数理解析研究所講究録 1422 巻 2005 年 12-25

12
Non-Colliding System of Brownian Particles
as Pfaffian Process
香取眞理

中央大学理工学部

Makoto Katori *Department of Physics, Faculty of Science and Engineering,
Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

In the paper [7] we studied the temporally inhomogeneous system of non-colliding Brownian motions and proved that multi-time correlation functions are generally given by thequaternion determinants in the sense of Dyson and Mehta. In this report we give anotherproof of the equivalent statement using Fredholm determinant and Fredholm pfaffian, andclaim that the present system is a typical example of pfaffian processes.

Non-Colliding Brownian Motions

1

By virtue of the Karlin-McGregor formula $[5, 6]$ , the transition density of the absorbing
Brownian motion in a Weyl chamber$\mathbb{R}_{<}^{N}=\{\mathrm{x}$$=$$(x_{1}, x_{2}, \ldots, x_{N})\in \mathbb{R}^{N}$; $x_{1}<$$<\cdots<x_{N}\}$$x_{2}$,

is given by$f_{N}.(t;\mathrm{x}, \mathrm{y})=\mathrm{d},\mathrm{e}\mathrm{t}1\leq i_{J}\leq N[p_{t}(x_{i_{7}}y_{j})]$where$p_{t}$,$\mathrm{x}$,$\mathrm{y}\in \mathbb{R}_{<}^{N}$,$t\in[0, \infty)$,

is the heat-kernel given by$p_{t}(x, y)= \frac{1}{\sqrt{2\pi t}}e^{-(x-y)^{2}/2t}$.

The integral$N_{N}(t; \mathrm{x})=\int_{\mathbb{R}_{<}^{N}}f_{N}(t;\mathrm{x}, \mathrm{y})d\mathrm{y}$,, gives the probability that a Brownian motion starting fromwhere
up to time $t>0$ .
does not hit the boundary of
For a given $T>0$ , we deine$d \mathrm{y}=\prod_{i=1}^{N}dy_{l}$$\mathrm{x}\in \mathbb{R}_{<}^{N}$$\mathbb{R}_{<}^{N}$(1.1)$g_{N,T}( \mathrm{S}, \mathrm{X}_{\}}^{\cdot}t, \mathrm{y})=.\frac{f_{N}(t-s,\mathrm{x},\mathrm{y})N_{N}(T-t,\mathrm{y})}{N_{N}(T-S_{)}\mathrm{X})}.\cdot$

for $0\leq s\leq t$
the state$\leq T$$\mathrm{x}\in \mathbb{R}_{<}^{N}$$|\mathrm{x}|$$arrow 0$

(1.2)

,

, ,
$\mathrm{x}$. It can bc regarded as the transition probability density fromto the state
at time . In $[8, 9]$ it was shown that asconverges to$\mathrm{y}\in \mathbb{R}_{<}^{N}$at time$g_{N}^{T}(0, \mathrm{x}; t, \mathrm{y})$

$s$

$t$

$\mathrm{y}\in \mathbb{R}_{<}^{N}$$g_{N,T}(0,0;t, \mathrm{y})\equiv C(N, T, t)h_{N}(\mathrm{y})\prod p_{t}(0, y_{i})N_{N}(T-Nt, \mathrm{y})$$i=1$’Electronic mail: lmtori@phys chuo-u. ac jp,

13where $C(N, T, t)= \pi^{N/2}(\prod_{j=1}^{N}\Gamma(j/2))^{-1}T^{N(N-1)/4}t^{-N(N-1)/2}$ , and$h_{N}( \mathrm{x})=\prod_{1\leq i<j\leq N}(x_{j}-x_{i})=\det[1\leq i,j\leq Nx_{j}^{i-1}]$.

The $N$ particle system of non-colliding Brownian motions
all started from the origin at$\mathrm{X}(0)=0$$=(0,0,\cdots,0)$, and conditioned not to collide with each other in a time
time 0, .
interval $(0, T]$ is defined by the process associated with the transition probability density
given by (1.1) and (1.2). This process is temporally inhomogeneous and it was obtained as a.
diffusion scaling limit of the vicious walker model in [9]. Figure 1 illustrates the process$\mathrm{X}(t)$$\mathrm{i}.e$$g_{N,T}$$\mathrm{X}(t)$Figure 1: Process$\mathrm{X}(t)$, $t\in[0, T]$ , with$\mathrm{X}(0)=0$.

Let be the space of countable subset 4 of satisfying $\#(\xi\cap K)<$ oc for any compact. Thc
simply by, we denote
subset $K$ . For
density$t\in[0,T]$
probabilitybydefinedthetransition
the
set
I
is
,
on
diffusion process, $0\leq s\leq t\leq T$ :$X$$\mathbb{R}$$\mathrm{x}=$$(x_{1}, x_{2}, \ldots, x_{n})\in\bigcup_{\ell=1}^{\infty}\mathbb{R}^{t}$$\{x_{i}\}_{i=1}^{n}\in X$$\{\mathrm{x}\}$$\{\mathrm{X}(t)\}$$\mathfrak{g}_{N,T}(s, \{\mathrm{x}\}_{\rangle}.t, \{\mathrm{y}\})$$g_{N,T}(s, \mathrm{x};t, \mathrm{y})$$\mathfrak{g}_{N,T}(s, \{\mathrm{x}\};t, \{\mathrm{y}\})=\{$$g_{N,T}(0, 0; t, \mathrm{y})$0,, if $s>0$ .
, if $s=0$ ,otherwise,$\#\{\mathrm{x}\}=\#\{\mathrm{y}\}=N$$\{\mathrm{x}\}=\{0\}$,, $\#\{\mathrm{y}\}=N$ ,. For the given time interval $[0, T]$ , we consider thewhere and are the elements of$M$ intermediate times $t_{0}=0<t_{1}<t_{2}<\cdots<t_{M}<t_{M+1}=T$ . The multi-time transitionis given bydensity of the process$\mathbb{R}_{<}^{N}$$\mathrm{x}$$\mathrm{y}$$\{\mathrm{X}(t)\}$$\mathrm{g}_{N_{1}T}(t_{0}, \{\mathrm{x}^{(0)}\};t_{1}, \{\mathrm{x}^{(1)}\};\ldots ; t_{M+1}, \{\mathrm{x}^{(M+1)}\})=\prod_{\mu=0}^{M}\mathfrak{g}_{N,T}(t_{\mu)}\{\mathrm{x}^{(\mu)}\};t_{\mu+1}, \{\mathrm{x}^{(\mu+1)}\})$.

14
Here we set $t_{0}=0$ , $t_{M+1}=T$ and(1.3)$\{\mathrm{x}^{(0)}\}\equiv\{0\}$$\mathfrak{g}_{N,T}(0, \{0\};t_{1}, \{\mathrm{x}^{(1)}\};\ldots).t_{M+1}$$=C(N, T, t_{1})h_{N}(\mathrm{x}^{(1)})$$\rangle\langle\prod_{i=1}^{N}p_{l_{1}}$For. From (1.1) and (1.2) we have$\mathrm{x}^{(\mu)}\in \mathbb{R}_{<}^{N}$$x_{i}^{(1)}$$\{N_{\mu}\}_{\mu=1}^{M+1}$$/)N,T(t_{1}$,$\{\mathrm{x}^{(M+1)}\})$$(h_{N}(\mathrm{x}^{(M+1)}))$$\prod_{\mu=1}^{M}1\leq i\mathrm{d},\mathrm{e}\mathrm{t}[j\leq Np_{t_{\mu+1}-t_{\mu}}(x_{i}^{(\mu)},$$x_{j}^{(\mu+1)})]$

.

).

,...,
of positive integers less than or equal to $N$ , we define the multi-time, $1\leq\mu\leq M$ ” 1, and $N’=1,2$ ,For a sequencecorrelation function by
(1.4)

(0, )

sgn

,

$\{\mathrm{x}_{N_{1}}^{(1)}\};t_{2}$,{$\mathrm{x}_{N_{2}}^{(2)}\mathrm{I}$...;$\ldots$, $N$ , we write$\mathrm{x}_{N}^{(\mu)},=(x_{1}^{(\mu)}$,$x_{2}^{(\mu)}$$x_{N}^{(\mu)},$$t_{M+1\mathrm{J}}\{\mathrm{x}_{N_{\Lambda I+1}}^{(M+1)}\})$$= \oint_{\Pi_{\mu=1}^{Nl+1}\mathbb{R}^{N-N\mu}}\prod_{\mu=1}^{M+1}\frac{1}{(N-N_{\mu})!}\prod_{i=N_{\mu}+1}^{N}dx_{i}^{(\mu)}$(0, {0}; ;,$\{\mathrm{x}_{N}^{(1)}\};t_{2}$$\mathrm{t}1$$\mathfrak{g}_{N,T}$$\{\mathrm{x}_{N}^{(2)}\};\ldots$; $t_{M+1}$ ,$\{\mathrm{x}_{N}^{(M+1)}\}$).

be the set of all continuous real functions with compact supports. For, and, the multi-time character,,, $X_{N}(t))$ as
istic function is defined for the processLet$\mathrm{f}=$$C_{0}(\mathbb{R})$$(f_{1)}f_{2}, \cdots, f_{M+1})\in C_{/0}(\mathbb{R})^{M+1}$$\theta=(\theta_{1)}\theta_{2}, \cdots, \theta_{M+1})\in \mathbb{R}^{M+1}$$\mathrm{X}(t)=\langle X_{1}(t)$$X_{2}(t,)$$\cdots$,$\Psi_{N,T}(\mathrm{f};\theta)=\mathrm{E}_{N,T}\ovalbox{\tt\small REJECT}_{\exp}\{\sqrt{-1}\sum_{\mu=1}^{M+1}\theta_{\mu}\sum_{i_{\mu}=1}^{N}f_{\mu}(X_{i_{\mu}}(t_{\mu}))\}\ovalbox{\tt\small REJECT}$

where$\mathrm{E}_{N.T}[\cdot ]$denotes the expectation determined by$\chi_{\mu}(x)=e^{\sqrt{-1}\theta_{\mu}f_{\mu}(x)}-1$,$\mathfrak{g}_{N,T}$. Let$1\leq\mu\leq M+1$ ,then by the definition of multi-time correlation function (1.4), we have$\Psi_{N,T}(\mathrm{f};\theta)$

(1.5)

2

$=$

$\sum_{N_{1}=0}^{N}\sum_{N_{2}=0N}^{N}\cdots\sum_{=NI+10}^{N}J_{\mathrm{R}^{N}1}^{\cdot}d\mathrm{x}_{N_{1}}^{(1)}J_{\mathbb{R}^{N_{2}}}^{\cdot}d\mathrm{x}_{N_{2}}^{(2)}\cdots\int_{\mathrm{R}^{N_{NJ+[perp]}}}.d\mathrm{x}_{N_{M+1}}^{(M+1)}$$\prod_{\mu=1}^{\lambda 4+1}\prod_{i_{\mu}=1}^{N_{\mu}}\chi_{\mu}(x_{i_{\mu}}^{(\mu)})\rho_{N,T}(t_{1},$$\{\mathrm{x}_{N_{1}}^{(1)}\};t_{2}$,$\{\mathrm{x}_{N_{9}}^{(2)}\};\sim$. . . ; $t_{M+1}$ ,$\{\mathrm{x}_{N_{Nl+1}}^{(M+1)}\})$.

Predholm Pfaffian Representation of Characteristic
Function

Lct$Z_{N,T}[\chi]$$=$$( \frac{1}{N!})^{M+1}f_{\mathrm{R}^{N(\mathrm{A}\mathit{1}+1)}}\prod_{\mu=1}^{M+1}d\mathrm{x}^{(\mu)}1\leq i,j\leq N\mathrm{L}\det\lceil M_{i-1}(x_{J}^{(1)})p_{t_{1}}(0, x_{j}^{(1)})(1+\chi_{1}(x_{j}^{(1)}))]$$\cross\prod_{\mu=1}^{M}1\leq i\mathrm{d},\mathrm{e}\mathrm{t}[j\leq Np_{t_{\mu+1}-t_{\mu}}(x_{i}^{(\mu)}, x_{j}^{(\mu+1)})(1+\chi_{\mu+1}(x_{j}^{\langle\mu+1)}))]$sgn$(h_{N}(\mathrm{x}_{N}^{(M+1\rangle}))$.

15
Here $M_{i}(x)$ is an arbitrary polynomial of with degree in the form $lVf_{i}(x)=b_{i}x^{i}+\cdots$ witha constant for $i=0$ , 1, 2, , and thus $h_{N}( \mathrm{x})=\det_{1\leq i,j\leq N}[M_{i-1}(x_{j})]/\prod_{k=1}^{N}b_{k-1}$ . Then
(1.3) gives$x$$\mathrm{i}$$\cdots$$b_{\dot{r}}$(2.1)$\Psi_{N,T}(\mathrm{f};\theta)=\frac{Z_{N,T}[\chi]}{Z_{N_{t}T}[0]}$where
is obtained from
By the well-known formula$Z_{N_{1}T}[0]$$Z_{N,T\lfloor}^{\mathrm{r}}|\chi$] by setting $\chi_{\mu}(x)=0$ for a1J .

$\oint_{\mathbb{R}_{<}^{N}}d\mathrm{x}\det 1\leq i,j\leq N[\phi_{i}(x_{j})]1\leq i\mathrm{d},\mathrm{e}\mathrm{t}j\leq N$

for square integrablc continuous functions$Z_{N,T}[\chi]$$=$$\oint_{\mathbb{R}_{<}^{N}}d\mathrm{y}\mathrm{d}\mathrm{e}\mathrm{t}1\leq i,j\leq N$

,
$\mu$

$[ \overline{\phi}_{i}(x_{j})]=\mathrm{d}\mathrm{e}\mathrm{t}1\leq i,j\leq N[\oint_{\mathrm{R}}$$\phi_{\iota},\overline{\phi}_{i}$,$1\leq \mathrm{i}\leq N$Jr$\phi_{i}(x)\overline{\phi}_{j}(x)]$, we have$\ovalbox{\tt\small REJECT}.\int_{\mathbb{R}^{\Lambda J+1}}\prod_{\mu=1}^{M+1}dx^{(\mu)}\{M_{i-1}(x^{(1)})p_{t_{1}}(0, x^{(1)})(1+\chi_{1}(x^{(1)}))\}$$\rangle \mathrm{e}\prod_{\mu=1}^{M}\{p_{t_{\mu+1}-t_{\mu}}(x^{(\mu)}, x^{(\mu+1\rangle})(1+\chi_{\mu+1}(x^{(\mu+1)}))\}p_{T-t_{M+1}}(x^{(M+1)}, y_{j})\ovalbox{\tt\small REJECT}$,

where $p_{T-t_{\mathit{1}\vee J+1}}(x, y)=p_{0}(x, y)=\delta(x-y)$ .
For simplicity of expressions, we will assume that the number of particles $N$ is even fromnow on. We use the formula [2]$\oint_{\mathrm{R}_{<}^{N}}d\mathrm{y}\mathrm{d}\mathrm{e}\mathrm{t}1\leq i,j\leq N[\phi_{i}(y_{j})]=\mathrm{P}\mathrm{f}_{1\leq i,g\leq N}[\int_{\mathbb{R}}dy\int_{\mathrm{R}}d\overline{y}\mathrm{s}\mathrm{g}\mathrm{n}(\tilde{y}-y)\phi_{i}(y)\phi_{j}(\overline{y})]$. Here, for an integer
for integrable continuous functions ,$2n$ matrix $A=(a_{ij}))$ the pfaffian is defined as$\phi_{l}$$1\leq \mathrm{i}\leq N$$n$and an antisymmetric$2n\mathrm{x}$$\mathrm{P}\mathrm{f}(A)=\mathrm{P}\mathrm{f}_{1\leq i<j\leq\sim n}?[a_{ij}]=\frac{1}{n!}\sum_{\sigma}\mathrm{s}\mathrm{g}\mathrm{n}(\sigma)a_{\sigma(1)\sigma(2)}a_{\sigma(3)\sigma(4)}\cdot$..

$a_{\sigma(2n-1)\sigma(2n)}$,where the summation is extended over all permutations a of $(1, 2, \ldots, 2n)$ with restriction$2n$$\sigma(2k-1)$ $<\sigma(2k)$ , $k=1,2$ ,for any antisymmetric, . Since
matrix , we have$\ldots$$n$$(\mathrm{P}\mathrm{f}(A))^{2}=\det$ $A$$2n\mathrm{x}$$A$$(Z_{N,T}[\chi])^{2}$$=$$1 \leq i,j\leq N\det[\oint dy’$ $d\tilde{y}\mathrm{s}\mathrm{g}\mathrm{n}(\overline{y}-y)$$\mathrm{x}$$\oint_{\mathrm{R}^{\Lambda I+1}}\prod_{\mu=1}^{M+1}dx^{(\mu)}\{M_{i-1}(x^{(1)})p_{t_{1}}(0, x^{(1)})(1+\chi_{1}(x^{(1)}))\}$$\mathrm{x}\prod_{\mu=1}^{M}\{p_{t_{\mu+1}-t_{\mu}}(x^{(\mu)}, x^{(\mu+1)})(1+\chi_{\mu+1}(x^{(\mu+1)}))\}p_{T-t_{M+1}}(x^{(M+1)}, y)$$\mathrm{x}$$J_{\mathbb{R}^{\mathrm{A}I+1}}^{\cdot} \prod_{\mu=1}^{M+1}d\overline{x}^{(\mu)}\{M_{j-1}(\tilde{x}^{\{1)})p_{t_{1}}(0,\tilde{x}^{(1)})(1+\chi_{1}(\tilde{x}^{(1)}))\}$

18$\mathrm{x}\prod_{\mu=1}^{M}\{p_{\mathrm{f}_{\mu+1}-t_{\mu}}(\tilde{x}^{(\mu)},\tilde{x}^{(\mu+1)})(1+\chi_{\mu+1}(\tilde{x}^{(\mu+1)}))\}p_{T-t_{M+1}}(\overline{x}^{(M+1)},\tilde{y})]$$=$$1\leq i_{1}j\leq N\det[(A_{0})_{ij}+(A_{1})_{ij}+(A_{2})_{ij}+(A_{3})_{ij}]$with$(A\mathrm{o})_{ij}$$=$$\int dy\oint d\tilde{y}\mathrm{s}\mathrm{g}\mathrm{n}(\tilde{y}-y)\int dxM_{i-1}$ $(x)p_{t_{1}}(0,$\mathrm{x}$$\oint d\overline{x}M_{j-1}(\tilde{x})p_{t_{1}}(0,\tilde{x})p_{T-t_{1}}(\tilde{x},\tilde{y})$x)p_{T-t_{1}}(x, y)$,$M+1 \sum$$(A_{1})_{ij}$$=$$\ell=1\sum_{1\leq\mu_{1}<\mu_{2}<\cdot<\mu\ell\leq M+1}\int dy\int d\tilde{y}\mathrm{s}\mathrm{g}\mathrm{n}(\overline{y}-y)$$\mathrm{x}$$\int_{\mathrm{R}^{\ell}}\prod_{k=1}^{\ell}dx^{(\mu_{k})}.\oint dxM_{i-1}(x)p_{t_{1}}(0, x)p_{t_{\mu_{1}}-t_{1}}(x, x^{(\mu_{1})})$$\mathrm{x}\prod_{k=1}^{\ell-1}\{\chi_{k}(x^{(\mu_{k})})p_{t-t_{\mathrm{A}}}\mu_{k+1k}’(x^{(\mu\kappa)}., x^{\{\mu_{k+1})})\}\chi_{f}(x^{\langle\mu\ell)})p_{T-t_{\mu_{\ell}}}(x^{\langle\mu_{t})}, y)$$\mathrm{x}$$(A_{2})_{ij}$$=$,$\oint d\tilde{x}\Lambda’I_{j-1}(\tilde{x})p_{t_{1}}(0,\tilde{x})p_{T-t_{1}}(\overline{x},\overline{y})$$\sum_{l=11\leq\mu_{1}<\mu 2}^{M+1}\sum_{<\cdot<\mu\ell\leq M+1}\oint$$dy \oint d\overline{y}\mathrm{s}\mathrm{g}\mathrm{n}(\overline{y}-y)$$\rangle\langle\int dxM_{\iota-1}(x)p_{t_{1}}(0, x)p_{T-t_{1}}(x, y)$$\rangle\langle\oint_{\mathrm{R}^{f}}\prod_{k=1}^{p}d\overline{x}^{(\mu_{k})}$$./\cdot$lx,$M_{j-1}(\tilde{x})p_{t_{1}}(0,\overline{x})p_{t_{\mu_{1}}-t_{1}}(\overline{x}.\tilde{x}^{(\mu_{1})})$$\mathrm{x}\prod_{k=1}^{\ell-1}\{\chi_{k}(\tilde{x}^{(\mu\iota)})p_{t_{\mu_{k+1}}-t_{\mu_{k}}}(\tilde{x}^{(\mu_{k})},\tilde{x}^{()}\mu k+1)\}\chi\ell(\overline{x}^{(\mu\ell)})p_{T-t_{\mu_{\ell}}}(\overline{x}^{(\mu\ell)},\tilde{y})$$(A_{3})_{ij}$$=$,$\sum_{l=1}^{M+1}\sum_{m=11\leq\mu_{1}<\mu_{2}}^{M+1}\sum_{<\cdot\cdot<\mu\ell\leq M+11\leq\nu_{1}<\nu_{2}}\sum_{<<\nu_{m}\leq M+1}\int dy\int d\tilde{y}\mathrm{s}\mathrm{g}\mathrm{n}(\tilde{y}-y)$$\rangle\langle\int_{\mathbb{R}^{l}}\prod_{k=1}^{\ell}$a$x^{(\mu_{k})} \int dxM_{i-1}(x)p_{t_{1}}(0, x)p_{t_{\mu_{1}}-t_{1}}(x, x^{(\mu_{1}\rangle})$$\mathrm{x}\prod_{k=1}^{f1}\{\chi_{k}(x^{(\mu_{k})})p_{t_{\mu_{k+1}}-t_{\mu_{k}}}(’x^{(\mu_{k})}, x^{\langle\mu_{k+1})})\}\chi_{f}(x^{(\mu\ell)})p_{T-t_{\mu\ell}}(x^{(\mu\ell)}, y)$$\rangle\langle\oint_{\mathrm{R}^{\mathit{1}}}\prod_{n=1}^{m}d\tilde{x}^{(\nu_{\pi\iota})}\oint d\overline{x}NI_{J}-1(\tilde{x})p_{t_{1}}(0,\tilde{x})p_{t_{\nu_{1}}-t_{1}}(\tilde{x},\tilde{x}^{(_{\ell/_{1}})})$$\mathrm{x}\prod_{n=1}^{m-1}\{\chi_{7b}(\tilde{x}^{(\iota J)})np_{t,-t}/_{n+[perp] n}$”$(\tilde{x}^{(\mathrm{I}/_{n})},\overline{x}^{(U_{\iota+1})},)\}\lambda m(\tilde{x}^{(\nu_{\tau n})})p_{T-t_{\nu_{7\mathrm{R}}}}(\overline{x}^{(\nu_{\tau n})},\tilde{y})$,17where we have used the Chapman-Kolmogorov equation for the heat-kernel$\oint dyp_{t-\mathrm{s}}(x, y)p_{u-t}(y, z)=p_{u-s}(x, z)$,$0<s<t$ $<u$ , ,and the fact that $p_{0}(x,y)= \lim_{tarrow 0}p_{t}(x, y)=\delta(x-y)$ .
We consider a vector space with the orthonormal basis
satisfy$\mathcal{V}$$\langle\mu\}x|\nu, y\rangle=\overline{\delta}_{\mu\nu}\delta(x-y)$

$/A$

,

$\nu$

$=1,2$ ,$\cdots$, $l1/I+1$ , ,$x$$y\in \mathbb{R}$$y\in \mathbb{R}$$x$$\{|\mu$,$x\rangle$

,

which

$\}_{\mu=1,2},$$,M+1,x\in \mathbb{R}$,. We introduce the operators$\hat{J},\hat{p},\hat{p}_{+},\hat{p}_{-}$

and$\hat{\chi}$acting on$\mathcal{V}$asfollows$\{\mu)x|\hat{J}|\nu$,$y\rangle=1_{(\mu=\mathrm{I}/=M+1)}\mathrm{s}\mathrm{g}\mathrm{n}(y-x)$$\langle\mu, x|\hat{p}|\iota/, y\rangle=p_{|t_{\nu}-t_{\mathit{1}^{4}}|}(x, y)$(2.2)$\langle\mu, x|\hat{p}_{+}|\nu, y\rangle=p_{t_{U}-t_{\mu}}(x, y)1_{(\mu<\nu)}=\langle\nu, y|\hat{p}_{-}|\mu, x\rangle$$\langle\mu, x|\hat{\chi}|\iota/, y\rangle=\chi_{\mu}(x)\mathit{5}_{\mu}\mathit{1}/\delta(x-y)$

is the indicator function: $1(\omega)=1$ if
where
will use the convention$1(\omega)$$\langle\mu, x|\hat{A}|\iota/, y\rangle\langle\nu$,$y|\hat{B}|\rho$,,

is satisfied and $1(\omega)=0$ otherwise, and we$\langle\mu, x|\hat{A}|\iota/y)\rangle=A(\mu, x;\iota/_{7}y)$$\hat{A},\hat{B}$$\{|i\rangle$(2.3)$\omega$$\mathrm{i}=1$, 2,$\cdots$

$\}$

in

$V$

, $\langle\mu, x|\hat{B}|\iota/, y\rangle=B(\mu, x;l’, y)$ .

and we assume that transformation matrix$\langle \mathrm{i}|\mu, x\rangle=\langle\mu, x|\mathrm{i}\rangle=\oint d\uparrow JM_{i-1}(y)p_{t_{1}}(0, y)p_{t_{\mu}-t_{1}}(y, x)$, 2,$\cdots$, $\mu=1,2$ ,$\cdots$,$z)= \sum_{\nu=1}^{M+1}\int_{\mathbb{R}}dy$ $A(\mu, x;\nu, y)B(\nu, y;\rho, z)=\langle\mu_{\mathrm{J}}x|\hat{A}\hat{B}|\rho, z\rangle$

withfor operators;
Consider another basis
between the two bases is given by$\mathrm{i}=1$

,

,

,

, $M+1$ , $x\in$ R. Then the quantity$\int dyl\mathrm{t}/I_{i-1}(y)p_{t_{1}}(0, y)p_{t_{\mu}-\mathrm{f}_{1}}(y, x)$$+ \sum_{m=1}^{\mu-1}\sum_{21\leq\mu_{1}<\mu<\cdot<\mu_{m}<\mu}.\oint_{\mathbb{R}^{m}}\prod_{j=1}^{m}dx^{(\mu,)}\oint$$dyM_{i-1}(y)p_{t_{1}}(0, y)p_{\mathrm{f}_{\mu_{1}}-t_{1}}(y, x^{(\mu_{1})})$$\mathrm{x}.\prod_{k=1}^{m-1}\{\chi_{\mu k}(x^{(\mu_{k})})p_{\tau_{\mu_{k+1}}.-t_{\mu_{k}}}(x^{(\mu_{k})}, x^{()}\mu k+1)\}\chi_{\mu_{m}}(x^{(\mu_{m})})p_{t_{J}-t_{\mu\gamma\prime \mathrm{t}}},(x^{(\mu_{m})}, x)$

for $\mu=1,2$ ,$\cdots$, $M+1$ ,$x\in \mathbb{R}$, $i=1_{7}2$ ,$\cdots$, can be written as$\langle \mathrm{i}|\mu, x\rangle+\sum_{m\geq 1}\langle \mathrm{i}|\mu_{1}, x^{(\mu_{1})}\rangle\langle\mu_{1}, x^{(\mu_{1})}|\hat{\chi}\hat{p}_{+}|\mu_{2}, x^{(\mu_{2})}\rangle$...

$\langle\mu_{n\iota-1}, x^{(\mu_{n\iota-1})}|\hat{\chi}\hat{p}_{+}|\mu_{m}, x^{(\mu_{m})}\rangle\langle\mu_{\tau n}, x^{(\mu_{m})}|\hat{\chi}\hat{p}_{+}|\mu, x\rangle$$=$$\langle \mathrm{i}|\mu, x\rangle+\sum_{m\geq 1}\langle\iota|(\hat{\chi}\hat{p}_{+})^{n\iota}|\mu, x\rangle$$=$$\langle \mathrm{i}|\frac{1}{1-\hat{\chi}\hat{p}_{+}}|\mu, x\rangle$

.

,

18

It is also expressed as$\langle\mu, x|\frac{1}{1-\hat{p}_{-}\hat{\chi}}|i\rangle$

It should be noted that the basis

we introduce an operator$\hat{\delta}$

(2.4)

.

$\{|\mathrm{i}\rangle$

;

, 2,

$\mathrm{i}=1$$\cdots$$\}$

is in general not orthonormal. Heresuch that$\langle \mathrm{i}|\hat{\delta}|j\rangle=\langle j|\hat{\delta}|\mathrm{i}\rangle=\delta_{ij}$,$\mathrm{i},\mathrm{J}$$=1,2$ ,.

$\cdots$

We will use the following convention$\langle$for

i|\^A|j$\rangle$$\langle j|\hat{B}|\mu, x\rangle=\sum_{j=1}^{\infty}A_{lj}B_{j}^{(\mu)}(x)$$A_{i_{J}}=\{\mathrm{i}|\hat{A}|j\rangle\equiv\langle i|\mu, x\rangle\langle\mu, x|\hat{A}|\nu, y\rangle\langle\nu, y|j\rangle$

but we will not write it as$\langle \mathrm{i}|\hat{A}\hat{B}|\mu, x\rangle$

basis. By this basis, any operator. If the matrix$\hat{A}$, since,$B_{j}^{(\mu)}(x)=\langle j|\hat{B}|\mu, x\rangle\equiv\langle j|\nu_{1}y\rangle\langle\iota/, y|\hat{B}|\mu, x\rangle$$\{|\dot{\mathrm{z}}\rangle$;$\mathrm{i}=1,2$,,

is in general not a complete$\cdots\}$

on may have a semi-infinite matrix representation
representing an operator i invertible, we define the$\mathcal{V}$$A$$A=((\mathrm{i}|\hat{A}|j\rangle)_{i,j=1,2},$$\hat{A}$$\mathrm{s}$$\cdot\cdot$

operator$\hat{A}^{\triangle}$such that its matrix representation is the inverse of(2.5)$(\{i|\hat{A}^{\triangle}|j\rangle)_{i_{?}j=1,2},\cdots=A^{-1}$

$A$

;

.

In other words,$\langle$i|\^A|j$\rangle$)$\langle j|\hat{A}^{\triangle}|k\rangle=\langle \mathrm{i}|\hat{A}^{\triangle}|j$

For any given operator$\hat{A}$,$i$, $k=1$ , 2,$\cdots$.

, the equality$\langle$holds for arbitrary , $\ell=1,2$ ,$\mathrm{i}$$\langle j|\hat{A}|k\rangle=\langle \mathrm{i}|\hat{\delta}|k\rangle$

i|\^A|j$\cdots$$\langle j|\hat{A}^{\triangle}|k\rangle\{k|\hat{B}|l\rangle=\langle \mathrm{i}|\hat{B}|\ell\rangle$$\rangle$

and

(2.6)

$\hat{B}$.

. Then the equality$\hat{A}|j\rangle\langle j|\hat{A}^{\triangle}|k\}\langle k|=1$

should be established for each . We will use this equality later.
Letbe a projection operator from the space spanned by$\hat{A}$$\mathcal{P}_{N}$-dimensional subspace spanned by$\mathrm{i}\mathrm{V}$$\{|\mathrm{i}\rangle$;,$\mathrm{i}=1,2$$\cdots$$\{|\mathrm{i}\rangle$; $i=1,2$,, $N\}$ , and thus$\langle i|\mu, x\rangle$, if $1\leq i\leq N$ ,$\langle \mathrm{i}|\mathcal{P}_{N}|\mu, x\rangle=\langle\mu, x|\mathcal{P}_{N}|\mathrm{i}\rangle=\{$Then we have the following expressions for$(A_{0})_{ij}$$(A_{\alpha})_{ij}$0otherwise.

,$=0,1,2$ , 3, $\mathrm{i},j=1,2$ ,$\alpha$$=$$\langle \mathrm{i}|\mathcal{P}_{N}|\mu, x\rangle\langle\mu, x|\hat{J}|\nu, y\rangle\langle\iota/y)|\mathcal{P}_{N}|j\rangle$$=$$\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}\mathcal{P}_{N}|j\rangle$,$\cdots$, $N$ ,$\cdots\}$

to itsI9$\langle \mathrm{i}|\mathcal{P}_{N}\hat{\chi}|\mu, x\rangle\langle\mu, x|\hat{p}\hat{J}\mathcal{P}_{N}|j\rangle\underline{1}$$(A_{1})_{ij}$$=$$1-\hat{\chi}\hat{p}_{+}$$\langle \mathrm{i}|\mathcal{P}_{N}\hat{\chi}|\mu, x\rangle\langle\mu, x|\hat{p}\hat{J}\mathcal{P}_{N}|j\rangle\underline{1}$$=$$1-\hat{p}_{+}\hat{\chi}$$(A_{2})_{ij}$,$=$$\langle \mathrm{i}|\mathcal{P}_{N}\hat{\chi}\frac{1}{1-\hat{p}_{+}\hat{\chi}}\hat{p}\hat{J}\mathcal{P}_{N}|j\rangle$$=$$\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}\hat{p}|\mu’.x\rangle\langle\mu, x|_{\hat{\lambda}}\frac{1}{1-\hat{p}_{-}\hat{\chi}}\mathcal{P}_{N}|j\rangle$$\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}\hat{p}\hat{\lambda}\underline{1}\mathcal{P}_{N}|j\rangle$

$=$

,

,

$1-\hat{p}_{-}\hat{\chi}$$\langle\iota|\mathcal{P}_{N}\wedge\hat{\chi}|\mu, x\rangle\langle\mu, x|\hat{p}\hat{J}\hat{p}|\iota^{J}, y\rangle\langle\nu, y|\hat{\chi}\mathcal{P}_{N}|j\rangle\underline{1},\underline{1}$$(A_{3})_{\iota j}$$=$$1-\hat{p}_{-}\hat{\chi}$$1-X\hat{\mathrm{P}}+$

$=$

$=$

That is, the matrices$\langle \mathrm{i}|\mathcal{P}_{N}\hat{\chi}\frac{1}{1-\hat{p}_{\neq}\hat{\chi}}|\mu\rangle x\rangle\langle\mu, x|\hat{p}\hat{J}\hat{p}|\iota/, y\rangle\langle\nu y)|\hat{\chi}\frac{1}{1-\hat{p}_{-}\hat{\chi}}\mathcal{P}_{N}|g\rangle$$\langle \mathrm{i}|\mathcal{P}_{N}\hat{\chi}\frac{1}{1-\hat{p}_{+}\hat{\chi}}\hat{p}\hat{J}\hat{p}\hat{\chi}\frac{1}{1-\hat{p}-\hat{\lambda}’}\mathcal{P}_{N}|j\rangle$$A_{\alpha}=((A_{\alpha})_{ij})_{i,j=1,2,\cdot,N}$

representations in the basis
Since
(2.7)

$\{|\mathrm{i}\rangle$;$\mathrm{i}=1,2$,$\cdots\}$.

. $ce=0,1,2,3$ , can be regarded as the matrixof the operators.

, (2.1) gives$(Z_{NT\rangle}[\mathrm{O}])^{2}=\det_{1\leq i,g\leq N}[(A_{0})_{ij}]$$\{\Psi_{N,T}(\mathrm{f};\theta)\}^{2}=\mathrm{d}\mathrm{e}\mathrm{t}1\leq i,j\leq N[\delta_{ij}+(A_{0}^{1}A_{1})_{ij}+(A_{0}^{-1}A_{2})_{ij}+(A_{0}^{-1}A_{3})_{\iota j}]$By our notation (2.5),$(A_{0}^{-1})_{ij}=\langle i|\mathcal{P}_{N}(\mathcal{P}_{N}\hat{J}7_{N}^{\mathit{2}})^{\triangle}\mathcal{P}_{N}|j\rangle)$since it satisfies the relation$\langle i|\mathcal{P}_{N}(\mathcal{P}_{N}\hat{J}\mathcal{P}_{N})^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\mathcal{P}_{N}\hat{\mathcal{J}}\mathcal{P}_{N}|k\rangle$

$=$

$=$

$1_{(1\leq i,k\leq N)} \sum_{j=1}^{N}(A_{0}^{-1})_{ij}(A_{0})_{jk}=1(1\leq i,k\leq N)\delta ik$$\langle \mathrm{i}|\mathcal{P}_{N}\hat{\delta}\mathcal{P}_{N}|j\rangle$.

We will use the abbreviation$\hat{A}_{N}=\mathcal{P}_{N}\hat{A}\mathcal{P}_{N}$for an operator(2.8)$\hat{A}$. Then it is easy to see that (2.7) is written in the form$\{\Psi_{N,T}(\mathrm{f};\theta)\}^{2}$

$=$

$=$

$\iota\leq i,j\leq N\det[\overline{\delta}_{ij}+\langle \mathrm{i}|\mathrm{B}|\mu, x\rangle\langle\mu, x|\mathrm{C}|j\rangle]$$1\leq i,j\leq N\det\langle \mathrm{i}|[\hat{\delta}_{N}+\mathrm{B}\mathrm{C}] |j\rangle 1$

.

20

where
given by$\langle \mathrm{i}|\mathrm{B}|\mu, x\rangle$$\langle i|\mathrm{B}|\mu, x\rangle$$=$$\langle\mu,x|\mathrm{C}|j\rangle$

$=$

and

(

$\langle\mu, x|\mathrm{C}|j\rangle$

are the two-dimensional row and column vectors, respectively,$\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|k\rangle\langle k|\hat{\chi}\frac{1}{1-p+\hat{X}}|\mu,x\rangle$$(\begin{array}{l}\langle\mu x|\hat{p}\hat{J}\mathcal{P}_{N}|j\rangle-\langle\mu,x|\frac{1}{1-\mathrm{P}-\hat{\mathrm{X}}}P_{N}|j\rangle\end{array})$$- \langle \mathrm{i}|\mathcal{P}_{N}J|k\rangle-\langle \mathrm{i}|\mathcal{P}_{N}J|k\rangle\langle k|\hat{J}\hat{p}\hat{\chi}|\mu,x\rangle \mathrm{x}\langle k|\hat{\chi}\frac{\hat\hat NN\triangle_{\mathcal{P}_{N}}\triangle_{\mathcal{P}_{N}}1}{1-\hat{p}+\hat{X}}|\nu,\tau/\rangle\langle\iota/,y|\hat{p}\hat{J}\hat{p}\hat{\chi}|\mu, x\rangle$),.

The determinant (2.8) is written using the Fredholm determinant,$\mathrm{D}\mathrm{e}\mathrm{t}\langle\mu, x|||I_{2}+\mathrm{C}\mathrm{B}\ovalbox{\tt\small REJECT}|\nu, y\rangle$$- \langle\mu,x|\frac{1}{1-p-\hat{\chi},\hat{J}_{N}^{\triangle}},|i\rangle+\langle\mu,x|\hat{p}\hat{J}|\mathrm{i}\rangle\langle\mu,x|\mathrm{I}/,y\rangle \mathrm{x}\langle \mathrm{i}|\mathcal{P}_{N}\mathcal{P}_{N}|j\rangle\langle j|\hat{\chi}\frac{1}{1-\hat{p}+\hat{X}}|\nu,y\rangle \mathrm{x}\langle i|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{\chi}\frac{1}{1-\hat{\mathrm{P}}+\hat{\mathrm{X}}}|\nu,y\rangle$$=$$\mathrm{D}\mathrm{e}\mathrm{t}\ovalbox{\tt\small REJECT}$$+( \mu,x|’,|\dot{x}\rangle\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{\chi}\frac{\hat{p}\hat{\chi}|\nu 1}{1-\hat{p}+\dot{X}},|\rho, z\rangle+\langle\mu_{2}x|\frac{y\rangle 1}{\frac,1-\hat{\mathrm{P}}-\hat{X}1-\hat{p}-\hat{\mathrm{X}}\mathrm{x}\langle\rho 1},|\iota\rangle\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{J}y\rangle\langle\mu,x|\nu-\langle\mu,x|\hat{p}\hat{J}|\mathrm{i}\rangle\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle(j|\hat{\chi}\frac{\hat{p}\hat{\chi}|\nu 1}{1-\mathrm{P}+\hat{X}}|\rho,z\rangle-\langle\mu,x|\hat{p}\hat{J}|\mathrm{i}\rangle\langle i|\mathcal{F}_{N}^{)}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{J},y\rangle \mathrm{x}\langle\rho,z|\hat{p}\hat{J}\hat{p}\hat{\chi}|\nu,y\rangle z|\hat{p}\hat{J}\hat{p}\hat{\chi}|l/\}y\rangle\ovalbox{\tt\small REJECT}$$+ \langle\mu,x|\hat{p}\hat{J}|\mathrm{i}\rangle-\langle\mu,x|\frac{1}{1-p-\hat{\mathrm{X}},\hat{J}_{N}^{\triangle}},|\mathrm{i}\rangle\langle\mu,x|\rho,z\rangle \mathrm{x}\langle \mathrm{i}|\mathcal{P}_{N}\mathcal{P}_{N}|j\rangle\langle j|\hat{\chi}\frac{1}{1-\hat{p}+\hat{X}}|\rho,\underline{7}\cross\langle\iota|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{F}_{N}^{\rangle}|j\rangle\langle j|\hat{\chi}\frac{1}{1-p_{\dagger\hat{\mathrm{X}}}}|\rho,z\rangle\rangle$$=$$\mathrm{D}\mathrm{e}\mathrm{t}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$+ \langle\mu,x|’\frac{z\rangle 1}{1-_{\mathrm{A}}p-,\overline{\chi},J_{N^{[mathring]_{J}}}^{\triangle}}|i\rangle+\langle\mu,x|\hat{p}\hat{J}p^{\mathrm{A}}\hat{\chi}|\beta_{\rangle}Z\rangle\langle\mu,x|\rho-\langle\mu,x|\hat{p}\hat{J}|i\rangle \mathrm{x}\langle \mathrm{i}|\mathcal{P}_{NN}|j\rangle\langle j|\hat{J}\hat{p}\hat{\chi}|\rho,z\rangle\cross\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{J}\hat{p}\hat{\chi}|\rho,z\rangle\ovalbox{\tt\small REJECT}$$\mathrm{x}\ovalbox{\tt\small REJECT}^{\langle\rho,z1\nu,y\rangle}0$$-\langle\rho,z|\hat{p}\hat{J}\hat{p}\hat{\chi}|\nu, y\rangle\langle\rho,z|\nu,y\rangle\ovalbox{\tt\small REJECT}$$- \langle\mu,x|\frac{1}{1-\hat{\mathrm{P}}-\hat{X},\hat{J}_{N}^{\triangle}},|\mathrm{i}\rangle+\langle\mu,x|\hat{p}\hat{J}|\mathrm{i}\rangle\langle\mu,x|l_{?}\mathrm{x}\langle \mathrm{i}|\mathcal{P}_{N}\mathcal{P}_{N}|j\rangle\langle j|\hat{\chi}\frac{1}{1-\mathrm{P}+\hat{X}}|\iota" y\rangle \mathrm{x}\langle \mathrm{i}|P_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{\chi}\frac{1}{1-p+\dot{\mathrm{X}}}|\nu,y\rangle/y\rangle$$=$$\mathrm{D}\mathrm{e}\mathrm{t}\ovalbox{\tt\small REJECT}$,$+ \langle\mu,x|’\frac{y\rangle 1}{1-\mathrm{P}-\hat{X},\hat{J}_{N}^{\triangle}},|i\rangle+\{\mu,x|\hat{p}\hat{J}\hat{p}\hat{\chi}|\nu,y\rangle\langle\mu_{7}x|\nu-\langle\mu,x|\hat{p}\hat{J}|\mathrm{i}\rangle \mathrm{x}\langle \mathrm{i}|\mathcal{P}_{N}\mathcal{P}_{N}|j\rangle\langle j|J^{\mathrm{A}}\hat{p}\hat{\chi}|\nu,y\rangle\cross\{i|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}P_{N}|j\rangle\langle j|\hat{J}\hat{p}\hat{\chi}|l/,y\rangle\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$

21

where

$I_{2}$

denotes the unit matrix with size 2. It is further rewritten as$\mathrm{D}\mathrm{e}\mathrm{t}\langle\mu,x|\{$ $I_{2}+\ovalbox{\tt\small REJECT}_{0}^{1}$$\frac{1}{1-p-\hat{\mathrm{X}}}\ovalbox{\tt\small REJECT} 0$$\ovalbox{\tt\small REJECT}^{\hat{p}\hat{J}|\mathrm{i}\rangle\langle \mathrm{i}|P_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{\chi}}-|i^{\iota}\rangle\langle i|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}P_{N}|j\rangle\langle j|\hat{\chi} (-\hat{p}\hat{J}|i\rangle\langle i|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{J}\hat{p}+\hat{p}\hat{J}\hat{p})\hat{\chi}|i\rangle\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{J}\hat{p}\hat{\chi}\ovalbox{\tt\small REJECT}$$\mathrm{x}$,$\frac{1}{1-p+\hat{X},0}$$\cross\ovalbox{\tt\small REJECT}$$=$$\mathrm{D}\mathrm{e}\mathrm{t}\langle\mu,x|(\ovalbox{\tt\small REJECT}_{0}^{1}$$+\{$$1-\hat{p}_{-}\hat{\chi}0\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}^{1-\hat{p}_{+}\hat{\chi}}0$$01$$01])|\iota/,y\rangle$$\ovalbox{\tt\small REJECT}$$\hat{p}\hat{J}|i\rangle\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{\chi} (-\hat{p}\hat{J}|\mathrm{i}\rangle\langle i|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{J}\hat{p}+\hat{p}\hat{J}\hat{p})\hat{\chi}\ovalbox{\tt\small REJECT} )|\nu, y\rangle$$-|i\rangle\langle i|\mathcal{P}_{N}\hat{J}_{N}^{\triangle},\mathrm{P}_{N}|j\rangle\langle j|\hat{\chi} |\mathrm{i}\rangle\langle i|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{J}\hat{p}\hat{\chi}$$=$$\mathrm{D}\mathrm{e}\mathrm{t}\langle\mu,x|(I_{2}+\ovalbox{\tt\small REJECT}$

,

$y\}$

$\hat{p}\hat{J}|\mathrm{i}\rangle\langle i|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|-\hat{p}_{+} -\hat{p}\hat{J}|i\rangle\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}P_{N}|j\rangle\langle j|\hat{J}\hat{p}+\hat{p}\hat{J}\hat{p}|i\rangle\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}_{N}^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{J}\hat{p}-\hat{p}-\ovalbox{\tt\small REJECT}\hat{\chi})|\nu$$-|i\rangle\langle i|\mathcal{P}_{N}\hat{J}_{N’}^{\triangle}P_{N}|j\rangle\langle j|$

Here we have used the facts that$\mathrm{D}\mathrm{e}\mathrm{t}\langle\mu\rangle x|\ovalbox{\tt\small REJECT}_{0}^{1}$, $y\rangle=1$ ,$1-\hat{p}_{-}\hat{\chi}0\ovalbox{\tt\small REJECT}|\nu$and$\mathrm{D}\mathrm{e}\mathrm{t}\langle\mu, x|\ovalbox{\tt\small REJECT}^{1-\hat{p}_{+}\hat{\chi}}0 01\ovalbox{\tt\small REJECT}|\nu, y\rangle=1$

which are consequences of definitions (2.2) of the operators(2.9)$\{\Psi_{N,T}(\mathrm{f};\theta)\}^{2}=\mathrm{D}\mathrm{e}\mathrm{t}(I_{2}\delta_{\mu\nu}\delta(x-y)+$$\hat{p}_{+}$,

and$\hat{p}_{-}$. Then we arrive at$\{\begin{array}{ll}\overline{S}^{\mu,\iota/}(x,y) \overline{I}^{\mu,\nu}(x,y)D^{\mu_{\prime}\nu}(x,y) \overline{S}^{\nu,\mu}(y,x)\end{array}\}$$\chi_{\nu}(y))$

,

where
(2.10)$D^{\mu,\nu}(x, y)$$S^{\mu,\nu}(x, y)$$I^{\mu,\nu}(x, y)$

$=$

$=$

$=$

$-\langle\mu, x|i\rangle\langle i|P_{N}(\mathcal{P}_{N}\hat{J}\mathcal{P}_{N})^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\nu, y\rangle$$\{\mu$,,$x|\hat{p}\hat{J}|\mathrm{i}\rangle\langle \mathrm{i}|\mathcal{P}_{N}(\mathcal{P}_{N}\hat{J}\mathcal{P}_{N})^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\nu, y\rangle$,$-\langle\mu, x|\hat{p}\hat{J}|\mathrm{i}\rangle\langle \mathrm{i}|\mathcal{P}_{N}(\mathcal{P}_{N}\hat{J}\mathcal{P}_{N})^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\hat{J}\hat{p}|\nu$, et),

and(2.11)$\overline{S}^{\mu,\nu}(x, y)$$=$$S^{/x,\iota/}(x, y)-\langle\mu)x|\hat{p}_{+}|\nu, y\rangle$$\langle\mu, x|\hat{p}J^{\Lambda}|i\rangle\langle i|\mathcal{P}_{N}(\mathcal{P}_{N}\hat{J}\mathcal{P}_{N})^{\triangle}\mathcal{P}_{N}|j\rangle\langle j|\nu, y\rangle$

$=$

,

if

$\mu\geq\nu$

,

$\{$

$-\langle\mu, x|\hat{p}\hat{J}|i\rangle\langle i|(\hat{J}^{\triangle}-\mathcal{P}_{N}(P_{N}\hat{J}\mathcal{P}_{N})^{\triangle}P_{N})|j\rangle\langle j|\iota/, y\rangle$, if $\mu<\nu$ ,.

22$I^{\overline{\mu},t/}(x, y)$$=$$I^{\mu,\nu}(x,y)+\langle\mu, x|\hat{p}\hat{J}\hat{p}|\nu, y\rangle$$=$$\langle\mu, x|\hat{p}\hat{J}|\mathrm{i}\rangle\langle \mathrm{i}|(\hat{J}^{\triangle}-\mathcal{P}_{N}(P_{N}\hat{J}\mathcal{P}_{N})^{\triangle}P_{N})|j\rangle\langle j|\hat{J}\hat{p}|\nu, y\rangle$,.
where we have used the equality (2.6) forIn [15] Rains introduced Fredholm pfaffian, denoted here byequality$=\hat{J}$$\hat{A}$(2.12)$\{\mathrm{P}\mathrm{F}(J_{2}+K)\}^{2}=\mathrm{D}\mathrm{e}\mathrm{t}(I_{2}+J_{2}^{-1}K)$$\mathrm{P}\mathrm{F}$, and proved a useful,for any antisym matrix $2><2$ matrix kernel $K$ , where(2.13).

$J_{2}=(\begin{array}{ll}0 1-1 0\end{array})$

Then (2.9) implies the Fredholm pfaffian representation of the multi-time characteristic function(2.14).
$\Psi_{N,T}(\mathrm{f};\theta)=\mathrm{P}\mathrm{F}(J_{2}\delta_{\mu}\mathrm{I}/\delta(x-y)+$3$\{\begin{array}{llll}D^{\mu,\iota/}(x y) \overline{S}^{\nu\mu}(y x)-\overline{S}^{\mu,\nu}(x,y) -\overline{I}^{\mu,\nu}(x y)\end{array}\}$$\chi_{\nu}(y))$

Pfaffian Process

Lct
(3.1)

$A^{\mu\iota\prime}\acute{(}x$and construct 2$A$,$y)=\{\begin{array}{llll}D^{\mu}I/(x y) \overline{S}^{l/,\mu}(y x)-\overline{S}^{\mu,\iota/}(_{X_{|}}y) -\overline{I}^{\mu,\iota/}(_{X_{)}}y) \end{array}\}$$\sum_{\mu=1}^{M+1}N_{\mu}\mathrm{x}2\sum_{\mu=1}^{M+1}N_{\mu}$( ,$\mathrm{x}_{N_{1}}^{(1)}$$\mathrm{x}_{N_{2}}^{(2)}$,$\cdots$,,

antisymmetric matrices$\mathrm{x}_{N_{NI+1}}^{\langle M+1)})=(A^{\mu,\nu}$( , )$x_{x}^{(\mu)}.$$x_{j}^{(\nu)}$$)_{1\leq i\leq N_{\mu},1\leq j\leq N_{\nu},1\leq\mu,\nu\leq M+1}$,$N$ , $1\leq m\leq M+1$
for $N_{m}=1,2$ ,By the definition of Fredholm pfaffian [15] and theequality (2.14), we can establish the following statement.
$\cdots$)

Theorem 1 The non-colliding system of Brownian motions
is a pfaffian process in thesense that any multi-time correlation function is given by a pfaffian$\mathrm{X}(t)$

(3.2)
(2.6)

pNiT

$(t_{1},$$=$$\{\mathrm{x}_{N_{1}}^{(1)}\};t_{2}$Pf,$[A(\mathrm{x}_{N_{1}}^{\langle 1)},$$\{\mathrm{x}_{N_{2}}^{(2)}\}$$\mathrm{x}_{N_{2}}^{(2)}$,;$\ldots$$\cdots$,; $t_{M+1}$ ,$\{\mathrm{x}_{N_{N\mathit{1}+1}}^{(M+1)}\})$$\mathrm{x}_{N_{M+1}}^{(M+1))]}$.

Remark. The pfaffian processes considered here may be regarded as the continuous spacetime version of the pfaffian point processes and pfaffian Schur processes studied by Sasamotoand Imamura [16] and by Borodin and Rains [1]. The processes studied in [4] are also pfaffian
processes, since the ‘quaternion dcterminantal expressions’, in the sense of Dyson and Mehta[3, 12, 13], of correlation functions are readily transformed to pfaffian expressions

23

Let

$H_{i}(x)$be the z-th Hermite polynomial$H_{i}(x)$$=$$e^{x^{2}}(- \frac{d}{dx})^{i}e^{-x^{2}}$$=$$\mathrm{i}!\sum_{j=0}^{[i/2]}(-1)^{j}\frac{(2x)^{i-2j}}{j!(i-2_{J})!}$

denotes the greatest integer not greater thanwhere
the orthogonal relations
$[a]$

(3.4)

$\int_{\mathrm{R}}dxe^{-x^{2}}H_{x}(x)H_{j}(x)=2^{i}\mathrm{i}!\sqrt{\pi}\delta_{i\dot{g}}$

$a$

,

.

$i$

,

The Hermite polynomials satisfy, $f=0,1,2$ ,$\cdots$.

Set,$c_{1}=\sqrt{\frac{t_{1}(2T-t_{1}1}{T}}$$z_{1}=\sqrt{\frac{2T-t_{1}}{t_{1}}}$

’

and

,

if is even,, if is odd.
$i$

(3.5)

$\alpha_{ij}=\{$$2^{-i}c_{1}^{i}\delta_{i}2^{-i}c_{1}^{i}’\{\begin{array}{ll}j \delta_{i_{J}} -2(i-1)\delta_{i-2j}\end{array}\}$$i$

Now we specify polynomials $\{M_{i}(x)\}$ as
,

(3.6)

with

$\mathrm{i}=0$$M_{i}(x)=b_{i}z_{1}^{-i} \sum_{j=0}^{i}\mathrm{o}i{}_{ijj}H(\frac{x}{c_{1}})z_{1}^{j}$$b_{\iota}=\{r_{[i/2]}\}^{-1/2}$, where, 1, 2,$r_{i}= \frac{1}{\pi}\Gamma(\mathrm{i}+1/2)\Gamma(i+1)\mathrm{t}\frac{t_{1}^{2}}{T})^{2i+1/2}$$J_{N}=I_{N/2}\otimes J_{2}$$\cdots$. Set,

is given by (2.13), and let be thedenotes the unit matrix with size $N/2$ andsemi-infinite matrix obtained as the $Narrow$ oo limit of . By the orthogonality of Hermite
polynomials (3.4), we can show through (2.3) with the choice (3.6) that $[14, 7]$
where

$J$

$J_{2}$$I_{N/2}$$J_{N}$$\langle i|\hat{J}|j\rangle=\langle \mathrm{i}|\mu, x\rangle\langle\mu, x|\hat{J}|\nu, y\rangle\langle\nu, y|j\rangle=J_{ij}$,$i$, $j=1$ , 2,

Since $J_{N}^{2}=-I_{N}$ for any even $N\geq 2$ , this implies that the matrix
and
(3.7)

(3.8)

$\langle \mathrm{i}|\hat{J}^{\triangle}|j\rangle$$\langle \mathrm{i}|\mathcal{P}_{N}(\mathcal{P}_{N}\hat{J}P_{N})^{\triangle}\mathcal{P}_{N}|j\rangle$$=$$=-J_{ij}$

$=$

$=$

$\{$

$-J_{ij}=-(J_{N})_{i\mathrm{y}}$for $i,j=1,2$ ,$\cdots$.

.

$(\langle \mathrm{i}|J^{\mathrm{A}}|j\rangle)_{i,j=1,2},\cdot$,$\langle \mathrm{i}|\mathcal{P}_{N}\hat{J}^{\triangle}\mathcal{P}_{N}|j\rangle$0,$\cdots$,

if $1\leq i$ , $j\leq N$ ,otherwise.

is invertible

24

If we write$\langle\mu, x|i\rangle=b_{i}R_{i-1}^{(\mu)}(x)$,$\langle i|\hat{J}\hat{p}|\mu, x\rangle=-\langle\mu, x|\hat{p}\hat{J}|\mathrm{i}\rangle=b_{i}\Phi_{i-1}^{(\mu)}(x)$$\mathrm{z}$$=1,2$ ,(3.9)$\cdots$, $\mu=1,2$ ,$\cdots$, $M+1$ ,$x\in \mathbb{R}$,, then the functions (2.10) are written as$D^{\mu,\nu}(x, y)$$=$$\sum_{i=0}^{N/2-1}\frac{1}{r_{i}}[R_{2i}^{(\mu)}(x)R_{2i+\mathrm{l}}^{(\iota/)}(y)-R_{2x+1}^{(\mu)}(x)R_{2\iota}^{(\nu)}(y)]$,$S^{\mu\nu}(x, y)$$=$$\sum_{i=0}^{N/2-1}\frac{1}{r_{i}}[\Phi_{2i}^{(\mu)}(x)R_{2i+1}^{(\nu)}(y)-\Phi_{2i+1}^{(\mu)}(x)R_{2i}^{(\nu)}(y)]$,$I^{\mu_{)}\nu}(x, y)$$=$-$\sum_{i=0}^{N/2-1}\frac{1}{r_{i}}[\Phi_{2\iota}^{(\mu)}(x)\Phi_{2i+1}^{(\nu)}(y)-\Phi_{2i+1}^{(\mu)}(x)\Phi_{2i}^{(\nu)}(y)])$and Equations (2. 11) become$\sum_{i=0}^{N/2-1}\frac{1}{r_{i}}[\Phi_{2i}^{(\mu)}(x)R_{2i+1}^{(\nu)}(y)-\Phi_{2i+1}^{(\mu)}(x)R_{2i}^{(\nu)}(y)]$(3.10)$\overline{S}^{\mu,\nu}(x, y)$

$=$

$=$

if

$\mu\geq\nu$

,

$\{$

-

$I^{\overline{\mu},\nu}(x, y)$,$\sum_{i=N/2}^{\infty}\frac{1}{r_{i}}[\Phi_{2i}^{(\mu)}(x)R_{2i+[perp]}^{(\nu)}(y)-\Phi_{2i+1}^{(\mu)}(x)R_{2i}^{\langle\nu)}(y)]$$\sum_{i=N/2}^{\infty}\frac{1}{r_{i}}[\Phi_{2\tau}^{(\mu)}$(ce)$\Phi_{2i+1}^{(U)}(y)-\Phi_{2i+1}^{(\mu)}(x)\Phi_{2i}^{(\nu)}(y)]$, if $\mu<\nu$ ,.

Theorem 1 with the expressions (3.9), (3.10) of the elements of matrix (3.1) is equivalent
with Theorem 3 reported in [7], although the latter was given in the form of quaternion
determinant. The present argument will be generalized to discuss other non-colliding systemsof diffusion particles reported in $[10, 11]$ .

Acknowledgment
This manuscript is based on the joint work with H. Tanemura. The author would like tothank T. Sasamoto for useful discussions on the papers [15] and [1].

References[1] A. Borodin and E. Rains, Eynard-Mehta theorem, Schur process, and their pfaffiananalogs, arXiv:math-ph/0409059.
[2] N. G. de Bruijn, On some multiple integrals involving determinants, J. Indian Math.
Soc. 19, 133-151 (1955)

25[3] D. J. Dyson, Correlation between the eigenvalues of a random matrix, Commun. Math.
Phys. 19, 235-250 (1970).

[4] P. J. Forrester, T. Nagao, and G. Honner, Correlations for orthogonal-unitary andsymplectiounitary transitions at the hard and soft edges, NucL Phys. B553 [PM], 601643 (1999).
[5] S. Karlin and J. McGregor, Coincidence properties of birth and death processes, Pacific
J. Math. 9, 1109-1140 (1959).
[6] S. Karlin and J. McGregor, Coincidence probabilities, Pacific J. Math. 9, 1141-1164(1959).
[7] 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”, 283-306 (2004), (Mathematical Society of Japan, Tokyo);.
$\mathrm{a}\mathrm{r}\mathrm{X}\mathrm{i}\mathrm{v}:\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{P}\mathrm{R}/0301143$[8] M. Katori and H. Tanemura, Scaling limit of vicious walks and two-matrix model, Phys.
Rev. E66,011105/1-12 (2002); arXiv:cond-mat/0203549.
[9] M. Katori and H. Tanemura, Functional central limit theorems for vicious walkers, Stock.
.
Stock. Rep. 75, 369-390 (2003); arXiv:math.
$\mathrm{P}\mathrm{R}/02032\mathrm{S}6$[10] M. Katori and H. Tanemura, Symmetry of matrix-valued stochastic processes and noncoliiding diffusion particle system s, J. Math. Phys. 45, 3058-3085 (2004); arXiv:math$\mathrm{p}\mathrm{h}/0402061$.

[11] M. Katori and H. Tanemura, in preparation.
[12] M. L. Mehta, Matrix Theory, Editions de Physique, (Orsay, 1989).
[13] M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, London, 1991).

[14] T. Nagao, M. Katori, and H. Tanemura, Dynamical correlations among vicious random
walkers, Phys. Lett. A307, 29-35 (2003); arXiv:cond-mat/020206S.

[15] E. M. Rains, Correlation functions for symmetrized increasing subsequences,
arXiv: math. CO /0006097.
[16] T. Sasamoto and T. Imamura, Fluctuations of the one-dimensional polynuclear growthmodel in half space, J. Stat Phys, 115, 749-803 (2004)