Nonintersecting Paths, Noncolliding Diffusion Processes and Representation Theory

0.23 MB·English

by  Makoto Katori

Nonintersecting Paths, Noncolliding Diffusion Processes
and Representation Theory
∗
Makoto Katori
†
Department of Physics, Faculty of Science and Engineering,
Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
Hideki Tanemura
‡
5
0 Department of Mathematics and Informatics, Faculty of Science,
0 Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan
2

n
a Abstract The system of one-dimensional symmetric simple random walks, in which none of walkers have
J
met others in a given time period, is called the vicious walker model. It was introduced by Michael Fisher
4
and applications of the model to various wetting and melting phenomena were described in his Boltzmann
1
medal lecture. In the present report, we explain interesting connections among representation theory, proba-

] bility theory, and random matrix theory using this simple diffusion particle system. Each vicious walk of N
R
walkers is represented by an N-tuple of nonintersecting lattice paths on the spatio-temporal plane. There
P is established a simple bijection between nonintersecting lattice paths and semistandard Young tableaux.
h. BasedonthisbijectionandsomeknowledgeofsymmetricpolynomialscalledtheSchur functions,wecangive
t a determinantal expressionto the partition function of vicious walks, which is regarded as a special case of the
a
m Karlin-McGregor formula in the probability theory (or the Lindstr¨om-Gessel-Viennot formula in the
enumerative combinatorics). Due to a basic property of Schur function, we can take the diffusion scaling limit
[
of the vicious walksand define a noncolliding system of Brownian particles. This diffusion process solves

1 the stochasticdifferentialequationswiththe drift termsactingas the repulsivetwo-bodyforcesproportionalto
v the inverse of distances between particles, and thus it is identified with Dyson’s Brownian motion model.
8
In other words, the obtained noncolliding system of Brownian particles is equivalent in distribution with the
1
eigenvalue process of a Hermitian matrix-valued process.
2
1
0
1 Vicious Walks, Young Tableaux and Schur Functions
5
0
/ x
h Let S(t) ,P be the N-dimensional Markov chain starting from
t=0,1,2,...
{ }
t
a x=(cid:16)(x1,x2,...,xN), su(cid:17)ch that the coordinates Si(t),i =1,2,...,N, are independent simple random walks on
m Z. We always take the starting point x from the set
:
v ZN = x=(x ,x ,...,x ) (2Z)N :x <x <...<x .
i < 1 2 N ∈ 1 2 N
X
n o
r We consider the condition that any of walkers does not meet other walkers up to time T >0, i.e.
a
S (t)<S (t)< <S (t), t=1,2,...,T. (1.1)
1 2 N
···
x x
We denote by Q the conditional probability of P under the event Λ = S (t) < S (t) < < S (t), t =
T T 1 2 ··· N
0,1,...,T . M. Fisher called the process S(t) ,Qx the viciouns walker model in his Boltzmann
{ }t=0,1,2,...,T T
medal lecoture [4]. (cid:16) (cid:17)
We will assume the initial positions as
S (0)=2(i 1), i=1,2, ,N (1.2)
i
− ···
∗This manuscript is based on the talks at the Institute of Applied Mathematics, Chinese Academy of Science, Beijing, China,
on27th September 2004, and inthe workshop ‘Combinatorial Methods in Representation Theory and Their Applications’ (19-22
October 2004) at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan; to be published in RIMS
Kokyuroku.
†Electronic mail: [email protected]
‡Electronic mail: [email protected]
1
t
T y1 y2 y3 y4
6 6 6 6
5
5
4
4 4 4
3 3
2 2
1
x
0 1 2 3 4 5 6 7 8 9 10 11
Figure 1: An example of vicious walk in the case N =4,T =6.
in Sections 1 in this report.
Each realization of vicious walk is represented by an N-tuple of nonintersecting lattice paths on the
1+1 spatio-temporal plane, Z 0,1, ,T . An example is given by Figure 1 in the case that four walkers
×{ ··· }
(N =4) perform a noncolliding walk up to time T =6.
2 3 4 6
4 4 6
5 6
(a) (b)
λ
L
Figure 2: (a) Young diagram and (b) Young tableau
T
corresponding to the vicious walk in Figure 1. Figure 3: A pair of conjugate YD’s.
Bijection between such nonintersecting lattice paths and semistandard Young tableaux (SSYT), =
T
( (i,j)), is established by the following procedure [8, 17].
T
(1) For 1 j N, let
≤ ≤
L the number of leftward steps among T steps of the j-th walker.
j
≡
Draw a collection of boxes with N columns, in which the number of boxes in the j-th column is L . (We
j
number columns from the left to the right.) Since L (L ,L ,L ,L )=(3,3,2,1) in the walk shown in
1 2 3 4
≡
Figure 1, we draw the collection of boxes as shown in Figure 2 (a) for this example.
(2) For each walker, we label each leftward step by the integer 1,2, ,T , which is the time when that
∈{ ··· }
leftward step was done. See Figure 1, in which labels of leftward steps are indicated by integers in small
circlesassociatedwiththelinesegmentsshowingleftwardsteps. Thenforthej-thcolumnofthecollection
of boxes, fill the boxes by the labels of leftward steps of the j-th walker, from the top to the bottom,
1 j N. For the walk given in Figure 1, we have the boxes with integers shown in Figure 2 (b). Let
≤ ≤
(i,j)=the integer in the box located in the i-th row and j-th column.
T
For example, (1,3)=4 and (3,1)=5 in this case.
T T
2
Remark 1. The above procedure with the nonintersecting condition (1.1) guarantees the inequalities
L L L , (1.3)
1 2 N
≥ ≥···≥
and
(i,j)< (i+1,j), strictly increasing in each column,
T T
(i,j) (i,j+1), weakly increasing in each row. (1.4)
T ≤T
Assume that the number of rows in the collection of boxes is ℓ. Let
λ the number of boxes in the i-th row, i=1,2, ,ℓ.
i
≡ ···
Then the inequalities (1.3) imply
λ λ λ . (1.5)
1 2 ℓ
≥ ≥···≥
The collections of boxes with such conditions concerning the numbers of boxes in rows (1.5) (and in
columns (1.3)) are called Young diagram (YD). The number of rows ℓ in the YD is called the length
of YD. In the present situation, ℓ T, in general. (In our example in Figure 2 (b), ℓ = 3 for T = 6.)
≤
YD’s with integers with the conditions (1.4) are called semistandard Young tableaux (SSYT).
Remark 2. YD with λ boxes in the i-th row, 1 i ℓ, is said to be the YD of shape λ = (λ , ,λ ).
i 1 ℓ
≤ ≤ ···
The YD with the shape L=(L , ,L ) is regardedas the conjugate of the YD with the shape λ and
1 N
···
denoted by
L=λ.
As shown in Figure 3, they are mirror images with respect to the diagonal line.
e
A sequence of integers with the condition (1.5), that is,
λ=(λ ,λ , ,λ ) with λ N x Z,x 0 , 1 i T, λ λ λ ,
1 2 T i 1 2 T
··· ∈ ≡ ∈ ≥ ≤ ≤ ≥ ≥···≥
n o
isregardedasapartitionofanintegern= T λ . WeintroduceasetofT variablesz= z ,z , ,z CT
i=1 i { 1 2 ··· T}∈
and define a monomial
P
zT = z
T(i,j)
(Yi,j)
T
#oftimesthattheinteger k occursinT
= z .
k
k=1
Y
For example, the monomial corresponding to the SSYT shown in Figure 2 (b) is
T
zT = z z z z
2 3 4 6
× × ×
z z z
4 4 6
× × ×
z z
5 6
× ×
= z z z3z z3.
2 3 4 5 6
Notice that for one YD with a given shape λ, there are different ways of filling boxes with integers to make
SSYT’s satisfying the conditions (1.4). For each YD with shape λ, we define a polynomial of z= z , ,z
1 T
{ ··· }
by summing zT over all SSYT defined on the YD:
s (z ,z , ,z )= zT.
λ 1 2 T
···
T:allSSYTwiththesameshapeλ
X
ThispolynomialiscalledtheSchur functionindexedby(the partition/YDwithshape)λonz=(z , ,z ).
1 T
···
We can prove the two formulae (Jacobi-Trudi formulae). The first one is the following.
3
Lemma 1
det zλj+T−j
i
1≤i,j≤T" #
s (z , ,z )= ,
λ 1 T
···
det zT−j
i
1≤i,j≤T" #
where the denominator is the Vandermonde determinant evaluated as the product of differences,
det zT−j = (z z ). (1.6)
1≤i,j≤T" i # i− j
1≤i<j≤T
Y
This formula clarifies that the Schur functions are symmetric polynomials in z=(z , ,z ).
1 T
···
For the second formula, we define the polynomials e (z , ,z )’s as the coefficients in the expansion
j 1 T
···
T T
(1+z ξ)= e (z , ,z )ξj. (1.7)
i j 1 T
···
i=1 j=0
Y X
Then
e (z , ,z ) = sum of all monomials in the form z z z
j 1 ··· T i1 i2··· ij
for all strictly increasing sequences 1 i <i < <i T.
1 2 j
≤ ··· ≤
e (z , ,z )’s are also symmetric polynomials in z , ,z and called the j-th elementary symmetric
j 1 T 1 T
··· ···
polynomials.
Lemma 2 Assume that the conjugate of λ is given by λ=(λ , ,λ ) with length N. Then
1 N
···
e e e
s (z , ,z )= det e (z , ,z ) .
λ 1 ··· T 1≤i,j≤N" λj+(i−j) 1 ··· T #
e
More details for YD, SSYT and symmetric polynomials, see e.g. Fulton (1997) [5].
Now we go back to the vicious walker model. We notice a simple relation between the partition L =
(L , ,L )andthefinalpositionsoftheN viciouswalkersattime T,y=(y , ,y ) (S (T), ,S (T)),
1 N 1 N 1 N
··· ··· ≡ ···
given by y = T 2L +2(i 1), 1 i N, on the initial condition (1.2). Then we have established the
i i
− − ≤ ≤
following relation between the vicious walks and YD/SSYT/Schur functions.
Set the initial positions as x = (0,2, ,2(N 1)). For y such that y ZN, if T 2N, y + 1
(y +1, ,y +1) ZN, if T0 +1 2N·,·l·et L=−(L , ,L ) with L =(T +∈x < y )/2,∈1 i N, an≡d
1 ··· N ∈ < ∈ 1 ··· N i 0i− i ≤ ≤
λ=L. Then
e positions y of vicious walkers
•
at the finial time T YD with the shape λ
⇐⇒
a realization of vicious walk
•
from S(0)=x to S(T)=y an SSYT with the shape λ
0
⇐⇒ T
a set of all vicious walks
•
from S(0)=x to S(T)=y a Schur functions (z , ,z ).
0 λ 1 T
⇐⇒ ···
Fory=(y ,y , ,y )suchthaty ZN,ifT 2N,y+1 ZN,ifT+1 2N,andx =(0,2, ,2(N 1)),
1 2 ··· N ∈ < ∈ ∈ < ∈ 0 ··· −
4
define the number
M (y) = total number of distinct realizations of vicious walk of N walkers
N,T
from the positions S(0)=x to S(T)=y.
0
The above relations prove the following identity.
T +2(i 1) y
Assume that L = − − i,L=(L , ,L ), and λ=L. Then
i 1 N
2 ···
e
M (y)=s (z , ,z ) .
N,T λ 1 T
··· (cid:12)
(cid:12)z1=z2=···=zT=1
(cid:12)
(cid:12)
(cid:12)
Lemma 1 gives the following estimate for M (y), via appropriate q-factorization and the formula for
N,T
the Vandermonde determinant (1.6).
Proposition 3
M (y) = lims (1,q,q2, ,qT−1)
N,T λ
q→1 ···
= limq Tk=1(k−1)λk qλi−λj+j−i−1 = λi−λj +j−i.
q→1 qj−i 1 j i
P 1≤i<j≤T − 1≤i<j≤T −
Y Y
On the other hand, by definition (1.7), it is easy to see that
T T!
e (z , ,z ) = zT zT.
j 1 T
··· (cid:12) j ≡ j!(T j)!
(cid:12)z1=···=zT=z (cid:18) (cid:19) −
(cid:12)
Then Lemma 2 gives the following. (cid:12)
(cid:12)
Proposition 4
T T
M (y) = det = det
N,T
1≤i,j≤N(cid:20)(cid:18)λj +i−j(cid:19)(cid:21) 1≤i,j≤N(cid:20)(cid:18)Lj +i−j(cid:19)(cid:21)
T
= det e .
1≤i,j≤N(cid:20)(cid:18){T +2(i−1)−yj}/2(cid:19)(cid:21)
2 Determinantal Formula for Nonintersecting Paths
Since we have assumed the initial positions as (1.2), we can see that the (i,j)-element of the matrix in the
determinant in Proposition 4 is
T
= # lattice path from 2(i 1) at time 0 to y at time T
j
T +2(i 1) y /2 −
(cid:18){ − − j} (cid:19) n o
= w(path) .
(cid:12)
alllatticepaths: (X2(i−1),0);(yj,T) (cid:12)w(path)=1
(cid:12)
If we define anappropriateweightfunction w(path) onsingle lattice paths, the summ(cid:12) ationofw(path) will give
(cid:12)
the Green function of single lattice paths,
G (x,0),(y,T) = w(path).
(cid:16) (cid:17) alllatticepathXs: (x,0);(y,T)
5
Proposition 4 can be regarded as a special case of the Karlin-McGregor formula in the probability theory
[10, 11], and the Lindstr¨om-Gessel-Viennot formula in the enumerative combinatorics (see [19, 17] and
referencestherein). Inordertoexplainthisfact,hereweintroducesomedefinitionsandnotationsfordescribing
lattice paths.
Let V = vertex ,E = directed edge , D =(V,E)= an acyclic directed graph, where acyclic means that
{ } { }
any cycles of directed edges are forbidden. For u,v V,
∈
a lattice path u v = a sequence of directed edges from u to v,
→
(u,v) = the set of all lattice paths from u to v.
P
A weight function w : E Z[[x : e E]] is introduced, where Z[[x : e E]] denotes a ring of formal power
e e
→ ∈ ∈
series of x : e E and the weight on a lattice path P is defined by w(P) w(e). Then the Green
e
{ ∈ } ≡
e∈P
Y
function of lattice paths from u to v is defined by
G(u,v)= w(P).
P:P∈XP(u,v)
Let I = u ,u , ,u ,J = v ,v , ,v with u ,v V,i = 1,2, ,N. The sets I,J are ordered;
1 2 N 1 2 N i i
{ ··· } { ··· } ∈ ···
u <u < <u ,v <v < <v . Then we consider a set of N-tuples of lattice paths
1 2 N 1 2 N
··· ···
(I,J)= P=(P , ,P ):P (u ,v ),i=1,2, ,N .
1 N i i i
P ··· ∈P ···
n o
The weight for each N-tuple of lattice paths is given by
N N
w(P)= w(P )= w(e).
i
iY=1 iY=1eY∈Pi
We say ‘lattice paths P and Q intersect’, if P and Q share at least one common vertex. Then, for ordered
sets of vertices I and J, we say ‘I is D-compatible with J’ in the case that, whenever u < u in I and
i j
v <v in J, every lattice path P (u ,v ) intersects every lattice path Q (u ,v ). A set of N-tuples of
i j i j j i
∈P ∈P
nonintersecting lattice paths is denoted by
(I,J)= P (I,J):any lattice paths in P do not intersect with others ,
0
P ∈P
n o
and the Green function of N-tuples of nonintersecting lattice paths is defined by
G (I,J)= w(P).
nonint
P∈PX0(I,J)
Theorem 5 Let I = (u , ,u ) and J = (v , ,v ) be two ordered sets of vertices in an acyclic graph
1 N 1 N
··· ···
D. If I is D-compatible with J, then the Green function of the nonintersecting N-tuples of lattice paths is
given by
G (I,J)= det G(u ,v ) ,
nonint i j
1≤i,j≤N" #
where G(u,v) denotes the Green function of single lattice paths from u to v on D.
The proofof Theorem5 is givenin Appendix A following Stembridge (1990)[19]. For our vicious walkermodel,
consider the directed graph D =(V,E), where
V = (x,t) Z2 :x+t=even, t=0,1,2, ,T ,
∈ ···
n o
6
and all edges connecting the nearest-neighbor pairs of vertices in V are oriented to the positive direction of t
axis. Set the weight function
1 for e= (x,t 1) (x 1,t)
h − → − i
w(e)= 1 for e= (x,t 1) (x+1,t)
 h − → i
0 otherwise.

Set T N and u = (x ,0),v = (y ,T) V,i = 1,2, ,N. Then the Green function of single lattice paths
i i i i
∈ ∈ ···
from u to v is
i i
T
G(u ,v )= (u v ) = .
i i i i
P → (T +x y )/2
(cid:12) (cid:12) (cid:18) i− i (cid:19)
(cid:12) (cid:12) N
Theorem5then givesthe Greenfunction of(cid:12)the N-tuple(cid:12)sofnonintersectinglattice paths fromI = (x ,0)
i
i=1
N T n o
to J = (y ,T) as det .
i
Fornx ZN,oai=n1d y s1u≤cih,j≤thNa(cid:20)t(cid:18)y(T +ZNx,i−if Tyj)/22(cid:19)N(cid:21), y+1 ZN, if T +1 2N, define
∈ < ∈ < ∈ ∈ < ∈
M (T,yx) = total number of distinct realizations of vicious walk of N walkers
N
|
from the positions x=(x , ,x ) to y=(y , ,y ) during time T.
1 N 1 N
··· ···
Proposition 4 is now generalized as follows.
T
Proposition 6 M (T,yx)= det .
N
| 1≤i,j≤N(cid:20)(cid:18)(T +xi−yj)/2(cid:19)(cid:21)
3 Diffusion Scaling Limit
x
Recall that ( S(t) ,Q ) denotes the vicious walk with the noncolliding condition up to time T > 0
startingfrom{thep}ots=it0i,1o,n2,s..x.,T ZTN. ForL 1,weconsiderprobabilitymeasuresµx onthespaceofcontinuous
∈ < ≥ L,T
paths C([0,T] RN) defined by
→
1
µx ()=Qx S(L2t) ,
L,T · L2T L ∈·
(cid:18) (cid:19)
where S(t),t 0, is now considered to be the interpolation of the N-dimensional random walk S(t),t =
≥ x
0,1,2,.... We study the limit of the probability measure µ , L .
L,T →∞
We put
RN = x RN :x <x < <x ,
< ∈ 1 2 ··· N
n o
whichcanbecalledtheWeylchamberoftypeA (see,forexample,[6]). ByvirtueoftheKarlin-McGregor
N−1
formula [10, 11], the transition density function f (t,yx) of the absorbing Brownian motion in RN and
the probability (t,x) that the Brownian motionNstart|ing from x RN does not hit the boundary of R<N up
NN ∈ < <
to time t>0 are given by
f (t,yx)= det 1 e−(xj−yi)2/2t , x,y RN,
N | 1≤i,j≤N √2πt ∈ <
(cid:20) (cid:21)
and (t,x)= dyf (t,yx), respectively. We put h (x)= (x x ), and let 0 = (0,0, ,0)
N N N j i
N RN | − ··· ∈
Z < 1≤iY<j≤N
RN, which describes the state that all N particles are at the origin 0.
7
Theorem 7 (i) For any fixed x ZN and T > 0, as L , µx () converges weakly to the law of
∈ < → ∞ L,T ·
the temporally inhomogeneous diffusion process X(t) = (X (t),X (t),...,X (t)),t [0,T], with
1 2 N
∈
transition probability density g (s,x;t,y);
N,T
g (0,0;t,y)=c TN(N−1)/4t−N2/2exp |y|2 h (y) (T t,y),
N,T N N N
− 2t N −
(cid:26) (cid:27)
f (t s,yx) (T t,y)
N N
g (s,x;t,y)= − | N − ,
N,T
(T s,x)
N
N −
for 0<s<t T, x,y RN, where c =2−N/2/ N Γ(i/2) with the gamma function Γ.
≤ ∈ < N i=1
(ii) The diffusion process X(t) solves the following equation:
Q
dX (t)=dB (t)+bT(t,X(t))dt, t [0,T], i=1,2,...,N,
i i i ∈
where B (t), i=1,2,...,N, are independent one-dimensional Brownian motions and
i
∂
bT(t,x)= ln (T t,x), i=1,2,...,N.
i ∂x NN −
i
Figure 4 illustrates the process X(t),t [0,T], when all N particles start from the origin; X(0)=0.
∈
t
T
O x
Figure 4: The process X(t),t [0,T], starting from 0.
∈
Although µx () is the probability measure defined on C([0,T] RN), it can be regarded as that on
L,T · →
C([0, ) RN) concentrated on the set w C([0, ) RN) : w(t) = w(T), t T . Next we consider the
∞ → { ∈ ∞ → ≥ }
case that T =T(L) goes to infinity as L .
→∞
Corollary 8 (i) Let T(L) be an increasing function of L with T(L) as L . For any fixed x ZN,
as L , µx () converges weakly to the law of the tempora→lly∞homog→en∞eous diffusion pr∈oce<ss
→ ∞ L,T(L) ·
Y(t)=(Y (t),Y (t),...,Y (t)),t [0, ), with transition probability density p (s,x;t,y);
1 2 N N
∈ ∞
p (0,0;t,y)=c′ t−N2/2exp |y|2 h (y)2,
N N − 2t N
(cid:26) (cid:27)
1
p (s,x;t,y)= f (t s,yx)h (y), (3.1)
N N N
h (x) − |
N
for 0<s<t< , x,y RN, where c′ =(2π)−N/2/ N Γ(i).
∞ ∈ < N i=1
(ii) The diffusion process Y(t) solves the equations of Dyson’s Brownian motion model with the param-
Q
eter β =2,
1
dY (t)=dB (t)+ dt, t [0, ), i=1,2,...,N.
i i
Y (t) Y (t) ∈ ∞
i j
1≤j≤N,j6=i −
X
8
Here we only give the proof of a key lemma used to prove Theorem 7, in order to demonstrate that the
Schur function plays an important role. See Katori and Tanemura [12] for the complete proofs of Theorem 7
and Corollary 8.
For L>0 we introduce the following functions:
Lx
φ (x)=2 , x R, and φ (x)= φ (x ),φ (x ),...,φ (x ) , x RN,
L L L 1 L 2 L N
2 ∈ ∈
(cid:20) (cid:21) (cid:16) (cid:17)
where[a]denotesthelargestintegernotgreaterthana. LetV (T,yx)=2−NTM (T,yx),whereM (T,yx)
N N N
| | |
is given by Proposition 6.
Lemma 9 For t>0, x ZN and y RN.
∈ < ∈ <
N
L
V φ (t),φ (y) x
2 N L2 L
(cid:18) (cid:19) (cid:16) (cid:12) (cid:17)
=c′ t−N2/2h (cid:12)(cid:12)x exp |y|2 h (y) 1+ |y| ,
N N L − 2t N O L
(cid:16) (cid:17) (cid:26) (cid:27) (cid:18) (cid:18) (cid:19)(cid:19)
as L .
→∞
Proof. Itwillbeenoughtoconsiderthecasethatx=2u=(2u , ,2u ) ZN,y=2v=(2v , ,2v ) ZN,
and φ (t)=2ℓ,ℓ Z , where Z = 1,2,3, . Then 1 ··· N ∈ < 1 ··· N ∈ <
L2 + +
∈ { ···}
2ℓ
M φ (t),φ (y) x =M (2ℓ,2v2u)= det ,
N(cid:16) L2 L (cid:12) (cid:17) N | 1≤i,j≤N(cid:20)(cid:18)ℓ+uj −vi(cid:19)(cid:21)
(cid:12)
and (cid:12)
2ℓ (2ℓ)!
=
ℓ+u v (ℓ+u v )!(ℓ u +v )!
(cid:18) j − i(cid:19) j − i − j i
(2ℓ)!
= A (ℓ,v,u),
ij
(ℓ v )!(ℓ+v )!
i i
−
with
(ℓ+v u +1)
A (ℓ,v,u)= i− j uj,
ij
(ℓ v +1)
− i uj
where we have used the Pochhammer symbol; (a) 1, (a) =a(a+1) (a+i 1), i 1. Then
0 i
≡ ··· − ≥
N
(2ℓ)!
M φ (t),φ (y) x = det A (ℓ,v,u) . (3.2)
N L2 L (ℓ vi)!(ℓ+vi)!1≤i,j≤N" ij #
(cid:16) (cid:12) (cid:17) iY=1 −
(cid:12)
(cid:12)
The leading term of det A (ℓ,v,u) in L is
ij
1≤i,j≤N" # →∞
ℓ+v uj
i
D (v,u) = det
1
1≤i,j≤N(cid:20)(cid:18)ℓ−vi(cid:19) (cid:21)
ℓ+v uN−j+1
= ( 1)N(N−1)/2 det i .
− 1≤i,j≤N(cid:20)(cid:18)ℓ−vi(cid:19) (cid:21)
Let ξ(u)=(ξ (u),...,ξ (u)) be a partition specified by the starting point 2u defined by
1 N
ξ (u)=u (N j), j =1,2,...,N.
j N−j+1
− −
We have
N−j
ℓ+v ℓ+v ℓ+v
D1(v,u) =(−1)N(N−1)/21≤di,ej≤tN"(cid:18)ℓ−vii(cid:19) #sξ(u)(cid:18)ℓ−v11,...,ℓ−vNN(cid:19)
9
ℓ+v ℓ+v ℓ+v ℓ+v
=(−1)N(N−1)/2 ℓ vi − ℓ vj sξ(u) ℓ v1,...,ℓ vN
1≤i<j≤N(cid:18) − i − j(cid:19) (cid:18) − 1 − N(cid:19)
Y
2ℓ(v v ) ℓ+v ℓ+v
j i 1 N
= (ℓ v )(−ℓ v )sξ(u) ℓ v ,...,ℓ v ,
1≤i<j≤N − i − j (cid:18) − 1 − N(cid:19)
Y
where we have used Lemma 1 for the Schur function associated to the partition ξ(u). Proposition 3 gives
ξ (u) ξ (u)+j i
i j
sξ(u)(1,1,...,1)= −j i − .
1≤i<j≤N −
Y
Therefore the leading term of D (v,u) in L is
1
→∞
2(v v )
j i
D2(v,u) = ℓ− ×sξ(u)(1,1,...,1)
1≤i<j≤N
Y
1
= ℓ−N(N−1)/22N(N−1)/2h (v)h (u)
N N
j i
1≤i<j≤N −
Y
N
v 1
= h h (2u) . (3.3)
N N
ℓ Γ(i)
(cid:16) (cid:17) iY=1
On the other hand, by Stirling’s formula we see that
N (2ℓ)! N v2 −ℓ−1/2 1 v /ℓ vi 1
=(ℓπ)−N/222Nℓ 1 i − i 1+ . (3.4)
(ℓ v )!(ℓ+v )! − ℓ2 1+v /ℓ O ℓ
i=1 − i i i=1(cid:18) (cid:19) (cid:18) i (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
Y Y
From (3.2), (3.3) and (3.4)
V φ (t),φ (y) x =2−2NℓM φ (t),φ (y) x
N L2 L N L2 L
(cid:16) 2 N/2 (cid:12)(cid:12) v(cid:17) (cid:16) v2 (cid:12)(cid:12) (cid:17) v
=c′ h (cid:12) h (2u)exp | | 1+(cid:12) | |
N ℓ N ℓ N − ℓ O ℓ
(cid:18) (cid:19) (cid:16) (cid:17) (cid:26) (cid:27)(cid:18) (cid:18) (cid:19)(cid:19)
=c′ 2 Nt−N2/2h x exp |y|2 h (y) 1+ |y| .
N L N L − 2t N O L
(cid:18) (cid:19) (cid:16) (cid:17) (cid:26) (cid:27) (cid:18) (cid:18) (cid:19)(cid:19)
Then we obtain Lemma 9.
Corollary 8 is obtained from Theorem 7 by the following evaluation of asymptotic [12]. Let t > 0 and
x RN, then
∈ <
1 x x x
(t,x)= h 1+ | | , in the limit | | 0,
N N
N cN (cid:18)√t(cid:19)(cid:18) O(cid:18)√t(cid:19)(cid:19) √t →
where c =πN/2 N Γ(i)/Γ(i/2) .
N i=1
n o
Q
4 Eigenvalue Process of Hermitian Matrix-valued Processes
We consider complex-valued processes ξ (t) C,1 i,j N,t [0, ), with the condition ξ (t)∗ =
ij ji
∈ ≤ ≤ ∈ ∞
ξ (t), and introduce Hermitian matrix-valued processes Ξ(t) = ξ (t) . We denote by U(t) =
ij ij
1≤i,j≤N
(cid:16) (cid:17)
u (t) the family of unitary matrices which diagonalize Ξ(t) so that
ij
1≤i,j≤N
(cid:16) (cid:17)
U(t)†Ξ(t)U(t)=Λ(t)=diag λ (t),λ (t), ,λ (t) , (4.1)
1 2 N
···
n o
N
where λ (t) are eigenvalues of Ξ(t) and we assume their increasing order
i
i=1
n o
λ (t) λ (t) λ (t).
1 2 N
≤ ≤···≤
10