy kernel, set ${\cal H}={\cal H}_{\rm Ai}$ with (6.4) and $\displaystyle{{\cal P}_{\rm Ai}=\int_{-\infty}^{0}d\lambda\,|\lambda;{\rm Ai}\rangle\langle\lambda;{\rm Ai}|.}$

(iii) for the Bessel kernel, set ${\cal H}={\cal H}_{J}$ with (6.8) and $\displaystyle{{\cal P}_{J}=\int_{0}^{1}d\lambda\,|\lambda;J\rangle\langle\lambda;J|.}$

2 Effective Hamiltonians and Matrix-Kernels

In the previous subsection, we claimed that the structure of the matrix-kernel of determinantal correlation functions is common both in finite particle systems and infinite particle systems. It should be noted, however, that even from the same finite system (e.g. noncolliding BM governed by the Hamiltonian ${\cal H}_{\varphi}$), depending on scaling limits, different kinds of infinite determinantal systems are derived, in which the Hamiltonian is replaced by appropriate effective Hamiltonians (e.g. ${\cal H}_{\rm sin}$ and ${\cal H}_{\rm Ai}$) and spectral projection operator is modified.

In the present subsection, we give a possible general consideration on the common structure of determinantal processes. First we note the following fact for a general form of effective Hamiltonian ${\cal H}=-a(x)\partial^{2}/\partial x^{2}-b(x)\partial/\partial x-c(x)$, where $a(x),b(x),c(x)$ are sufficiently smooth functions with $a(x)\not=0$ in an interval $\Lambda\subset{\bf R}$. If we change the variable $x\mapsto z$ following

${\cal H}\to\widetilde{\cal H}=-\partial^{2}/\partial z^{2}-\widetilde{b}(z)\partial/\partial z-c(z)$ with $\widetilde{b}(z(x))=a(x)z^{\prime\prime}(x)+b(x)z^{\prime}(x)=a(x)^{-1/2}\{-a^{\prime}(x)/2+b(x)\}$. Then we define

and if we perform a similarity transformation $\widetilde{\cal H}\to\overline{\cal H}=r^{-1}\widetilde{\cal H}r$, the term of first derivative can be eliminated and we have the form $\overline{\cal H}=-\partial^{2}/\partial z^{2}+q(z)$. The transformation ${\cal H}\to\overline{\cal H}$ is called the Liouville transformation. Then, without loss of generality, we can assume effective Hamiltonians of the form (the Sturm-Liouville operator )

Example 1. The effective Hamiltonians ${\cal H}_{\varphi},{\cal H}_{\rm sin},{\cal H}_{\rm Ai}$, and ${\cal H}_{J}$ are transformed to the form (6.9) with the following $q(x)$, respectively

where $\Lambda={\bf R}$ for the first three cases and $\Lambda={\bf R}_{+}\equiv\{x\in{\bf R}:x>0\}$ for the last case.

Let $\widehat{\cal H}$ be the operator corresponding to (6.9); $\langle x|\widehat{\cal H}|y\rangle=\delta(x-y){\cal H}$, and define

Assume that $\widehat{\cal H}$ has a distribution $\omega(d\lambda)$ of spectrum $\sigma(\widehat{\cal H})=\{\lambda:\widehat{\cal H}|\lambda\rangle=\lambda|\lambda\rangle\}$ with the complete set of eigenvectors $\{|\lambda\rangle:\lambda\in\sigma(\widehat{\cal H})\}$. Then the spectral representation of $\delta_{t}(x,y)$ is given by

with $\Phi_{\lambda}(x)=\langle x|\lambda\rangle=\langle\lambda|x\rangle$. We assume that

Example 2. For the four examples (6.10), we have the following explicit expressions of $\delta_{t}(x,y)$;

where $I_{\nu}(z)$ is the modified Bessel function given by $I_{\nu}(z)=\sum_{n=0}^{\infty}(z/2)^{2n+\nu}/\{n!\Gamma(\nu+n+1)\}.$ We can confirm that (6.14) is satisfied in these four cases.

Now we consider a subset of spectrum $\sigma(\widehat{\cal H})$, $\sigma_{-}(\widehat{\cal H})=\{\lambda\in\sigma(\widehat{\cal H}):\lambda\leq\lambda_{*}\}$ with a specified level $\lambda_{*}\in\sigma(\widehat{\cal H})$, and define the projection operator onto $\sigma_{-}(\widehat{\cal H})$

Moreover, by the completeness of $\{|x\rangle:x\in\Lambda\}$, $\int_{\Lambda}dx|x\rangle\langle x|=1$, we have the relations

and consider the determinantal process \Xi(t)=\{\mbox{\boldmathX}(t)\}, whose multitime correlation function is given by

for any $M\geq 0$, any sequence $\{N_{m}\}_{m=0}^{M}$ of positive integers, and any series of observation times $0<t_{1}<\cdots<t_{M}$.

The invariant measure of the process is the determinantal point field $\mu$ associated with $K(x,y)$ given by (6.17).

The two-time correlation function \rho(0,\{\mbox{\boldmathx}_{m}\};t,\{\mbox{\boldmathy}_{n}\}) of the system is given by

For a matrix $A=(a_{ij})_{i\in I,j\in J}$ with index sets $I,J$, we denote its submatrix as $A_{I^{\prime}J^{\prime}}\equiv(a_{ij})_{i\in I^{\prime},j\in J^{\prime}}$ for $I^{\prime}\subset I,J^{\prime}\subset J$, and the complementary submatrix as $A^{I^{\prime}J^{\prime}}\equiv(a_{ij})_{i\in I\setminus I^{\prime},j\in J\setminus J^{\prime}}$. By using the relation (6.19), \rho(0,\{\mbox{\boldmathx}_{m}\};t,\{\mbox{\boldmathy}_{n}\}) is expanded as

It was shown by Shirai and Takahashi that the Palm measure $\mu^{z}$ coincides with the determinantal point field associated with the kernel $K^{z}$ defined by

and set $\alpha,\beta\in{\bf C}$. Define $F(\xi)=e^{\alpha\langle f,\xi\rangle}$, $G(\xi)=e^{\beta\langle g,\xi\rangle}$. Then the two-time generating function is defined by

CHARACTERIZATION OF DETERMINANTAL PROCESSES

In the previous section, we introduced a class of determinantal processes associated with spectral projections defined by effective Hamiltonians. Here we give properties of determinantal processes of this class, which can be derived from the common structure of correlation functions.

Let $C_{0}^{\infty}({\bf R})$ be the set of all infinitely differentiable real functions with compact supports and set $\langle f,\xi\rangle\equiv\sum_{x\in\xi}f(x)$ for $f\in C_{0}^{\infty}({\bf R}),\xi\in\mathfrak{X}$. By a criterion of Kolmogorov (see, for example,), the following lemma implies that $\sum_{x\in\Xi(t)}f(x)$ is continuous in time with probability one for any $f\in C_{0}^{\infty}({\bf R})$. Since $\mathfrak{X}$ is separable with the vague topology, we can choose a countable set $\{f_{j}\}_{j=1}^{\infty}\subset C_{0}^{\infty}({\bf R})$ such that $\xi_{n}\to\xi$ in $n\to\infty$ on $\mathfrak{X}$ if and only if $\langle\xi_{n},f_{j}\rangle\to\langle\xi,f_{j}\rangle$ in $n\to\infty,\,\forall j\geq 1$. Then it implies that the determinantal process $\Xi(t)$ is continuous in the vague topology with probability one. That is, if the condition (6.14) is satisfied, it can be expressed by

with some real-valued continuous processes $X_{j}(t),j\in{\bf N}$.

Let $\Xi(t)$ be the determinantal process, whose multitime correlation functions are given by (6.25) with (6.16) and (6.24) associated with an effective Hamiltonian (6.9) defined on $\Lambda\subset{\bf R}$. Assume that (6.14) is satisfied. Then for any $f\in C_{0}^{\infty}({\bf R})$

Remark 4. Theorems 5.1 and 5.2 are the limit theorems of the processes

Proof of Lemma 7.1. Here we use the notation

The left-hand-side of (7.1) is given by $\partial^{4}\Phi_{t}(-f,f;\alpha,\alpha)/\partial^{4}\alpha|_{\alpha=0}$ from the two-time generating function given by (6.29) and (6.30), which equals

Since $\rho_{3,0}(x_{1},x_{2},x_{3})=\rho_{0,3}(x_{1},x_{2},x_{3}),\rho_{2,1}(x_{1},x_{2},x_{3})=\rho_{1,2}(x_{3},x_{2},x_{1}),$ etc., the above quantity is twice of

We put {\bf M}_{m,n}(\mbox{\boldmathx}_{m+n})={\bf M}(t,x_{m+1},\dots,x_{m+n}|x_{1},\dots,x_{m}), {\bf D}(\mbox{\boldmathx}_{n})={\bf D}(t,\mbox{\boldmathx}_{n}|\mbox{\boldmathx}_{n}) and

We divide $I$ into four terms $I=\sum_{j=1}^{4}I_{j}$ with

for the proof, where $C_{j}$’s do not depend on $t$. Using eqs. (6.22) and (6.23), we obtain

By the assumption (6.14) we obtain (7.3) for $j=2$.

Then we have (7.3) for $j=3$. This completes the proof. ∎

2 Bilinear Forms

Since the Fredholm determinant of the dual operator coincides with that of the original operator, the reversibility (and also the stationarity) of the present determinantal processes is guaranteed.

Osada constructed $\mathfrak{X}$-valued reversible processes, which have determinantal point fields as their reversible stationary measures by Dirichlet form approach. With the Palm measure $\mu^{z}$ the Dirichlet form of Osada can be written as

for local smooth functions $F,G$ (see also ). By the general theory of Dirichlet forms , his processes are diffusion processes (i.e., continuous strong Markov processes). For our class of determinantal processes, we found the following fact (Proposition 7.2). It suggests that our determinantal processes are identified with Osada’s processes.

A function $F$ on the configuration space $\mathfrak{X}$ is said to be polynomial, if it is written of the form $F(\xi)=\widetilde{F}(\langle f_{1},\xi\rangle,\langle f_{2},\xi\rangle,\cdots,\langle f_{k},\xi\rangle)$ with a polynomial function $\widetilde{F}$ on ${{\bf R}}^{k},k\in{\bf N}_{0}$, where $f_{j}\in C_{0}^{\infty}({\bf R}),1\leq j\leq k$. Let $\wp$ be the set of all polynomial functions on $\mathfrak{X}$, which is a dense subset of $L^{2}(\mathfrak{X},\mu)$ ; the space of square integrable functions on $\mathfrak{X}$ with the determinantal point field $\mu$.

Let $\Xi(t)$ be the determinantal process, whose multitime correlation functions are given by (6.25). Then for $F,G\in\wp$ we have

Markov property of determinantal processes has been studied by Borodin and Olshanski . But we can not prove that the infinite determinantal processes in our class are Markovian. The equivalence between our processes and Osada’s is not yet established.

To show this proposition, it is enough to consider the case that $F$ and $G$ are of the form

with $k\in{\bf N}_{0}$, $\alpha_{m},\beta_{m}\in{\bf R}$, $f_{m},g_{m}\in C_{0}^{\infty}({\bf R}),1\leq m\leq k$. Then if we set $\chi_{1}(x)=\exp\left(\sum_{m=1}^{k}\alpha_{m}f_{m}(x)\right)-1$ and $\chi_{2}(x)=\exp\left(\sum_{m=1}^{k}\beta_{m}g_{m}(x)\right)-1$,

Since the Palm measure $\mu^{z}$ is a determinantal point field associated with \rho^{z}(\{\mbox{\boldmathx}_{m}\}) (see (6.27) ) the right-hand-side of (7.5) equals

Hence Proposition 7.2 can be derived from the following lemma.

Proof of Lemma 7.3. We use the expansion formula (LABEL:eqn:expansion1) of the two-time correlation function to calculate the integral

where we have used the abbreviation $\partial^{2}_{y}=\partial^{2}/\partial y^{2}$. By partial integration

Combining (7.6), (7.7), (7.8) and (7.9), we obtain Lemma 7.3. ∎

CONCLUDING REMARKS

The study on noncolliding BM and determinantal processes reported in this paper will be extended in several directions. We would like to give some of future problems below.

(1) In Section 4 we let the initial configuration $\nu_{0}$ be the GUE-eigenvalue distribution, and shown that the system is determinantal. If we let $\nu_{0}$ be the eigenvalue distribution in the Gaussian orthogonal ensemble (GOE),

Instead of the Heine identity (4.11), we should use the de Bruin identity

for integrable continuous functions $g_{j},1\leq j\leq N$. As shown in Appendix A of , for example, the generating function of multitime correlation functions is then expressed by the Fredholm Pfaffian and the system becomes a Pfaffian process, in the sense that any multitime correlation function is given by a Pfaffian. Such Pfaffian processes have been studied by many authors . The systems studied in are also Pfaffian processes, since the ‘quaternion determinantal expressions’ of correlation functions, introduced and developed by Dyson, Mehta, Forrester, and Nagao , are readily transformed to Pfaffian expressions. As implied by Proposition 3.2, the system exhibits a transition from GOE distribution to GUE distribution . Continuity of sample paths and general characterization of infinite Pfaffian processes will be interesting problems. The case with other initial distribution (in particular, when it has continuous parameters) will be interesting .

(2) As explained in Sections 1 and 3, the present noncolliding BM is the $h$-transform of the absorbing BM in the Weyl chamber (1.10) of type $A_{N-1}$. We can find appropriate $h$-transforms of the absorbing BMs in the Weyl chambers of types $C_{N}$ and $D_{N}$. The obtained noncolliding diffusion processes are stochastic versions of non-standard random matrix ensembles, which were called the class C and class D, respectively, by Altland and Zirnbauer . The stochastic version of the chiral GUE, realized by the noncolliding squared Bessel process, was studied by König and O’Connell , which is also obtained as an $h$-transform of the absorbing BM in the Weyl chamber of type $C_{N}$. See for more details. Systematic classifications of determinantal and Pfaffian processes will be important.

(3) There are many other examples of finite and infinite determinantal processes, which are not considered in the present paper. Markov processes on partitions (Young diagrams) have been studied and determinantal processes associated with other types of projections than ours have been reported . The determinantal processes, whose kernels are expressed using multiple orthogonal functions (e.g. the Pearcey kernel), are discussed in . The consideration given in Sections 6 and 7 in the present paper should be generalized.

The present authors would like to thank H. Osada, Y. Takahashi, A. Borodin and G. Olshanski for useful discussions on determinantal processes. M.K. is supported in part by the Grant-in-Aid for Scientific Research (KIBAN-C, No.17540363) of Japan Society for the Promotion of Science. H.T. is supported in part by the Grant-in-Aid for Scientific Research (KIBAN-C, No.19540114) of Japan Society for the Promotion of Science.

References