数理解析研究所講究録 1227 巻 2001 年 145-153

145

The Symmetric Groups andAlgebraic Central Limit Theorems
Akihito HORA]$\mathrm{t}^{\approx}\sqrt\ulcorner_{9}^{-}$$\wedge:\yen \mathrm{Y}$$\mathrm{A}\sim$)Okayama University
Okayama 700-8530, Japan
Abstract
In this note, we review some of our results on central limit theorems in algebraic
probability and report an attempt to develop their quantum aspects. We illustrate
our approach with materials concerning the symmetric groups.

1Introduction
generate $G$ with $S^{-1}=S$ (as aset) andLetbe adiscrete group and(theidentity element in ). $(G, S)$ forms aCayley graph , in whichis the vertex set and, $y\in G$ are adjacent (denoted by $x\sim y$ ) if and only ifsuch that $sx=y$ . The
acts by definition on asuitable function space on $G$ asadjacency operator on
$S$

$G$

$S\not\supset$

$G$

$e$

$G$

$\mathcal{X}$$\exists s\in S$

$x$

$A$

$\mathcal{X}$

(A )
$f$

$(\ovalbox{\tt\small REJECT} X)$$=. \sum_{y\cdot y\sim x}f(y)$$(f\in Fun(G))$ ,,which is aformal expression when the degree $=|S|=\infty$ . Let us take anormalised
positive-definite function on or let be astate on asuitable algebra $A(G)$ generated
by . We are interested in asymptotic spectral structure of on large
through somewith $|S^{(n)}|<\infty$ and $S^{(n)-1}=S^{(n)}$ . The
scaling limit. To be more precise, let
at afinite level isadjacency operator$\kappa$$\varphi$$G$$\varphi$

$A$

$G$

$\mathcal{X}$$S^{(n)}\nearrow S$$A^{(n)}$$(A^{(n)}f)(x)= \sum_{y:y\sim x,yx^{-1}\in S^{(n)}}f(y)$,$(f\in Fun(G))$ .

We can formulate our central limit theorem by considering convergence of moments orspectral distribution of$(A^{(n)}-\varphi(A^{(n)}))/\sqrt{\varphi((A^{(n)}-\varphi(A^{(n)}))^{2})}$

with respect to . The asymptotic is taken along the size and possibly other addtionalparameters contained in the state . (See later sections.) More generally, we can discu ss$n$$\varphi$$\varphi$

146
,
,
,
associated with subsets , , ,ofseveral adjacency operatorsand their mixed moments or joint distribution (if ’s are commuting) with respect to. It is straightforward to extend the consideration to other regular graphs than Cayley
graphs.
and distance-regular
In this note, we treat Cayley graphs of the symmetric groups
graphs appearing as homogeneous spaces of the symmetric groups. Spectral structureof these groups is at finite level studied well by using combinatorial and representationtheoretical technique. The algorithmic results, however, become very complicated as thesize of the graph grows. In order to make the limiting procedure more transparent, we
intend to apply quantum decomposition of an adjacency operator, which is abasic idea
widely used in quantum probability.
$A_{S_{1}}$$A_{S_{2}}$$A_{S_{3}}$$S_{1}$$\cdots$$S_{2}$$S_{3}$$\cdots$$S$$A_{:}$$\varphi$$S_{n}$

2Working on Johnson Graph
AJohnson graph is an important distance-regular graph as well-known as aHamming
graph. For , $d\in N$ , let $X=\{x\subset\{1,2, \cdots, v\}||x|=d\}$ be the subsets of au-set.
( $2d\leq v$ without loss of generality.) By definition two vertices , $y\in X$ are adjacent if$|x\cap y|=d-1$ in Johnson graph $J(v,d)$ . It has diameterand degree $=d(v-d)$ .
$J(v, d)$ is regarded as ahomogeneous space. We fix abase point $x_{0}\in X$ .
.
where
The vacuum state is defined as
In [7], we showed the following central limit theorem by using spectral data of Johnson
graphs(e.g. seen in Bannai-Ito [3]).
$d$

$v$

$x$

$d$

$\kappa$$S_{d}\cross S_{v-d}\backslash S_{v}$$\Phi(0)=\delta_{x_{0}}$$\langle\Phi(0), \cdot\Phi(0)\rangle_{\ell^{2}(X)}$

Theorem 1For agrowing family of $J(v, d)$ , the distribution of normalised adjacency
with respect to the vacuum state converges weakly to:operator$A/\sqrt{\kappa}$

(i)

(i)

$e^{-(x+1)}I1-1,\infty)(x)dx$

as$darrow\infty$

and$\sum_{l=0}^{\infty}\frac{2(1-p)}{2-p}(\frac{p}{2-p})^{l}\delta_{-\frac{\mathrm{p}}{\sqrt{\mathrm{p}(2-\mathrm{p})}}+\frac{2(1-\mathrm{p})}{\sqrt{\mathrm{p}(2-\mathrm{p})}}l}$$\frac{2d}{v}arrow 1$

as,$darrow\infty$

and $\frac{2d}{v}arrow p\in(0,1]$ .

(The original statement in [7] contained an extra condition in (ii), which proves to beinessential.)

We can extend Theorem 1to aquantum central limit theorem by introducing quantum
decomposition of the adjacency operator: $A=A^{+}+A^{-}$ Letbeis a
the finite-dimensional Fock space associated with aJohnson graph , whereLet
normalised number vector.
$\Gamma(\mathcal{X})=\oplus_{n=0}^{d}\Phi(n)$$\mathcal{X}$$\Phi(n)$$\Gamma=\{(\xi_{n})=\sum_{n=0}^{\infty}\xi_{n}e_{n}\in C^{\infty}|\sum_{n=0}^{\infty}(n!)^{2}|\xi_{n}|^{2}<\infty\}$,
and $N$ denote the creator, the annihilator
be a1-mode interacting Fock space. Let
and the number operator on . For an interacting Fock space and operators on it, weand Accardi-Obata [2]refer to$B^{+}$$\Gamma$$\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{i}- \mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}[1]$$B^{-}$

147
Theorem 2For agrowing family of $J(v, d)$ such that $darrow\infty$ and $2d/varrow p\in(0,1]$ ,we have$\langle\Phi(t), \frac{A^{\epsilon_{1}}}{\sqrt{\kappa}}\frac{A^{\epsilon_{2}}}{\sqrt{\kappa}}\cdots\frac{A^{\epsilon_{m}}}{\sqrt{\kappa}}\Phi(j)\rangle_{\Gamma(\mathcal{X})}arrow\langle e_{l}, C^{\epsilon_{1}}C^{\epsilon_{2}}\cdots C^{\epsilon_{m}}e_{j}\rangle_{\Gamma}$

for $\forall m\in N$ ,$\forall\epsilon_{1}$,$\epsilon_{2}$,$\cdots$,$\epsilon_{m}\in\{+$

,-

$\}$

, $\forall t,j\in\{0,1,2, \cdots\}$ , where$C^{\pm}=C_{p}^{\pm}=B^{\pm}+ \frac{1}{\sqrt{p(2-p)}}N$.

Theorem 2yields Theorem 1as classical reduction with an interesting observation ofrelations to orthogonal polynomials. Including the definition of quantum decomposition$A=A^{+}+A^{-}$ , full details of Theorem 2and more general version about distance-regulargraphs will be included in [11] (partly announced in an IIAS workshop 20-22/2/2001).
As for aquantum central limit theorem on Hamming graphs, we refer to HashimotOObata-Tabei [6].
on the adjacency algebra
Motivated by Hashimoto [4], we introduced Gibbs statein [9]:of adistance-regular graph$\Phi_{q}$$\mathcal{X}$$A(\mathcal{X})$$\Phi_{q}(Q)=\langle\Phi(0), (\sum_{h=0}^{d}q^{h}A_{h})Q\Phi(0)\rangle$$(Q\in A(\mathcal{X}))$.

is the th adjacency operator on .
becomes actually astate onfor$0\leq q\leq 1$ if the graphis nice, e.g. if
is quadratically embedded into aHilbert
space. Then the temperature $T$ of
is introduced as $T$ oc $-1/\log q$ . In [9], we showed
the following central limit theorem (low temperature limit).

Here

$i$

$A_{i}$$\mathcal{X}$$A(\mathcal{X})$$\Phi_{q}$$\mathcal{X}$$\mathcal{X}$$\mathcal{X}$

Theorem 3For agrowing family of $J(2d, d)$ , the distribution of$(A-\Phi_{q}(A))/\sqrt{\Phi_{q}((A-\Phi_{q}(A))^{2})}$

with respect toas$\Phi_{q}$converges weakly to:

as

and

( $r\geq 0$ , $\alpha>1$ : fixed)(i)$e^{-(x+1)}I_{[-1,\infty)}(x)dx$(ii)$\sqrt{2r+1}e^{-(x\sqrt{2r+1}+2r+1)}J_{0}(i2\sqrt{r(x\sqrt{2r+1}+r+1)})I_{[-\frac{r+1}{\sqrt{2r+1}},\infty)}(x)dx$$darrow\infty$$darrow\infty$$q=r/d^{\alpha}arrow \mathrm{O}$and $q=r/darrow \mathrm{O}$ ( $r\geq 0$ : fixed), where$J_{0}(z)= \sum_{k=0}^{\infty}\frac{(-z^{2}/4)^{k}}{(k!)^{2}}$$(z\in C)$

is the Oth Bessel function.

Seen from the viewpoint of Theorem 2, Theorem 3can be interpreted as convergenceof asuperposition of matrix elements. Finding the limit distribution of (ii) is equivalent
to computing the moments$\sum_{n=0}^{\infty}\frac{r^{n}}{(n!)^{2}}\langle(B^{+}+B^{-}+2N)^{p}e_{0}, e_{n}\rangle_{\Gamma}$$(p\in\{0,1,2, \cdots\})$148,
and $N$ on the interacting Fock space . It can be done through afor operators
combinatorial argument by using an appropriate Bratteli diagram. It turns out that the,,,limit distribution is atranslation of that of $X_{0}+X_{1}+\cdots+X_{M}$ where$e^{-x}dx$
$M$
and
is
are independent random variables obeying the exponential distribution.
parameter
with
Poissondistribution
also independent random variable of ’s obeying
Details of these observations and computation of the moments will be contained in [10].
See Hashimoto [5] for the discussion of Haagerup states on the free group algebras.
$B^{+}$$B^{-}$$\Gamma$$X_{0}$$X_{1}$$X_{2}$$\cdots$$r$$X_{\dot{\iota}}$

3Working on the Infinite Symmetric Groupbe the infinite symmetric group with the identity element . The
Let$(\neq\{e\})$are parametrised by $D$ , the set of the Young
conjugacy classes of
nontrivial
denote the conjugacy class
diagrams without arow consisting of only one box. Let
which. We use the cycle notation $\rho=$corresponding to$|\rho|=\sum_{j}jk_{j}(\rho)$$\rho\in
V$
, the
of
-rows.
Set
numbercontainsmeans that diagram
denote the number of rows of and $l(\rho)=|\rho|-r(\rho)$ thenumber of boxes of . Let,“length function”. In fact, for given $\rho\in D$ , taking sufficiently large and$g\in
S_{n}$,
theminimalnumber
of
we
see
the
cyclesofofthenumberdenoteand letting
expressing as their product is equal totranspositions in$e$$S_{\infty}= \bigcup_{n=1}^{\infty}S_{n}$$S_{\infty}$$C_{\rho}$$(2^{k_{2}(\rho)}3^{k_{3}(\rho)}\cdots j^{k_{\mathrm{j}}(\rho)}\cdots)$$\rho\in \mathrm{V}$$j$$k_{j}(\rho)$$\rho$$r(\rho)$$\rho$$g\in C_{\rho}\cap S_{n}$$n$$[g]_{n}$$S_{n}$$g$$n-[g]_{n}=n-\#$ of rows$=$$(|\rho|+|\mathrm{l}\mathrm{e}\mathrm{g}|)-(r(\rho)+|\mathrm{l}\mathrm{e}\mathrm{g}|)$$=$$l(\rho)$.

, which is induced by adding
It is convenient to arrange the diagrams in $V$ according to
one column to the left side (as indicated below) of each diagram in the usual arrangementof the Young lattice.
$l(\rho)$[OI]$\mathrm{R}$$\mathrm{r}$$\mathrm{m}$$\mathrm{B}$$\ovalbox{\tt\small REJECT}$

01$\phi$$\mathrm{B}$$\mathrm{P}$$\mathrm{O}$

ffl ....

$–>$$\Phi$$ff$$\mathrm{m}$$\ovalbox{\tt\small REJECT}$

ffl$\mathrm{B}$....

$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\mathrm{f}\mathrm{f}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\#(\mathrm{P})\overline{-}0$$\iota$2

3

4$\ldots$$.\wedge$

The assignment of edges is in adifferent way from the Young lattice, as is mentione$\mathrm{d}$149
An adjacency operator is formally written as
for $\rho\in D$ . Taking $n\geq|\rho|$and setting, we get an adjacency operator at -level,.
Letbe the vacuum state. In [8], we showed the following central$H_{k}(x)$limit theorem.
denotes the Hermite polynomial of degree obeying the recurrence
formula$xH_{k}(x)=H_{k+1}(x)+kH_{k-1}(x)$ ,$H_{0}(x)=1$ ,$H_{1}(x)=x$ .
$A_{\rho}= \sum_{g\in C_{\rho}}g$$C_{\rho}^{(n)}=C_{\rho}\cap S_{n}$$n$$A_{\rho}^{(n)}=\Sigma_{g\in C_{\rho}^{(n)}}g$$\Phi=\langle\delta_{e}, \cdot\delta_{e}\rangle_{\ell^{2}(\mathrm{S}_{\infty})}$$k$

Theorem 4For $\forall m\in N$ ,$\forall\rho_{1}$,$\rho_{2}$,$\cdots$, $\rho_{m}\in D$ ,$\forall r_{1}$,$r_{2}$,$\cdots$, $r_{m}\in\{0,1,2, \cdots\}$ , we have$\lim_{narrow\infty}\Phi((\frac{A_{\rho 1}^{(n)}}{\sqrt{|C_{\rho_{1}}^{(n)}|}})^{r_{1}}(\frac{A_{\beta 2}^{(n)}}{\sqrt{|C_{\beta 2}^{(n)}|}})^{\mathrm{r}_{2}}\cdots(\frac{A_{\rho_{m}}^{(n)}}{\sqrt{|C_{\rho_{m}}^{(n)}|}})^{\mathrm{r}_{m}})$$= \prod_{j\geq 2}\int_{R}\frac{e^{-x^{2}/2}}{\sqrt{2\pi}}(\frac{H_{k_{j}(\rho_{1})}(x)}{\sqrt{k_{j}(\rho_{1})!}})^{f}1(\frac{H_{k_{\mathrm{j}}(\rho_{2})}(x)}{\sqrt{k_{j}(\rho_{2})!}})^{r_{2}}\cdots(\frac{H_{k_{j}(\rho_{m})}(x)}{\sqrt{k_{j}(\rho_{m})!}})^{\mathrm{r}_{m}}dx$

This result extended Kerov’s theorem in [12]. Indeed, restricted tothe spectral decomposition ofacting on
is given bysuch that $n\geq|\rho|$ ,$\ell^{2}(S_{n})$$A_{\rho}$$A_{\rho}= \sum_{\lambda\in \mathcal{Y}n}\frac{|C_{\rho}^{(n)}|\chi_{\rho}^{\lambda}}{\dim\lambda}E_{\lambda}$$S_{n}$.

,$\Phi(E_{\lambda})=\frac{\dim^{2}\lambda}{n!}$(Plancherel measure)where
denotes the set of Young diagrams withboxes andis the value of theirreducible character corresponding to Ataken on.
As an attempt to develop aquantum aspect of Theorem 4, let us discuss at first a
decomposition offor simplicity. For $e\neq g\in S_{n}$ , we define operators
and
on
$n$

$\mathcal{Y}_{n}$$\chi_{\rho}^{\lambda}$$C_{\rho}$$g^{+}$$A_{\Pi}$$\ell^{2}(S_{n})$$g^{-}$

asif $[gx]<[x]$0otherwise,$\delta_{gx}$$g^{+}\delta_{x}=\{$

if $[gx]>[x]$0otherwise.

$\delta_{gx}$$g^{-}\delta_{x}=\{$

Set. Clearly $A\mathrm{m}=A^{+}+A^{-}$ Let
denote the intersection number$y\in
S_{n}$of the group association scheme, namely, if ,and,$A^{\pm}=\Sigma_{g\in\oplus\cap S_{n}}g^{\pm}$$p_{\tau\rho}^{\sigma}$$\mathcal{X}(S_{n})$$x^{-1}y\in C_{\sigma}$$x$$p_{\tau\rho}^{\sigma}=|\{z\in S_{n}|x^{-1}z\in C_{\tau}, z^{-1}y\in C_{\rho}\}|$(This quantity does not depend on the choice of ,ofto number vectors is as follows.
$x$

$y$

whenever.
$x^{-1}y\in C_{\sigma}.$) The action$A^{\pm}$

Proposition 1Set

$v_{\rho}= \sum_{x\in C_{\rho}\cap \mathrm{S}_{n}}\delta_{x}$for $\rho\in D$ . We have$A^{\pm}v_{\rho}= \sum_{\sigma:l(\sigma)=l(\rho)\pm 1}p_{\mathrm{m}^{\rho}}^{\sigma}v_{\sigma}$.

Proof Note that, if $g=(ij)$ ,and$[gx]>[x]\Leftrightarrow[gx]=[x]+1\Leftrightarrow i$ and$[gx]<[x]\Leftrightarrow[gx]=[x]-1\Leftrightarrow i$

$j$

$j$

are contained in different cycles of ,are contained in the same cycle of .
$x$

$x$

150

$A^{+}v_{\rho}$$=$$\sum_{x\in C_{\rho}}\sum_{g\in c_{\mathrm{n}}}g^{+}\delta_{x}$

$=$

$=$

$=$

$\sum_{y\in S_{\hslash}}|\{(g, x)|x\in C_{\rho}, g\in C_{\Phi}, y=gx, [gx]=[x]-1\}|\delta_{y}$$\sum_{\sigma}\sum_{y\in C_{\sigma}}|\{(g, x)|x\in C_{\rho}, g\in C_{\mathrm{m}}, y=gx, [gx]=[x]-1\}|\delta_{y}$$|\delta_{y}$$\sum_{\sigma:l(\sigma)=l(\rho)+1}\sum_{y\in C_{\sigma}}|\{(g, x)|x\in C_{\rho}, g\in\%, et=gx\}$$=$$\sum_{\sigma:l(\sigma)=l(\rho)+1}p_{\Phi\rho}^{\sigma}v_{\sigma}$

The action of$A^{-}$.

is similar. (Alternatively, use the adjoint relation.)$\mathrm{Q}\mathrm{E}\mathrm{D}$in sufficiently
Let us assume that , $\rho\in D$ satisfy $l(\sigma)=l(\rho)+1$ and consider
splits into
-row
of
a
that
, then we can specify the number suchlarge . If
two rows of (possibly 1-rows). According to the cases: (i) $|\sigma|=|\rho|+2$ , (ii) $|\sigma|=|\rho|+1$ ,, we have(iii)$p_{\Phi^{\rho}}^{\sigma}$$\sigma$$S_{n}$$j$$j$$p_{\mathrm{Q}\rho}^{\sigma}>0$$\sigma$$\rho$$|\sigma|=|\rho|$$p_{\mathrm{m}\rho}^{\sigma}=\{$(i) $k_{2}(\sigma)=k_{2}(\rho)+1$(ii) $jk_{j}(\sigma)=j(k_{j}(\rho)+1)$(iii) $C_{j}k_{j}(\sigma)=Cj(kj(\rho)+1)$

is the number of cutting a -polygon into two pieces so that each has at least two
where
does not depend on if $l(\sigma)=l(\rho)+1$ . This property enables
vertices. In particular,such that $l(\sigma)=l(\rho)+1$ in V. (Especially,for pair
us to assign multiplicity) We define anormalised number vector $\Phi(\rho)=$they are joined by definition if
and afinite-dimensional Fockwith respect to the usual inner product in. Proposition 1yields
space
$j$

$C_{j}$$n$$p_{\mathrm{n}\rho}^{\sigma}$$(\rho, \sigma)$$p_{\mathrm{m}\rho}^{\sigma}$$\mathrm{a}_{\rho}^{\sigma}>0.$$\ell^{2}(S_{n})$$v_{\rho}/\sqrt{|C_{\rho}|}$$\Gamma(S_{n})=C\Phi(\emptyset)\oplus\oplus_{\rho\in D,|\rho|\leq n}C\Phi(\rho)$$\frac{1}{\sqrt{|C_{0}|}}A^{\pm}\Phi(\rho)$. If $l(\sigma)=l(\rho)+1$ , only the term of $|\sigma|=|\rho|+2$ survives in the right
Clearly
hand side sum as $narrow\infty$ , for which we have$|C_{\rho}|\vee\wedge n^{|\rho|}$$arrow\sqrt{k_{2}(\rho)+1}$

Usingsatisfies$|C_{\rho}|p_{\mathrm{O}^{\sigma}}^{\rho}=|C_{\sigma}|p_{\mathrm{q}\rho}^{\sigma}$, we see, if$1(\mathrm{a})=1(\mathrm{p})-1$$(\sigma=\rho \mathrm{u}\mathrm{m})$.

, the only surviving term of $|\rho|=|\sigma|+2$$arrow\sqrt{k_{2}(\rho)}$$(\sigma=\rho\backslash \mathrm{m})$.

151, whereLet us introduce aFock space t(the vacuum state)Oandare normalised vectors. The above discussion leads to the following assertion.
$\ovalbox{\tt\small REJECT}$$C\mathrm{V}(\mathrm{O})$$\mathrm{V}(\mathrm{O})$$\mathrm{e}_{\ovalbox{\tt\small REJECT}^{\mathrm{p}}}C^{\ovalbox{\tt\small REJECT}}\mathrm{I}^{\ovalbox{\tt\small REJECT}}(\mathrm{p})$$\ovalbox{\tt\small REJECT} \mathrm{I}^{\ovalbox{\tt\small REJECT}}(\mathrm{p})’ \mathrm{s}$Proposition 2Associated with the decomposition $A_{\mathrm{Q}}=A^{+}+A^{-}$ onquantum central limit theorem holds:$S_{n}$, the following$\langle\Phi(\sigma), \frac{A^{\epsilon_{1}}}{\sqrt{|*|}}\frac{A^{\epsilon_{2}}}{\sqrt{|*|}}\cdots\frac{A^{\epsilon_{m}}}{\sqrt{|C_{\mathrm{D}}|}}\Phi(\rho)\rangle_{\Gamma(S_{n})}arrow\langle\Psi(\sigma), a^{\epsilon_{1}}a^{\epsilon_{2}}\cdots a^{\epsilon_{m}}\Psi(\rho)\rangle_{\Gamma}$

as$narrow \mathrm{o}\mathrm{o}$operators$a^{\pm}$

for $\forall m\in N$ ,are defined as$\forall\epsilon_{1}$,$\epsilon_{2}$,$\cdots$,$\epsilon_{m}\in\{+$$a^{+}\Psi(\rho)=\sqrt{k_{2}(\rho)+1}\Psi(\rho \mathrm{u}\mathrm{m})$

,

,-

$\}$

, Va,

$\rho\in D\cup\{\emptyset\}$, where the limit$a^{-}\Psi(\rho)=\sqrt{k_{2}(\rho)}\Psi(\rho\backslash \mathrm{m})$.

Hence, if we start from the vacuum state,
do not take us out ofand act on it in the same way as the creator and the annihilator on a
Boson Fock space. Obviously, classical reduction of Proposition 2yields awell-known
Gaussian limit (included in Kerov’s theorem [12]).
Beyond the vacuum expectation, let us discuss acentral limit theorem measured by thestate associated with Kerov-Olshanski-Vershik’s generalised regular representation of the. In [13], Kerov-Olshanski-Vershik introduced an interesting
infinite symmetric group
deformation of the regular representation of
by using a1-cocycle containing acomplex
parameter . The representation space is the -space on aprojective limit of probability
spaces. Here $\mu_{t}^{n}(\{x\})=$.
for $x\in S_{n}$ , $t=|z|^{2}$ and $(t)_{n}=t(t+1)\cdots(t+$$n-1)$ . This representation gives acentral positive-definite function, which canon
be expressed onas$\Psi(\emptyset)$$a^{\pm}$$C\Psi(\emptyset)\oplus$$\oplus_{k=1}^{\infty}C\Psi((2^{k}))$$S_{\infty}$$S_{\infty}$$L^{2}$$z$$t^{[x]}/(t)_{n}$$(S_{n}, \mu_{t}^{n})$$S_{\infty}$$\phi_{z}$$S_{n}$$\phi_{z}|_{S_{n}}(x)=\sum_{u\in S_{n}}z^{[x^{-1}u]-[u]_{\frac{t^{[u]}}{(t)_{n}}}}=\sum_{\lambda\in \mathcal{Y}n}M_{z}(\lambda)\frac{\chi_{\rho}^{\lambda}}{\dim\lambda}$$M_{z}( \lambda)=\frac{1}{(t)_{n}}\prod_{(ij)\in\lambda}|z+\dot{g}-i|^{2}\frac{\dim^{2}\lambda}{n!}$$(x\in C_{\rho}\cap S_{n})$$(z\in C\backslash \{0\})$,.

by $C$-linear extension. The limiting case
We obtain atracial statecorresponds to the regular representation, and thenis interpreted as the vacuum vector
state. Set.
where
Let us work on . Our state
has density matrix. Asimple compu$=(z+\overline{z})/(t+1)$ . If $z+\overline{z}=0$ , thenation yields
mean 0and variance$\phi_{z}|_{S_{n}}$$zarrow \mathrm{o}\mathrm{o}$$\phi_{\infty}$$\phi_{z}(\rho)=\phi_{z}(w)$$S_{n}$$\phi_{z}(\mathrm{m})$$w\in C_{\rho}$$\phi_{z}$$\sum_{|\rho|\leq n}\phi_{z}(\rho)A_{\rho}$$A_{\mathrm{m}}\mathrm{h}\mathrm{a}\mathrm{s}$$\phi_{z}(A_{\mathrm{m}}^{2})=\frac{n(n-1)}{2}\{1+\frac{2(n-2)}{t+2}+\frac{(n-2)(n-3)}{(t+2)(t+3)}\}$.

This suggests acentral limit theorem for the adjacency operator
respect toas$narrow\infty$ and $n/t$ converges to anonzero value. (Compare it with Theorem 3.)
It is interesting to understand the limiting expectation with respect toas asuperposition of matrix elements similarly to the discussion following Theorem 3. In that
$A_{\mathrm{m}}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}$$\phi_{z}$$\phi_{z}$152situation,
culty that
that

and q
0. However, we have now adifficonverged as. Assumewith tdoes not converge in general as
0. For special diagrams (21), we have$d\ovalbox{\tt\small REJECT}^{1}\rangle\circ \mathrm{p}$$\ovalbox{\tt\small REJECT}$$q_{\ovalbox{\tt\small REJECT}}^{h}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}$$r/d\ovalbox{\tt\small REJECT}$$o_{\ovalbox{\tt\small REJECT}}(p)\ovalbox{\tt\small REJECT}$$z+\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$n-=\ovalbox{\tt\small REJECT}+\mathrm{o}\mathrm{o}$$\ovalbox{\tt\small REJECT}$$|\mathrm{s}|^{2}\ovalbox{\tt\small REJECT} \mathrm{x}n$if is even$l$$\frac{1}{(t)_{2l}}(\frac{l!}{(l/2)!}t^{l}+O(t^{l-1}))$$\phi_{z}((2^{l}))=\{$

0if Iis odd.
Hence
have

$\phi_{z}((2^{l}))\sqrt{|C_{(2^{l})}|}$

converges as$t\wedge\vee narrow\infty$. On the other hand, for the cycles, we$\phi_{z}((2k-1))=\frac{\mathrm{p}\mathrm{o}1\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{a}1\mathrm{o}\mathrm{f}\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}k\mathrm{i}\mathrm{n}t}{(t)_{2k-1}}$

Actually, we conjecture that, for $\rho=$$\phi_{z}(\rho)$$(2^{k_{2}}3^{k_{3}}4^{k_{4}}\cdots)$,$\phi_{z}((2k))=0$,$=$$O(t^{k_{2}+2k_{3}+2k_{4}+3k_{6}+3k_{6}+4k_{7}+}\ldots)/(t)_{|\rho|}$$=$$O(1/t^{k_{2}+k_{\}+2k_{4}+2k_{6}+3k_{6}+3k_{7}+4k_{8}+}\ldots)$may possibly diverge byholds and hence
vectornumbertake another normalised$\phi_{z}(\rho)\sqrt{|C_{\rho}|}$.

$n^{(k_{S}+k_{6}+k_{7}+\cdots)/2}$$\Phi’(\rho)=v_{\rho}/\sqrt{\frac{|C_{\rho}|}{n^{k,(\rho)+k_{6}(\rho)+}}}\ldots$

as$t_{\wedge}\vee narrow\infty$. If we,this problem is overcome and we can still control the convergence of the branching
the appropriate normalisation. However, thecoefficients of the action of
in Proposition 2have more complicated actions on the Fock space
limit operators. Details will appear elsewhere later on.
$A\pm \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}$$a^{\pm}$$C\Psi(\emptyset)\oplus\oplus_{\rho\in \mathcal{D}}C\Psi(\rho)$

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measures, Infinite Dimen. Anal. Quantm Prob. 1(1998), 663 –670.

[2] L.Accardi, N.Obata, Introduction to algebraic probability (in Japanese), Nagoya
Mathematical Lectures 2, Nagoya Univ., 1999.

[3] E.Bannai, T.Ito, Algebraic combinatorics I, Menlo Park,, 1984.

California,

Ben-$\mathrm{j}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{n}/\mathrm{C}\mathrm{u}\mathrm{m}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}$[4] Y.Hashimoto, Deformations of the semicircle law derived from random walks on freegroups, Prob. Math. Stat. 18 (1998), 399-410.
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