On the distribution of the length of the second row of a Young diagram under Plancherel measure

25 Pages·0.28 MB·English

by  Jinho Baik

On the distribution of the length of the second row
9
9
of a Young diagram under Plancherel measure
9
1

n
a Jinho Baik
J
Courant Institute of Mathematical Sciences, New York
6
2 [email protected]

] Percy Deift
O
Courant Institute of Mathematical Sciences, New York
C
. and
h
t
a Institute for Advanced Study, New Jersey
m
[email protected]
[

Kurt Johansson
1
v Royal Institute of Technology, Sweden
8
1 [email protected]
1
1
January 15, 1999
0
9
9
/
h Abstract
t
a
Weinvestigate theprobability distribution of thelength of the second row of a Youngdiagram
m
of size N equipped with Plancherel measure. We obtain an expression for the generating function
:
v
of thedistribution in termsof aderivativeof an associated Fredholm determinant,which can then
i
X beused toshowthat asN →∞thedistributionconverges totheTracy-Widomdistribution [TW]
r for thesecond largest eigenvalue of arandom GUEmatrix. This paperis asequel to[BDJ], where
a
we showed that as N → ∞ the distribution of the length of the first row of a Young diagram, or
equivalently, the length of the longest increasing subsequence of a random permutation, converges
to theTracy-Widom distribution [TW] for thelargest eigenvalue of a random GUE matrix.
1 Introduction
LetY denotethesetofallYoungdiagramsofsizeN. Asiswellknown,Y maybeviewed,equivalently,
N N
as the set of all partitions of N. We write µ = (µ ,µ , ) Y and/or µ N, depending on the
1 2 N
··· ∈ ⊢
context. General references for properties of the Young diagrams are, for example, [Sa] and Section
5.1.4in [Kn]. For µ Y , d is defined to be the number of (standard)Young tableaux ofshape µ. By
N µ
∈
1
Plancherel measure on Y , we mean the probability measure defined by
N
d2
µ
Prob µ := . (1)
N!
(cid:0) (cid:1)
Under the Robinson-SchenstedcorrespondencebetweenS ,the groupofpermutationof 1,2, ,N ,
N
{ ··· }
and the set of pairs of the Young tableaux of the same shape, Plancherel measure (1) is just the push
forwardoftheuniformprobabilitydistributiononS . Letl(k)(µ)denotethenumberofboxesinthekth
N N
rowofµ Y . Schensted[Sc]showedthatthelengthofthelongestincreasingsubsequenceofπ S is
N N
∈ ∈
l(1)(µ) where µ is the diagramof either of the Young tableaux corresponding to π under the Robinson-
N
Schenstedcorrespondence. Furthermore,Greene[Gr]obtainedacombinatorialinterpretationsofl(k) for
N
generalk 2,namelyl(1)(µ)+l(2)(µ)+ +l(k)(µ)isthelengthofthelongestk-increasingsubsequence
≥ N N ··· N
of π. (Recall that a k-increasing subsequence is a union of k disjoint increasing subsequences in π.)
Similarly the sum of the lengths of the first k columns gives the length of the longest k-decreasing
subsequence.
For N 1,n 0, define
≥ ≥
1
q(k) :=Prob l(k) n = d2. (2)
n,N N ≤ N! µ
µ N
(cid:0) (cid:1) l(k)X(⊢µ) n
N ≤
Set q(k) :=1, for n 0. Define the exponential generating function (or Poissonization)of q(k) by
n,0 ≥ n,N
φ(k)(λ):= ∞ e−λλNq(k) . (3)
n N! n,N
N=0
X
First, we summarize the results of [BDJ] using the above notation. For 0<t 1, let u(x;t) be the
≤
unique solution of the Painlev´e II equation
u =2u3+xu, (4)
xx
with the boundary condition
u(x;t) √tAi(x) as x + . (5)
∼− → ∞
The proof of the existence and the uniqueness of this solution, as well as the asymptotics as x ,
→−∞
can be found, for example, in [DZ2]. Define the Tracy-Widom [TW] distributions
F(x;t)=exp ∞(y x)(u(y;t))2dy . (6)
− −
(cid:18) Zx (cid:19)
From the properties of u(x;t) in [DZ2], it is easy to see that F(x;t), 0<t 1, is indeed a distribution
≤
function. One of the main results in [BDJ] is that for fixed x R,
∈
l(1) 2√N
lim Prob χ(1) := N − x =F(x;1). (7)
N N N1/6 ≤
→∞ (cid:18) (cid:19)
2
Convergence of the moments of χ(1)is also proved in [BDJ], implying that
N
Var l(1) 2
lim N = ∞ t2dF(t;1) ∞ tdF(t;1) =0.8132 . (8)
N→∞ N(cid:0)1/3 (cid:1) Z−∞ −(cid:18)Z−∞ (cid:19) ···
and
Exp l(1) 2√N
lim N − = ∞ tdF(t;1)= 1.7711 . (9)
N→∞ (cid:0) N1(cid:1)/6 Z−∞ − ···
The above results are intimately connected to the Gaussian Unitary Ensemble (GUE) of random
matrix theory. In GUE, one considers N N Hermitian matrices, and the probability density for the
×
eigenvalues in an infinitesimal multi-interval about points x , ,x is given by
1 N
···
ZN−1e− Nj=1x2j |xj −xk|2dx1···dxN, (10)
P 1 j<k N
≤Y≤
where Z is the normalization constant. Let λ (M) be the largest eigenvalue of M in GUE. In 1994
N 1st
in [TW], Tracy and Widom showed that if one scales λsc := λ √2N √2N1/6, then for any fixed
1st 1st −
x R,
(cid:0) (cid:1)
∈
lim Prob λsc x =F(x;1). (11)
N 1st ≤
→∞
(cid:0) (cid:1)
Inotherwords,properlycenteredandscaled,thelengthofthelongestincreasingsubsequenceforπ S
N
∈
(or l(1) under Plancherel measure on Y ), behaves statistically for large N like the largest eigenvalue
N N
of a random GUE matrix. Moreover, in [TW] the authors also computed the limiting distribution
of the kth largest eigenvalue λ of a random GUE matrix for general k 2. Indeed, again scaling
kth ≥
λsc := λ √2N √2N1/6, Tracy and Widom showed that for fixed x R, and for k 2,
kth kth − ∈ ≥
(cid:0) (cid:1) (k 1)
1 ∂ −
lim Prob λsc x = lim Prob λsc x + F(x;t). (12)
N→∞ kth ≤ N→∞ (k−1)th ≤ (k−1)!(cid:18)−∂t(cid:19) (cid:12)t=1
(cid:0) (cid:1) (cid:0) (cid:1) (cid:12)
(cid:12)
In this paper, we prove that l(2) behaves statistically, as N , like the secon(cid:12)d largest eigenvalue
N →∞
of a random GUE matrix. This result was conjectured in [BDJ] : the obvious analogous result should
be true for all the rows. For the second row, the conjecture was strongly supported by Monte Carlo
(2)
simulationsofOdlyzkoandRains. Tocompute theasymptoticsofl ,wefirstobtainanexpressionfor
N
the generating function φ(2)(λ) in terms of a derivative of a Fredholm determinant. For the case k =1
n
(1)
in [BDJ], it was already known (see, for example, [Ge], [Ra]) that the generating function φ (λ) is a
n
Toeplitz determinant :
φ(1)(λ)=e λdet T , n 0, (13)
n − n−1 ≥
(cid:0) (cid:1)
whereT = (T ) isthen nToeplitzmatrixwithrespecttotheweighte2√λcosθdθ/(2π),
n−1 n−1 j,k 0≤j,k≤n−1 ×
(cid:0) (cid:1)
2π dθ
T = e i(j k)θe2√λcosθ , n 1, (14)
n 1 − −
− (cid:18)Z0 2π(cid:19)0 j,k n 1 ≥
≤ ≤ −
3
and T is the 1 1 matrix with entry equal to 1. Therefore the main part of the calculation in
1
− ×
[BDJ] was the computation of the asymptotics of φ(1)(λ) as λ,n . In this paper, however, we
n
→ ∞
(2)
focus onderiving anappropriateexpressionfor the generatingfunction φ (λ). As we will explain, the
n
asymptoticsasλ,n canthenbeobtainedinasimilarmannerto[BDJ]. Aninterestingby-product
→∞
of the calculation of φ(2)(λ) is a new expression for φ(1)(λ) (see (17) below).
We state our results. Set
ϕ(z):=e√λ(z−z−1). (15)
and let Σ be the unit circle in the complex plane oriented counterclockwise. Let K be the integral
n
operator acting on L2(Σ, dw), whose kernel is defined by
| |
z nwn ϕ(z)ϕ(w) 1
− −
K (z,w):= − , K f (z)= K (z,w)f(w)dw. (16)
n n n
2πi(z w)
− ZΣ
(cid:0) (cid:1)
Theorem 1. For n 0, we have
≥
φ(1)(λ)=2 ndet(I K ), (17)
n − − n
and
∂
φ(2) (λ)=φ(1)(λ)+ (1+√t) ndet(I √tK ) . (18)
n+1 n −∂t − − n
(cid:18) (cid:19)(cid:12)t=1(cid:20) (cid:21)
(cid:12)
(cid:12)
(cid:12)
Theorem 2. For fixed x R, we have
∈
l(2) 2√N
lim Prob χ(2) := N − x =F(2)(x), (19)
N N N1/6 ≤
→∞ (cid:18) (cid:19)
where F(2)(x)=F(x;1)+ ∂ F(x;t) is the Tracy-Widom second eigenvalue distribution formula
−∂t t=1
given in (12) above. (cid:0) (cid:1)(cid:12)
(cid:12)
As in [BDJ], we also have convergence of the moments. Indeed, let χ(2) be a random variable with
distribution function F(2). The we have the following result.
Theorem 3. For m=1,2, ,
···
lim Exp χ(2) m =Exp χ(2) m . (20)
N
N
→∞
(cid:0)(cid:0) (cid:1) (cid:1) (cid:0)(cid:0) (cid:1) (cid:1)
These results show that l(2) behaves statistically for large N like the second largest eigenvalue of a
N
random GUE matrix, under appropriate centering and scaling. Furthermore, in view of the previous
remarks,bytheRobinson-Schenstedcorrespondencetheseresultsshowthat,afterappropriatecentering
andscaling,the difference betweenthe length ofthe longest2-increasingsubsequenceandthe lengthof
the longest increasing subsequence of a random permutation of N numbers, also behaves statistically
for large N like the second largest eigenvalue of a random GUE matrix.
4
As indicated above, the bulk of the paper is devoted to proving formula (18) for φ(2) (λ). We
n+1
indicate briefly in Section 6 below how Theorems 2 and 3 then follow by the Riemann-Hilbert/steepest
descent methods of [BDJ] : further details on the computations will appear in a later publication. In
[BDJ],theauthorsexpressφ(1)(λ)via(13)intermsofthesolutionofaRiemann-Hilbertproblem(RHP)
n
for polynomials orthogonal with respect to the weight e2√λcosθdθ/(2π) on the unit circle (this RHP
is the analog for orthogonal polynomials on the circle of the RHP introduced in [FIK] for orthogonal
polynomials on the line), and then apply the steepest descent method for RHP’s introduced by Deift
and Zhou in [DZ1], further developed in [DZ2], [DVZ1], and finally placed in a systematic form by
Deift, Venakides and Zhou in [DVZ2], to compute the asymptotics as λ,n . A general reference
→ ∞
for RHP’s is, for example, [CG]. The calculations in [BDJ] have many similarities to the calculations
in [DKMVZ]. The methods of [BDJ] apply here because the operator K in (16) is an example of
n
a so-called integrable operator, whose resolvent can be computed in terms of a canonically associated
RHP(seeSection2below). IntegrableoperatorswereintroducedasadistinguishedclassbyIts,Izergin,
Korepin and Slavnov in [IIKS], and have since been applied to a broad and rapidly growing array of
problems in pure and applied mathematics (see, for example, [De]).
The proof of Theorem 1 is based, in large part, on manipulations of RHP’s. The proof of equality
(17) for φ(1)(λ) is given in Section 2 (see Proposition 6). In Section 2, the RHP’s (v ( ;k),Σ) and
n Y
·
(v( ;k;t),Σ) are also introduced and their connections to the Fredholm determinant det(I K ) are
N
· −
established. In Section 3, to prove (18) for φ(1) (λ), we first use the Frobenius-Young formula for d
n+1 µ
in (1) to obtain an intermediate form for φ(2) (λ) in terms of the inverse of a Toeplitz matrix together
n+1
with certain binomial sums (see Proposition 7). The calculation of this intermediate form is similar to
thederivationof (13)aboveintheAppendixin[BDJ]. Theidentificationofthisintermediateformwith
the righthandside of (18)is madefirstwhenn=0 (Section4),andthen forgeneraln 1(Section5).
≥
Finally,inSection6,weindicatehowtoproveTheorems2and3followingthemethodsin[BDJ]. Inthe
Appendix, we discuss the spectralpropertiesof the operatorK in (16), and the (unique) solvability of
n
the RHP (24) below.
The proof that we give for the basic formula (18) appears rather ad hoc. At the end of the paper,
in Section 6, we provide a motivation for our calculations.
Acknowledgments. TheauthorswouldliketothankAndrewOdlyzkoandEricRainsfortheirMonte
Carlo simulations on the statistics of the second row. The authors also greatly appreciate many useful
conversations and communications with Persi Diaconis, Andrew Odlyzko, Eric Rains, Craig Tracy and
Harold Widom. The work of the authors was supported in part by a Sloan Doctoral Dissertation
Fellowship [J.B.], by NSF grant #DMS-9500867 and the Institute of Advanced Study [P.D.], and by
the Swedish National Research Council (NFR) [K.J.].
5
2 Riemann-Hilbert problems (RHP’s)
As above, let Σ denote the unit circle in the complex plane, oriented counterclockwise. Set
ψ(z):=e√λ(z+z−1). (21)
Inthispaper,weusethefollowingtwo(matrix)RHP’s. First,fork 0,letY(z;k),a2 2matrix-valued
≥ ×
function of z, be the solution of the RHP (v ( ;k),Σ),
Y
·
Y(z;k) is analytic in z C Σ,
∈ \
 1 1 ψ(z)
Y+(z;k)=Y−(z;k)0 zk 1 , on z ∈Σ, (22)
Y(z;k) z−k 0 =I+O(1) as z .
The notation Y (z;k) (resp., Y (cid:18)) d0enzokt(cid:19)es the limitzing value →lim∞ Y(z ;k) with z < 1 (resp.,
+ z z ′ ′
− ′→ | |
z >1). Notethatk playstheroleofanexternalparameterin(22);inparticular,thetermO(1)does
| ′| z
not imply a uniform bound in k. For a general RHP, the existence and uniqueness of the solution is,
of course, not clear a priori. But for the case at hand, we can simply write down the (unique) solution
explicitly in terms of orthogonalpolynomials, as follows.
Let πn(z) = zn + pn=−01ηpnzp denote the n-th monic orthogonal polynomial with respect to the
measureψ(z)dz/(2πiz)ontheunitcircle,andintroducethepolynomialπ (z):=znπ (1/z)=zn(z n+
P n∗ n −
pn=−01ηpnz−p (see[Sz]). Forthemeasureathand,ψ(z)dz/(2πiz)=e2√λcosθdθ/(2π),allthecoefficients
of π (z) are real and π (z) = znπ (1/z). The solution of the RHP (v ( ;k),Σ) is given by the
P n n∗ n Y ·
following formula (see Lemma 4.1 in [BDJ] ; here we change the notation f(z) to ψ(z) and use the
monic orthogonalpolynomial π (z) instead of the normalized orthogonal polynomials p (z)),
k k
π (z) πk(s)ψ(s)ds
Y(z;k)= k Σ s z 2πisk , k 1. (23)
−κ2k−1πk∗−1(z) −κ2k−R1 Σ−πk∗s−−1z(s)ψ2(πsi)sdks! ≥
R
For k =0, the solution is given by (82) below.
Let ϕ(z) be defined as in (15). For 0<t 1 and for k 0, the second RHP (v( ;k;t),Σ) is to find
≤ ≥ ·
m(z;k;t) satisfying
m(z;k;t) is analytic in z C Σ,
∈ \
 1 t √tz kϕ(z) 1
 m+(z;k;t)=m−(z;k;t)√tzk−ϕ(z) − −1 −  on z ∈Σ, (24)
 
m(z;k;t) I as z .
In this RHP, there are two ex→ternal param→et∞ers k and t. For t=1, this RHP is equivalent to the RHP
(v ( ;k),Σ), in the sense that a solution of one RHP implies a solution of the other RHP. Indeed one
Y
·
6
can easily check that
0 1 Y(z;k) e√λz 0 , z <1,
m(z;k;1)=(cid:18)1−0 (cid:19) (cid:18) 0 e−√λz(cid:19) | | (25)
(cid:18)10−01(cid:19)Y(z;k)(cid:18)z−ke√0λz−1 zke−√0λz−1(cid:19)(cid:18)−0101(cid:19), |z|>1,
solves (24) if and only if Y(z;k) solves (22). The (unique) solvability of the RHP (24) for 0 < t < 1
is discussed in the Appendix. As noted earlier, the motivation for the introduction of (24) for t <1 is
given in Section 6 below.
The solution m(z;k;t) of the RHP (v(z;k;t),Σ) is related to the Fredholm determinant of the
integraloperatorK in(16)actingonL2(Σ, dw). Foreachn, it is easyto verifythat K is traceclass
n n
| |
andhencedet(I K )exist. TheproofofthefollowingLemmaismodeledonProposition6.13in[DIZ].
n
−
Lemma 4. Let m(z;k;t) be the solution of the RHP (v(z;k;t),Σ) given in (24), and let K be defined
k
as above. If we denote the 11-component of m(z;k;t) by m (z;k;t), we have at z =0,
11
det(I √tK )
m11(0;k;t)=(1+√t) − k−1 , k 0. (26)
det(I √tK ) ≥
k
−
Proof. TheoperatorK hasnormlessthanorequalto1and1isnotaneigenvalueofK (seeAppendix).
n n
So det(I √tK ) never vanishes for any k, 0<t 1.
k
− ≤
First note that
1
K (z,w)=K (z,w)+ z kwk 1 =:K (z,w)+E (z,w). (27)
k 1 k − − k k
− 2πi
Then we have
det(I √tK ) 1
− k−1 =det I √tEk . (28)
det(I √tK ) − I √tK
− k (cid:18) − k (cid:19)
Since E is a rank 1 operator,
k
det(I √tK ) 1 √t 1
− k−1 =1 tr √tEk =1 f1 (z) zk−1dz, (29)
det(I √tK ) − I √tK − 2πi I √tK ·
− k (cid:18) − k (cid:19) ZΣ(cid:18) − k (cid:19)
where f (z)=z k as in (33) below.
1 −
On the other hand, we define M(z,k;t) as follows,
m(z;k;t) 1/(1+√t) 0 , z <1
M(z;k;t):= 0 1+√t | | (30)
 (cid:18) (cid:19)
m(z;k;t), z >1.
| |
Then it is easy to check that M(z;k;t) solves a new RHP (V( ;k;t),Σ),
·
M(z;k;t) is analytic in C Σ,
\
 1 √t √t(1+√t)z kϕ(z) 1
 M+(z;k;t)=M−(z;k;t)1+√√tt−zkϕ(z) − 1+√−t −  on z ∈Σ, (31)
 
M(z;k;t) I as z .
 → →∞
7
Note that the jump matrix V(z;k;t) can be written in the form
V(z;k;t)=I 2πi√tf(z;k;t) g(z;k;t) T, (32)
−
(cid:0) (cid:1)
where f,g are column vectors defined by
f(z;k;t)=(f ,f )T = z k, 1 ϕ(z) T, g(z;k;t)=(g ,g )T = 1 zk,(1+√t)ϕ(z) 1 T.
1 2 − 1 2 −
−1+√t 2πi
(cid:0) (cid:1) (cid:0) (cid:1)(33)
From the general theory of RHP’s, the solution M(z;k;t) of the RHP (V(z;k;t),Σ) satisfies
1 M+(s;k;t) I V(s;k;t)−1
M(z;k;t)=I + − ds
2πi s z
ZΣ (cid:0) − (cid:1) (34)
=I √t M (s;k;t) 1 (1+√t)s−kϕ(s)−1 ds .
− 2πiZΣ + (cid:18)−1+1√tskϕ(s) −1 (cid:19)s−z
Recalling the definition of f(z;k;t), we have
√t
M11(0;k;t)=1− 2πi M+(z;k;t)f(z;k;t) 1·zk−1dz. (35)
ZΣ
(cid:0) (cid:1)
Now from the theory of integrable operators (see for example [De]), the integral operator S acting on
k
L2(Σ, dw) with kernel
| |
(f(z;k;t))Tg(w;k;t)
S (z,w):= , (36)
k
z w
−
satisfies the relation
T
1 F(z;k;t) G(w;k;t)
S (z,w)= , (37)
k
I √tS z w
(cid:18) − k (cid:19) (cid:0) −(cid:1)
where F,G are column vectors given by
1
F(z;k;t)= f (z)=M (z;k;t)f(z;k;t),
+
I √tS
(cid:18) − k (cid:19) (38)
G(w;k;t)= 1 g (w)= M+(w;k;t)T −1g(w;k;t).
I √tS
(cid:18) − k (cid:19)
(cid:0) (cid:1)
But from the definitions of f and g in (33),
S (z,w)=K (z,w). (39)
k k
Therefore (29), (35) and the first part of (38) give us that
det(I √tK )
k 1
M11(0;k;t)= − − . (40)
det(I √tK )
k
−
The relation (30) completes the proof.
Now for fixed t, λ, we compute the asymptotics of det(I √tK ) as p . The calculationbelow
p
− →∞
is similar to that of [De] where a new proof of Szego¨’s strong limit theorem is given.
8
Lemma 5. For fixed 0<t 1 and λ>0, we have
≤
lim (1+√t) pdet(I √tK )=1. (41)
− p
p −
→∞
Proof. First note that
√t 1
logdet(I √tK )= tr K ds. (42)
p p
− − I sK
Z0 (cid:18) − p (cid:19)
From (37) and (39), the right hand side of (42) is given by
√t F(z;p;s2) TG(w;p;s2) √t
tr ds= ds F′(z;p;s2) TG(z;p;s2)dz (43)
− z w −
Z0 (cid:18)(cid:0) −(cid:1) (cid:19) Z0 ZΣ
(cid:0) (cid:1)
where the prime indicates differentiation with respect to z. From (38), in order to compute the
′
asymptotics of det(I √tK ), we need the asymptotics of M (z;p;s2) asp uniformly in 0<s
p +
− →∞ ≤
√t.
Note that the jump matrix V(z;p;s2) has the following factorization
1 sz pϕ 1 1 0 1 0
V(z;p;s2)= − − − 1+s . (44)
0 1 ! 0 1+s! s zpϕ 1!
(1+s)2
Let 0<ρ<1 be any number. Define
M˜(z):=M(z;p;s2), z <ρ,
| |
 1
−
MM˜˜((zz))::==MM((zz;;pp;;ss22))1(1+ss−)12szzp−ϕpϕ−101 , , 1ρ<<|zz|<<ρ1, 1, (45)
−
  | |
0 1
and set M˜(z):=M(z;p;s2),  |z|>ρ−1,
1 0
V˜(z)= , z =ρ,
  s zpϕ 1 | |
V˜(z)=(11+01+ss)21+0 s, |z|=1, (46)
1 sz pϕ1
Then M˜ solves the new RHP (V˜V˜,Σ˜(z))w=here0Σ˜−:= −1z =− ρ, |zz|==ρ1−1. z =ρ 1 , oriented counter-
−
{| | }∪{| | }∪{| | }
clockwise on each of the three circles,
M˜(z) is analytic in C Σ˜,
\
M˜ (z)=M˜ (z)V˜(z), on z Σ˜, (47)
M˜+(z) I, − as z . ∈
→ →∞

9
Here M˜ (z) = lim M˜(z ) and M˜ (z) = lim M˜(z ) for z on each of the three
+ z′→z,|z′|<|z| ′ − z′→z,|z′|>|z| ′
circles. Nowasp ,weseethatV˜(z) I forz z =ρ z =ρ 1 . Moreovertheconvergence
−
→∞ → ∈{| | }∪{| | }
is exponential at a rate which is uniform in z and s. Therefore it follows that M˜ M˜ where M˜ is
∞ ∞
→
the solution of the RHP (V˜∞,Σ), V˜∞ := 1+1s 0 . This RHP has the explicit solution
0 1+s
(cid:18) (cid:19)
M˜∞ = 1+1s 0 , z <1,
0 1+s | | (48)
 (cid:18) (cid:19)
M˜∞ =I, z >1.
| |
Therefore as p , we have the following asymptotics for z Σ,
→∞ ∈
1 0 1 0 1 0
M+(z;p;s2)=M˜+(z) s zpϕ 1!∼M˜+∞(z) s zpϕ 1!= s1+zspϕ 1+s!, (49)
(1+s)2 (1+s)2 1+s
where the error is exponentially small, uniformly for z and s.
Now from (38), the above asymptotic result yields for z Σ,
∈
z p ϕ T
F(z;p;s2)=M (z;p;s2)f(z;p;s2) − , − ,
+
∼ 1+s 1+s
(cid:18) (cid:19)
F′(z;p;s2) −pz−p−1,−√λ(1+z−2)ϕ T, (50)
∼ 1+s 1+s
(cid:18) (cid:19)
G(z;p;s2)= M+(z;p;s2)T −1g(z;p;s2) 1 zp,ϕ−1 T.
∼ 2πi
(cid:0) (cid:1) (cid:0) (cid:1)
Inserting these asymptotics, which have exponential error terms, into (43), equation (42) becomes
√t ds
logdet(I √tKp)= pz−1 √λ(1+z−2) dz+O e−cp
− − 2πi(1+s) − − (51)
Z0 ZΣ
(cid:2) (cid:3) (cid:0) (cid:1)
=plog(1+√t)+O e cp ,
−
where the exponential error term is uniform for 0<(cid:0) t (cid:1)1.
≤
Combining Lemma 4 and Lemma 5, we summarize the results in this section as follows.
Proposition 6. Letm(z;k;t)bethesolutionoftheRHP(v( ;t),Σ)givenin (24),andletK bedefined
n
·
as in (16). Denoting the 11-component of m(z;k;t) by m (z;k;t), we have
11
∞
m (0;k+1;t)=(1+√t) ndet(I √tK ), n 0. (52)
11 − n
− ≥
k=n
Y
In particular, we have, for the generating function of l(1),
N
φ(1)(λ)=2 ndet(I K ), n 0. (53)
n − − n ≥
Also for n 0, the right hand side of (18) is given by
≥
φ(1)(λ)+ ∂ (1+√t) ndet(I √tK ) = 1 ∞ m˙11(0;k+1;1) φ(1)(λ), (54)
n −∂t − − n − m (0;k+1;1) n
(cid:18) (cid:19)(cid:12)t=1(cid:20) (cid:21) (cid:20) k=n 11 (cid:21)
(cid:12) X
(cid:12)
where the dot indicate(cid:12)s differentiation with respect to t.
·
10