Acta Mathematica Sinica, English Series
Jun., 2008, Vol. 24, No. 6, pp. 971–982
Published online: June 4, 2008
DOI: 10.1007/s10114-007-6365-8
Http://www.ActaMath.com
Acta Mathematica Sinica,
English Series
The Editorial Office of AMS &
Springer-Verlag 2008
Precise Asymptotics for Random Matrices
and Random Growth Models
Zhong Gen SU
Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
E-mail: suzhonggen@zju.edu.cn
Abstract The author considers the largest eigenvalues of random matrices from Gaussian unitary
ensemble and Laguerre unitary ensemble, and the rightmost charge in certain random growth models.
We obtain some precise asymptotics results, which are in a sense similar to the precise asymptotics
for sums of independent random variables in the context of the law of large numbers and complete
convergence. Our proofs depend heavily upon the upper and lower tail estimates for random matrices
and random growth models. The Tracy–Widom distribution plays a central role as well.
Keywords
Gaussian unitary ensemble, Laguerre unitary ensemble, largest eigenvalues, random
growth models, Tracy–Widom distribution
MR(2000) Subject Classification 60G50, 60F15; 60G70
1
Introduction and Results
In the recent years, important developments have taken place in the analysis of the spectrum
of large random matrices and of various random growth models. In particular, universality
questions at the edge of the spectrum has been conjectured, and settled, for a number of
apparently disconnected examples. Just recently, König [1] and Ledoux [2] aimed at nonexperts, surveyed a number of fine results and technical tools from various parts of mathematics.
In this paper we study the precise asymptotics for the largest eigenvalues of Gaussian
unitary ensemble and Laguerre unitary ensemble, which are two most prominent examples of
orthogonal polynomial ensembles. We also consider the rightmost charge in a certain random
growth model, directed last passage percolation with specific weights.
N
Let, for each integer N , X N = (Xij
)1≤i,j≤N be a N ×N complex Hermitian matrix such that
the entries above the diagonal are independent, complex-centered Gaussian random variables
with variances σ 2 . Denote by λ1 , λ2 , . . . , λN the real eigenvalues of X N . There is a nice,
fundamental formula for the joint probability density of the random vector λ = (λ1 , λ2 , . . . , λN )
as follows:
N
x2
j
1
2
pN (x1 , x2 , . . . , xN ) =
ΔN (x)
e− 2σ2 ,
(1.1)
ZN
j=1
Received July 5, 2006, Accepted June 11, 2007
Supported partly by NSF of China (No. 10371109, 10671176) and the Royal Society K. C. Wong Education
Foundation
Su Z. G.
972
where ZN =
√ N N 2 N
−1
2π σ
, ΔN (x) is the well-known Vandermonde determinant, i.e.,
j=1 j!
ΔN (x) =
(xj − xi ).
1≤i<j≤N
2
Under the normalization σ =
that, almost surely,
1
4N
of the variance, it is a classical result due to Wigner [3]
N
1 δλ → ν
N i=1 i
(1.2)
in distribution as N → ∞, where ν is the semicircle law with density
to Lebesgue measure on (−1, +1).
2
2 1/2
π (1 − x )
with respect
It is also well known that the largest eigenvalue λN
max = max1≤i≤N λi converges almost
surely to the right-end point of the support of the semicircle law, i.e., 1 in the normalization
here; see [4–6] for a proof.
As one of the main recent achievements of the theory of random matrices, it has been shown
by Forrester [7], and Tracy and Widom [8], that the fluctuations of the largest eigenvalue λN
max
around its expected value 1 take places at the rate N 2/3 . More precisely,
2N 2/3 λN
(1.3)
max − 1 → F2
in distribution, where F2 is as usual the so-called Tracy–Widom distribution.
The new distribution F2 occurs naturally as a Fredholm determinant
F2 (s) = det(I − KA )|L2 (s,∞) ,
s ∈ (−∞, ∞)
of the integral operator associated with the Air kernel KA . It is Tracy and Widom [8] that
provides a description of this new distribution F2 in terms of some differential equations as
∞
F2 (s) = exp −
(x − s)u(x)2 dx , s ∈ (−∞, ∞),
s
where u(x) is the solution of the Painlevé II equation u = 2u3 + xu with the asymptotics
1
− 23 x2/3
u(x) ∼ 2√πx
as x → ∞.
1/4 e
This is a nonstandard central limit theorem. A few characteristics of the distribution F2
are known. It is non-centered, with a mean around −1.758, and its respective behaviors at ±∞
are given by
3
C −1 e−Cs ≤ F2 (−s) ≤ Ce−s
3
/C
and
C −1 e−Cs
3/2
≤ 1 − F2 (s) ≤ Ce−s
3/2
/C
,
for large s and C numerical.
We now add a precise asymptotics result.
Theorem 1
2
Let λN
max be as above with σ =
lim (2ε)3/2
ε→0
∞
N =1
P λN
max
1
4N .
We have
3 ∞ 1/2
>1+ε =
y (1 − F2 (y))dy
2 0
(1.4)
Precise Asymptotics for Random Matrices
and
lim (2ε)3/2
ε→0
∞
N =1
973
3
P λN
max < 1 − ε =
2
0
−∞
|y|1/2 F2 (y)dy.
(1.5)
During the last years there has been a lot of activity around the problem of distribution of
the length of a longest increasing subsequence of a random permutation. Let π be a random
permutation from the symmetric group SN with uniform distribution PN . The length of the
longest increasing subsequence of π is the maximal n such that there are indices 1 ≤ i1 < i2 <
· · · < in ≤ N satisfying π(i1 ) < π(i2 ) < · · · < π(in ). We denote the length by LN (π). In
the early 1960’s Ulam raised the question about the large N behavior of LN . On the basis of
computer simulations, he conjectured that
ELN
c = lim √
(1.6)
N →∞
N
exists in (0, ∞). The verification of this statement and the identification of c have become
known as Ulam’s problem. A long list of researchers contributed to this problem, including
Hammersley, Logan and Shepp, Vershik and Kerov, Aldous and Diaconis, Seppäläinen. An
excellent survey on the history of Ulam’s problem may be found in [9] and [10].
By the end of the 1990’s, several completely different methods, both hard combinatorial
analysis and purely probabilistic argument, have been developed to show the limit in (1.6)
exists with c = 2. Indeed a kind of strong law of large number holds for LN , i.e.,
LN
(1.7)
lim √ = 2, a.e.
N →∞
N
In a striking contribution [11], Baik, Deift and Johansson proved in 1999 that the Tracy–Widom
distribution governs the fluctuation of LN using the methods from Topelitz determinants, integrable differential equations of the Painlevé type and the closely related Riemann–Hilbert
techniques. Particularly, they showed
√
LN − 2 N
→ F2
(1.8)
N 1/6
in distribution, together with its moments.
This result and its generalizations has also been independently proved by Johansson [12],
Borodin, Okounkov and Olshanski [13] and Okounkov [14]. Note that the normalization is given
√
N
by the third power of the mean order 2 N , as it would be the case if we replace λN
max by N λmax
in the above Gaussian unitary ensemble.
Our precise asymptotics about LN reads as follows:
Theorem 2
Assume that LN is defined as above. Then we have
∞
∞
√
lim ε3
P (LN ≥ N (2 + ε)) = 3
y 2 (1 − F2 (y))dy
ε→0
and
3
lim ε
ε→0
∞
N =1
(1.9)
0
N =1
P (LN
√
≤ N (2 − ε)) = 3
0
−∞
y 2 F2 (y)dy.
(1.10)
It is the explicit form (1.1) that makes possible the deep and thorough asymptotic analysis,
both inside the bulk, and at the edge of the spectrum, of the eigenvalue distribution of Gaussian
Su Z. G.
974
unitary ensemble. There are other well-known ensembles with such a determinantal point
process representation in random matrices like the Laguerre unitary ensemble for the spectrum
of Wishart matrices. Let X = X M,N = (Xi,j ) be a complex M × N , M ≥ N , random matrix
the entries of which are independent complex Gaussian random variables with mean zero and
variance σ 2 (if and only if Re(Xi,j ), Im(Xi,j ) form a family of independent real Gaussian
2
random variables each with mean value 0 and variance σ2 ). Let Y N = X ∗ X. Then Y N can be
viewed as a sample covariance matrix of M samples of N -dimensional random vectors and it is
of fundamental importance in multivariate statistical analysis.
The complex Wishart matrices were first studied by Goodman [15] and Khatri [16]. Denote
by ρ1 , ρ2 , . . . , ρN the real eigenvalues of Y N . The joint probability density of (ρ1 , ρ2 , . . . , ρN ) is
as follows:
N
x
1
−N − σ2i
pN (x1 , x2 , . . . , xN ) =
ΔN (x)2
xM
e
, x1 , x2 , . . . , xN > 0,
(1.11)
i
ZM,N
i=1
2 M N N
where ZM,N = Mσ N
j=1 j!(M − j)!.
The Marchenko–Pastur [17] theorem states that, under the condition σ 2 = N1 , as N → ∞
such that M
N → γ, almost surely
N
1 δρ → μγ
(1.12)
N i=1 i
in distribution, where
1 μγ =
(b − x)(x − a), a < x < b,
2πx
√
√
and a = ( γ − 1)2 and b = ( γ + 1)2 when γ ≥ 1. When 0 < γ < 1, there is an additional
Dirac measure at x = 0 of mass 1 − γ. Moreover, the largest eigenvalue ρN
max = max1≤i≤N ρi
converges almost surely to the right edge of the support of μγ :
√ 2
ρN
γ) a.e.
(1.13)
max → (1 +
(see [5] and [6] for a proof).
While studying a random growth model of interest in probability, Johansson [18] derived
a limiting distribution of ρN
max after being properly scaled, which is again the Tracy–Widom
distribution F2 discovered in the Gaussian unitary ensemble. Specifically speaking, define
1/3
√
√ 2
√
√
1
1
.
μM N = ( M + N ) , σM N = ( M + N ) √ + √
M
N
Then, under the condition σ 2 =
1
N,
M
N → γ ≥ 1,
N ρN
max − μM N
as
σM N
→ F2
(1.14)
in distribution.
We can also obtain the precise asymptotics result for ρN
max as follows:
Theorem 3
we have
2
Assume that ρN
max is as above with σ =
1
N.
If M = [γN ] for some γ ≥ 1, then
∞
N
3 ∞ 2
γ 1/4
√ 2
3/2
lim √
ε
P
ρ
>
(1
+
γ)
+
ε
=
y (1 − F2 (y))dy
max
ε→0 ( γ + 1)2
2 0
N =1
(1.15)
Precise Asymptotics for Random Matrices
and
975
∞
N
3 0 2
γ 1/4
√ 2
3/2
lim √
ε
P
ρ
<
(1
+
γ)
−
ε
=
y F2 (y)dy.
max
ε→0 ( γ + 1)2
2 −∞
(1.16)
N =1
Next we consider a certain random growth model. Let wij , i, j ∈ N be independent geometric random variables with parameter q, 0 < q < 1. For any M ≥ N ≥ 1, set
W (M, N ) = max
wi,j ,
π
(i,j)∈π
where the maximum runs over all up/right paths π in N2 from (1, 1) to (M, N ). An up/right
path π from (1, 1) to (M, N ) is a collection of sites {(ik , jk )}1≤k≤M +N −1 such that (i1 , j1 ) =
(1, 1), (iM +N −1 , jM +N −1 ) = (M, N ) and (ik+1 , jk+1 ) − (ik , jk ) is either (1, 0) or (0, 1). This
random growth model may be interpreted as a directed last passage time in the percolation
model.
It is known that, for each 0 < q < 1 and γ ≥ 1,
√
(1 + qγ)2
W ([γN ], N )
→
− 1 =: ω(γ, q) a.e.
(1.17)
N
1−q
Note that the existence of the limit (1.17) follows by a simple and standard subadditivity
argument, so it is the explicit form of the limit constant that is interesting.
Using the Robinson–Schensted–Knuth correspondence between permutations and Young
tableaux, Johansson [18] proved that, for every t ≥ 0,
N
hi + M − N
1
2
P (W ([γN ], N ) ≤ t) =
ΔN (h)
(1.18)
q hi ,
ZM,N
h
i
h ,...,h
i=1
1
N
max{hi }≤t+N −1
where ZM,N is the normalization constant (partition function).
This remarkable formula should be compared with the formula for the distribution function
for the largest eigenvalue, λN
max , of an N ×N random matrix from the Gaussian unitary ensemble.
Indeed, this is just the Meixner orthogonal polynomial ensemble with parameters q and M −
N + 1. Provided with this correspondence, Johansson showed the following fluctuation of
W ([γN ], N ) around its mean: as N → ∞, for every γ ≥ 1,
W ([γN ], N ) − ω(γ, q)N
→ F2
(1.19)
N 1/3
in distribution. The convergence of moments is also proved in Baik, Deift, McLaughlin, Miller
and Zhou [19], using the refined Riemann–Hilbert steepest descent methods.
Notice that, if w is a geometric random variable with parameter 0 < q < 1, then as q → 1,
(1 − q)w converges in distribution to an exponential random variable with parameter 1. Thus
if W (M, N ) is understood as a maximum over up/right of independent exponential random
variables, the identity (1.18) then translates into
N
1
−N −xi
P (W (M, N ) ≤ t) =
ΔN (x)2
xM
e dx1 · · · dxN .
(1.20)
i
ZM,N [0,t]N
i=1
The right-hand side in (1.20) is the probability that the largest eigenvalue in the Laguerre
unitary ensemble is ≤ t.
Su Z. G.
976
Still, as q = Nθ2 , N → ∞, the Meixner orthogonal polynomial ensemble converges to
the θ-Poissonization of the Plancherel measure on partitions. Since the Plancherel measure is
the push forward of the uniform distribution on the symmetric group SN by the Robinson–
Schensted–Knuth correspondence which maps a permutation π ∈ SN to a pair of standard
Young tableaux of the same shape, the length of the first row is equal to the length LN (π) of
the longest increasing subsequences in π. As a consequence,
∞
θ n −n
e P (Ln ≤ t) = lim P (W (N, N ) ≤ t).
N →∞
n!
n=0
(1.21)
Thus the orthogonal approach may be used to produce a new proof of (1.8).
Our precise asymptotics about W (M, N ) reads as follows:
Theorem 4 Let W (M, N ) be defined as above with geometric weights. If M = [γN ], for
some γ ≥ 1, then we have
∞
3 ∞ 2
3/2
lim ε
P (W ([γN ], N ) > N (ω(γ, q) + ε)) =
y (1 − F2 (y))dy
(1.22)
ε→0
2 0
N =1
and
lim ε3/2
ε→0
∞
P (W ([γN ], N ) < N (ω(γ, q) − ε)) =
N =1
3
2
0
−∞
y 2 F2 (y)dy.
(1.23)
We shall give the proofs of theorems in the next section. Below is a few words about the
motivation of this paper. In a sense, our results are similar to precise asymptotics for sums
of independent random variables in the context of the law of large numbers and complete
N
convergence. Let X, X1 , X2 , . . . be i.i.d. random variables, and set SN = i=1 Xi , N ≥ 1. The
classic Kolmogorov’s law of large numbers states that
SN
→ μ a.e.,
N
if and only if E|X| exists and EX = μ. Hsu and Robbins introduced the concept of complete
convergence and proved that the sequence of arithmetic means converges completely provided
the mean and the variance exist. The converse was proved by Erdös. More generally, it was
shown in Baum and Katz [20] that, for 0 < p < 2 and r ≥ p,
∞
nr/p−2 P (|Sn | ≥ εn1/p ) < ∞
n=1
r
if and only if E|X| < ∞, and when r ≥ 1, EX = 0.
An interesting observation is that the sum tends to infinity as ε 0. Heyde [21] first proved
lim ε2
ε0
2
∞
P (|Sn | ≥ εn) = EX 2
n=1
whenever EX = 0 and EX < ∞. In this setting the classic central limit theorem holds true,
and one has a Cramér type of exponential estimate for P (|Sn | ≥ εn). The square root in the
√
normalizing constant n correctly explains the growth size ε12 of the infinite sum. For analogous
results in a more general case, see Gut and Spǎtaru [22], [23].
Precise Asymptotics for Random Matrices
2
977
Proofs
Our proofs depend heavily on some non-asymptotic exponential deviation inequalities on the
largest eigenvalues or rightmost charge of random matrix and random growth models at the
1/3
order (mean)
of the fluctuation results. It actually turns out that several results are already
available in the literature, motivated by the convergence of moments in nonstandard central
limit theorems or moderate deviation principles interpolating between fluctuations and large
deviations. However, we remark that these non-asymptotic exponential deviation inequalities
require a rather heavy analysis and only concern some rather specific models. To my knowledge,
there is no unified method of dealing with these tail probability estimates. On the other hand,
since the Tracy–Widom distribution is asymmetric, it is not surprising that upper tails on the
right of the mean is different than lower tails on the left of the mean. But it is worthwhile to
note that the growth size of the infinite sum is related only to the order 1/3 (or 2/3) of the
normalizing constant.
Proof of Theorem 1
∞
We shall use the following equation:
P (λN
max
> 1 + ε) =
N =1
∞
2/3
[P (λN
))]
max > 1 + ε) − (1 − F (2εN
N =1
∞
+
(1 − F (2εN 2/3 )).
(2.1)
N =1
First, by the Euler–Maclaurin sum formula, we have
∞
∞
∞
1
(1 − F (2εN 2/3 )) =
(1 − F (2εx2/3 ))dx +
x − [x] +
d(1 − F (2εx2/3 )), (2.2)
2
1
1
N =1
where [x] stands for the largest integer less than x.
Making a change of variables gives
∞
(1 − F (2εx2/3 ))dx =
1
and
1
∞
3
2(2ε)3/2
∞
2ε
y 1/2 (1 − F (y))dy
∞ 3/2 3/2 y
y
1
1
2/3
−
x − [x] +
+
d(1 − F (2εx ) = −
dF (y).
2
2ε
2ε
2
2ε
Inserting (2.3) and (2.4) into (2.2), and then letting ε → 0 yields
∞
3 ∞ 1/2
3/2
2/3
(1 − F (2εN )) =
y (1 − F (y))dy.
lim (2ε)
ε→0
2 0
(2.3)
(2.4)
(2.5)
N =1
Next we turn to the first sum on the right-hand side in (2.1). Let N (ε, M ) = [ εM
3/2 ], where ε > 0
and M > 0. Then we have
∞
2/3
[P (λN
))]
max > 1 + ε) − (1 − F (2εN
N =1
N (ε,M )
=
N =1
2/3
[P (λN
))]
max > 1 + ε) − (1 − F (2εN
Su Z. G.
978
∞
+
∞
P (λN
max ≥ 1 + ε) −
N =N (ε,M )+1
(1 − F (2εN 2/3 )).
(2.6)
N =N (ε)+1
Note that
2N 2/3 (λN
max − 1) → F2
in distribution and F2 is a continuous distribution function. Then
ΔN =:
sup
−∞<x<∞
|P (2N 2/3 (λN
max − 1) > x) − (1 − F2 (x))| → 0
as N → ∞.
Using the Kronecker lemma, we have, for fixed M > 0,
N (ε,M )
3/2
lim ε
ε→0
2/3
[P (2N 2/3 (λN
) − (1 − F2 (2εN 2/3 ))] = 0.
max − 1) > 2εN
(2.7)
N =1
On the other hand, it follows from Auburn [24] that, for some numerical constant C > 0,
all N ≥ 1 and ε > 0,
−N ε3/2 /C
P (λN
.
(2.8)
max > 1 + ε) < Ce
We remark that Auburn [24] first obtained such a small deviation inequality by carefully following the proof of Tracy–Widom theorem and controlling the various Fredholm determinants
by appropriate bounds on orthogonal polynomials. Ledoux [2] recovers this upper-tail estimate
by using the large deviation of W (M, N ) in the random growth model (see below) and taking
the limit in the explicit rate function.
Now using (2.8) we have
∞
∞
P (λN
max > 1 + ε) ≤ C
N =N (ε,M )+1
e
N =N (ε,M )+1
∞
≤C
−N ε3/2
C
N (ε,M )
e
−xε3/2
C
dx.
(2.9)
So there holds for fixed ε > 0 that
lim ε3/2
M →∞
∞
P (λN
max > 1 + ε) = 0.
N =N (ε,M )+1
Also, it is easy to see that
∞
(1 − F (2εN 2/3 )) ≤
N =N (ε,M )+1
=
∞
N (ε,M )
(1 − F (2εx2/3 ))dx
3
2(2ε)3/2
Thus we have
(2ε)3/2
∞
n=N (ε,M )+1
(1 − F (2εN 2/3 )) ≤
3
2
∞
M 1/3
∞
M 1/3
y 1/2 (1 − F (y))dy.
(2.10)
y 1/2 (1 − F (y))dy,
(2.11)
which in turn tends to 0 as M → ∞. Taking (2.1) into account, we can finish the proof of (1.4).
The proof of (1.5) is similar to that of (1.4), with the only change of the upper tail of λN
max
on the left of the mean.
Precise Asymptotics for Random Matrices
979
It easily follows, for 0 < ε < 1, that
∞
P (λN
max < 1 − ε)
N =1
=
∞
∞
2/3
[P (λN
)] +
max < 1 − ε) − F2 (−2εN
N =1
F2 (−2εN 2/3 ).
(2.12)
N =1
Using the Euler–Maclaurin sum formula again, we have
∞
∞
∞
1
F2 (−2εN 2/3 ) =
F2 (−2εx2/3 )dx +
x − [x] +
dF2 (−2εx2/3 ).
2
1
1
(2.13)
N =1
Making a change of variables gives
∞
F2 (−2εx2/3 )dx =
1
3
2(2ε)3/2
−2ε
−∞
|y|1/2 F2 (y)dy
(2.14)
and
3/2 3/2 ∞
−2ε y
1
y
1
2/3
−
−
x − [x] +
−
+
dF2 (−2εx ) =
dF2 (y). (2.15)
2
2ε
2ε
2
1
−∞
So substituting (2.14) and (2.15) into (2.13) yields
lim (2ε)3/2
ε→0
∞
F2 (−2εN 2/3 ) =
N =1
3
2
0
−∞
|y|1/2 F2 (y)dy.
(2.16)
Next we turn to the first sum on the right-hand side of (2.12).
Let N (ε, M ) = [ εM
3/2 ] where ε > 0 and M > 0. Then we have
∞
2/3
[P (λN
)]
max < 1 − ε) − F2 (−2εN
N =1
N (ε,M )
=
2/3
[P (λN
)]
max < 1 − ε) − F2 (−2εN
N =1
∞
+
P (λN
max < 1 − ε) −
N =N (ε,M )+1
∞
F2 (−2εN 2/3 ).
(2.17)
N =N (ε,M )+1
A similar argument to that for (2.7) and (2.11) shows that, for fixed M > 0,
N (ε,M )
lim (2ε)
3/2
ε→0
2/3
[P (λN
)] = 0
max < 1 − ε) − F2 (−2εN
(2.18)
N =1
and
lim (2ε)3/2
ε→0
∞
F2 (−2εN 2/3 ) = 0.
(2.19)
P (λN
max < 1 − ε) = 0.
(2.20)
N =N (ε,M )+1
Now it is sufficient to show
lim (2ε)3/2
ε→0
∞
N =N (ε,M )+1
To this end, we need a bound on the lower tail of λN
max on the left of the mean. It turns out to
be much more delicate than the upper tail on the right of the mean. Here is a simple but quite
Su Z. G.
980
accessible inequality, which is due to Ledoux [2]. For some C > 0, every N and every ε such
that N −2/3 ≤ ε ≤ 1,
−N ε
P (λN
max < 1 − ε) ≤ Ce
3/2
/C
.
We remark that this deviation inequality does not reflect the N 2 rate of the large deviation
asymptotics nor is it in accordance with the decay rate of F at −∞. However, it is powerful
enough to show
∞
P (λN
(2.21)
lim (2ε)3/2
max < 1 − ε) = 0.
M →∞
N =N (ε,M )+1
Now we complete the proof of (1.5), and hence the proof of Theorem 1.
Proofs of Theorems 2, 3 and 4 The proofs are almost identical to the proof of Theorem 1
with only change of tail estimates in different settings. Below are the tail estimates we need in
the proofs. Since they are known in the literature, only some technical remarks are given. The
interested reader is referred to the original papers or Ledoux’s notes [2].
Lemma 1
Let LN be, as above, the length of the longest increasing subsequences. Then :
(1) There is a numerical constant C > 0 such that, for every N ≥ 1 and every ε > 0,
√
√
3/2
P (LN > N (2 + ε)) ≤ Ce− N ε /C ;
(2.22)
(2) There are numerical constants c2 , C > 0 such that, for sufficiently small ε > 0 and
(2−ε)c3
N ≥ ε3 2 ,
√
3
(2.23)
P (LN < N (2 − ε)) ≤ e−ε N/C .
(2.22) follows by taking q =
θ
N2 , N
→ ∞, in (2.26) of Lemma 2 below. To see (2.23), we use
Theorem 3.3 in Löwe, Merkl and Rolles [25], which gives the following speed of convergence:
For every fixed 0 < α < 1/2, there exist positive constants c0 , c1 , c2 and c3 (α) such that, for all
√
√
N −l
natural numbers l ≤ N with 1 < 2 l N ≤ 1 + c3 (α) and 2 l1/3
≥ c2 ,
α √
√
3
√
3/2 1
2 N
2 N −l
2 N −l
− c0
−1
P (LN ≤ l) ≤ exp −
+ c1
. (2.24)
6
l
l1/3
l1/3
√
In particular, letting l = (2 − ε) N , then, for sufficiently small ε > 0, we have
√
√
2 N
ε
2 N
2
−1=
, 1<
=
≤ 1 + c3 (α)
l
2−ε
l
2−ε
and
√
2 N −l
εN 1/3
=
≥ c2
1/3
l
(2 − ε)1/3
as N ≥
(2 − ε)c32
.
ε3
Hence it follows from (2.24) that
α 3
√ 1
ε
ε
ε3/2 N 1/2
− c0
N + c1
P LN ≤ (2 − ε) N ≤ exp −
6
2−ε
2−ε
(2 − ε)1/2
≤ e−ε
3
(if necessary take c2 large enough).
N/C
(2.25)
Precise Asymptotics for Random Matrices
981
Lemma 2 Let W (M, N ) be defined as above with geometric weights. Then, for γ ≥ 1 and
0 < q < 1, there is a numerical constant C = C(γ, q) > 0 such that, for every N ≥ 1 and
0 < ε < 1,
P (W ([γN ], N ) > N (ω(γ, q) + ε)) ≤ e−N ε
3/2
P (W ([γN ], N ) < N (ω(γ, q) − ε)) ≤ Ce−N
2 3
/C
(2.26)
and
ε /C
.
(2.27)
(2.26) is a direct consequence of the following stronger deviation result due to Johansson [18]:
P (W ([γN ], N ) > N (ω(γ, q) + ε)) ≤ e−N J(ε) ,
where the function J(ε) is explicitly known as
x
dy
γ−q
1 − γq
1
+
J(ε) =
(x − y)
,
1−q 1
y+B
y+D
y2 − 1
where
x=1+
γ(1 − q)
,
√
2 γq
γ+q
B= √ ,
2 γq
D=
1 + γq
√ .
2 γq
(2.27) was proved in [19], using refined Riemann–Hilbert steepest descent methods.
Lemma 3
2
Let ρN
max be the the largest eigenvalue of complex Wishart matrices with σ =
1
N.
Then there is a numerical constant C > 0 such that, for every N ≥ 1 and every 0 < ε < 1,
√
2
−N ε3/2 /C
P (ρN
(2.28)
max > ( γ + 1) + ε) ≤ e
and
√
2
−N ε3/2 /C
.
P (ρN
max < ( γ + 1) − ε) ≤ e
(2.29)
The explicit form of J(ε) again allows us to obtain (2.28) by taking q → 1 on both sides of
(2.26). (2.29) can be proved in a similar way to that of (5.16) in Ledoux [2], since Götze and
Tikhomirov [26] also gave the rate of convergence of the spectrum of the Wishart matrix to the
M–P distribution in (1.12).
References
[1] König, W.: Orthogonal polynomial ensembles in probability theory. Probability Surveys, 2, 385–447 (2005)
[2] Ledoux, M.: Deviation inequalities on largest eigenvalues. Concentration between probability and geometric
functional analysis, GAFA Seminar Notes, in press
[3] Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math., 62, 548–564
(1955)
[4] Bai, Z. D.: Methodologies in the spectral analysis of large dimensional random matrices: a review. Statist.
Sinica, 9, 611–661 (1999)
[5] Geman, S.: A limit theorem for the norm of random matrices. Ann. Probab., 8, 252–261 (1980)
[6] Haagerup, U., Thorbjørnsen, S.: Random matrices with complex Gaussian entries. Expo. Math., 21,
293–337 (2003)
[7] Forrester, P. J.: The spectrum edge of random matrix ensemble. Nuclear Phys., B 402, 709–728 (1993)
[8] Tracy, C. A., Widom, H.: Level-spacing distribution and the Airy kernel. Comm. Math. Phys., 159,
151–174 (1994)
[9] Aldous, D., Diaconis, P.: Longest increasing subsequences: from patience sorting to the Baik–Deift–Johan–
sson theorem. Bull. Amer. Math. Soc., 36, 413–432 (1999) .
[10] Odlyzko, A. M., Rains, E. M.: On the longest increasing subsequences in random permutations. Analysis,
Geometry, Number theory: The Mathematics of Leon Ehrenpreis, E. L. Grinberg, S. Berhanu, M. Knopp,
G. Mendoza, E. T. Quinto, eds. AMS. Contemporary Math., 251, 439–451 (2000)
982
Su Z. G.
[11] Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence
in a random permutation. J. Amer. Math. Soc., 12, 1119–1178 (1999)
[12] Johansson, K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math., 259–
296 (2001)
[13] Borodin, A., OKounkov, A., Olshanski, G., Asymptotics of Plancherel measures for sysmetric groups. J.
Amer. Math. Soc., 13, 481–515 (2000)
[14] Okounkov, A.: Random matrices and random permutations. Internat. Math. Res. Notices, 20, 1043–1095
(2001)
[15] Goodman, N. R.: Statistical analysis based on a certain multivariate complex Gaussian distribution (an
introdutcion). Ann. Math. Stat., 34, 152–177 (1963)
[16] Khatri, C. G.: Classical statistical analysis based on certain multivariate complex distributions. Ann. Math.
Statist., 36, 98–114 (1965)
[17] Marchenko, V., Pastur, L.: The distribution of eigenvalues in certain sets of random matrices. Math. Sb.,
72, 507–536 (1967)
[18] Johansson, K.: Shape fluctuations and random matrices. Comm. Math. Phys., 209, 437–476 (2000)
[19] Baik, J., Deift, P., McLaughlin, K., Miller, P., Zhou, X.: Optimal tail estimates for directed last passage
site percolation with geometric random variables. Adv. Theor. Math. Phys., 5, 1207–1250 (2001)
[20] Baum, L. E., Katz, M.: Convergence rates in the law of large numbers. Trans. Amer. Math. Soc., 120,
108–123 (1965)
[21] Heyde, C. C.: A supplement to the strong law of large numbers. J. Appl. Probab., 12, 173–175 (1975)
[22] Gut, A., Spǎtaru, A.: Precise asymptotics in the Baum–Katz and Davis law of large numbers. J. Math.
Anal. Appl., 248, 233–246 (2000)
[23] Gut, A., Spǎtaru, A.: Precise asymptotics in the law of the iterated logarithm. Ann. Probab., 28, 1870–1883
(2000)
[24] Aubrun, G.: An inequality about the largest eigenvalues of a random matrix. Lecture Notes in Math.,
1857, 320–337 (2003)
[25] Löwe, M., Merkl, F., Rolles, S.: Moderate deviations for longest increasing subsequences: the lower tail. J.
Theoret. Probab., 15, 1031–1047 (2002)
[26] Götze, F., Tikhomirov, A.: The rate of convergence for the spectra of GUE and LUE matrix ensembles.
Central European J. Math., 3, 666–704 (2005)