Higher order analogues of the Tracy-Widom distribution and the Painleve II hierarchy

0.39 MB·English

by  T. Claeys

Higher order analogues of the Tracy-Widom distribution
and the Painlev´e II hierarchy
T. Claeys, A. Its, and I. Krasovsky
January 16, 2009
9
0
0
2
Abstract

n
We study Fredholm determinants related to a family of kernels which describe
a
J the edge eigenvalue behavior in unitary random matrix models with critical edge
points. The kernels are natural higher order analogues of the Airy kernel and
6
1 are built out of functions associated with the Painlev´e I hierarchy. The Fredholm
determinantsrelatedtothosekernelsarehigherordergeneralizationsoftheTracy-

] Widom distribution. We give an explicit expression for the determinants in terms
h
p of a distinguished smooth solution to the Painlev´e II hierarchy. In addition we
- compute large gap asymptotics for the Fredholm determinants.
h
t
a
m 1 Introduction
[

In unitary random matrix ensembles with a probability measure of the form
1
v 1
3 e−ntrV(M)dM, (1.1)
Z
7 n
4 on the Hermitian n n matrices, where V is real analytic on R with sufficient growth
2 ×
at infinity, various correlation functions of eigenvalues can be expressed in terms of the
.
1
kernel
0
09 K (x,y) = e−n2V(x)e−n2V(y)κn−1(p (x)p (y) p (y)p (x)), (1.2)
n n n−1 n n−1
v: x−y κn −
i constructed from the polynomials
X
r p (x) = κ xk + , κ > 0,
a k k k
···
orthonormal with respect to the weight e−nV on R.
The limiting mean eigenvalue density is known, see e.g. [15], to be given as the
density of an equilibrium measure µ minimizing the logarithmic energy
V
1
I (µ)= log dµ(x)dµ(y)+ V(x)dµ(x), (1.3)
V
x y
ZZ | − | Z
among all probability measures µ on R. The equilibrium density depends on the po-
tential V but can in general be written in the form [19]
dµ (x) 1 Q (x), if Q (x) < 0,
ψ (x) = V = Q−(x), Q−(x) = − V V
V dx π V V (0, otherwise,
q
whereQ is areal analytic function determined byV. Generically Q has simplezeros
V V
attheendpointsofsuppψ ,sothatψ vanishesasasquareroot[36]. Forspecial(non-
V V
regular or critical) V’s however, the limiting mean eigenvalue density vanishes faster.
In general, Q has a zero of order 4k+1, k = 0,1,... at an endpoint of the support.
V
1
Remark 1.1 Q cannot have zeros of order 4k+3. These endpoint behaviors would
V
contradict variational conditions that follow from the minimization property of the
equilibrium measure [20].
Example 1.2 Thesimplestnon-regularcasek = 1isrealized,e.g.,foracriticalquartic
potential
1 4 1 8
V(x) = x4 x3+ x2+ x. (1.4)
20 − 15 5 5
Here the limiting mean eigenvalue density is supported on [ 2,2] and given by
−
1
ψ (x) = (x+2)1/2(2 x)5/2χ (x). (1.5)
V [−2,2]
10π −
It is easy to verify (1.4) by substituting this ψ (x) into the variational conditions.
V
In fact, it is much easier to recover V starting from a given limiting mean eigenvalue
density than to find the limiting mean eigenvalue density corresponding to a given
potential V directly.
In order to construct polynomial potentials V for which k 1, it is necessary that
≥
the degree of V is at least 2k+2.
In the generic case where k = 0 for the rightmost endpoint b = b of suppψ , it
V V
follows from the results of Deift et al [20, 16] that, for any fixed u and v,
1 u v
lim K (b+ ,b+ ) = K(0)(u,v), (1.6)
n
n→∞cn2/3 cn2/3 cn2/3
where
Ai(u)Ai′(v) Ai(v)Ai′(u)
K(0)(u,v) = − (1.7)
u v
−
is the Airy kernel, and c = c is a constant depending on V. Let λ be the largest
V n
eigenvalue of a random matrix M. It was proved in many cases [39, 16] and is believed
to hold for k = 0 in general that the limiting distribution of λ is given by
n
lim Prob cn2/3(λ b) < s = det(I K(0)), (1.8)
n→∞ n− − s
(cid:16) (cid:17)
where K(0) is the Airy-kernel trace-class operator acting on L2(s, ).
s
∞
Thefunctionat ther.h.s. of (1.8) isknownas theTracy-Widom distribution. Tracy
and Widom [39] discovered a representation
+∞
det(I K(0)) = exp (y s)q2(y)dy (1.9)
− s − − 0
(cid:18) Zs (cid:19)
in terms of the Hastings-McLeod solution q of the Painlev´e II equation
0
q = xq+2q3 (1.10)
xx
characterized by the asymptotic behavior
q (x) Ai(x), as x + , (1.11)
0
∼ → ∞
x 1
q (x) = − 1+ +O x−6 , as x . (1.12)
0 2 8x3 → −∞
r (cid:18) (cid:19)
(cid:0) (cid:1)
2
(0)
The Tracy-Widom function det(I K ) does not only describe the largest eigenvalue
s
−
distribution in random matrix ensembles, but appears also in several combinatorial
models, for example, it provides the distribution of the longest increasing subsequence
of random permutations [3].
In the general case of arbitrary k, a family of limiting kernels K(k)(u,v) appears in
place of the Airy kernel. Consider a potential V(x) such that Q (x) has a zero of order
V
4k+1, k = 0,1,... at therightmost endpointof suppψ . Furthermore, to obtain more
V
general results, consider for suitable V (x) the deformation
j
2k−1
V(x) = V(x)+ T V (x),
j j
j=0
X
e
where T are constants. The natural analogues of (1.6) are double-scaling limits where
j
n and simultaneously T 0 at an appropriate rate in n characterized by
j
→ ∞ →
parameters t (in the simplest case we can take all T = t = 0). We refer to [12]
j j j
for more details about those double-scaling limits in the case k = 1. In these limits
we expect the following expressions for the kernel (corresponding to V(x)) and the
distribution of the largest eigenvalue, according to conjectures in the physics literature
[6, 8]: e
1 u v
lim K (b+ ,b+ )= K(k)(u,v;t ,...,t ), (1.13)
n→∞cn2/(4k+3) n cn2/(4k+3) cn2/(4k+3) 0 2k−1
for certain constants b and c and any fixed u, v. Here
(2k) (2k) (2k) (2k)
Φ (u)Φ (v) Φ (v)Φ (u)
K(k)(u,v;t ,...,t )= 1 2 − 1 2 , (1.14)
0 2k−1
2πi(u v)
− −
(2k) (2k)
where the functions Φ (w) = Φ (w;t ,...,t ) are described below. Moreover,
j j 0 2k−1
lim Prob cn2/(4k+3)(λ b)< s
n
n→∞ −
(cid:16) = lim det(I K(cid:17)χ ) = det(I K(k)), (1.15)
n→∞ − n (b+s/[cn2/(4k+3)],+∞) − s
where K χ is the operator with kernel K acting on L2(a,b), and K(k) is the trace-
n (a,b) n s
class operator with kernel (1.14) acting on L2(s, ).
∞
Note that (1.13) for the case k = 1 was proved in [12]. Our goal in this paper is not
(k)
to prove (1.13) and (1.15). Rather, we study the properties of det(I K ).
s
−
(2k) (2k) (2k) (2k)
The functions Φ = Φ (ζ;t ,...,t ) and Φ = Φ (ζ;t ,...,t ) ap-
1 1 0 2k−1 2 2 0 2k−1
pear as solutions of the Lax pair associated with a distinguished solution to the 2k-th
member of the Painlev´e I hierarchy [6, 8, 12]. They are most easily characterized in
terms of the following Riemann-Hilbert (RH) problem.
RH problem for Φ
(a) Φ =Φ(2k) : C Γ C2×2 is analytic, with
\ →
Γ = ∪4j=1Γj∪{0}, Γ1 = R+, Γ3 = R−, Γ2 =e4−k+iπ3R−, Γ4 = e4kiπ+3R−,
oriented as in Figure 1.
3
H
H
H Γ
H 2
H
HjH
H
Γ H Γ
3 - (cid:8)H - 1
(cid:8)
(cid:8) 0
(cid:8) r
(cid:8)*
(cid:8)
(cid:8)
(cid:8)(cid:8) Γ4
(cid:8)
Figure 1: Contour for the Φ-RH problem.
(b) ΦhasL2 boundaryvaluesΦ asζ approaches Γfromtheleft(w.r.t. thedirection
+
shown by an arrow), and Φ , from the right. They are related by the jump
−
conditions
0 1
Φ (ζ)= Φ (ζ) , for ζ Γ , (1.16)
+ − 1 0 ∈ 3
(cid:18)− (cid:19)
1 1
Φ (ζ)= Φ (ζ) , for ζ Γ , (1.17)
+ − 0 1 ∈ 1
(cid:18) (cid:19)
1 0
Φ (ζ)= Φ (ζ) , for ζ Γ Γ . (1.18)
+ − 1 1 ∈ 2∪ 4
(cid:18) (cid:19)
(c) Φ has the following behavior as ζ :
→ ∞
Φ(ζ)= ζ−41σ3N I +Φ∞ζ−1/2+ (ζ−1) e−θ(ζ)σ3, (1.19)
O
(cid:16) (cid:17)
whereΦ isindependentofζ (itsexplicitexpressionwillnotbeimportantbelow),
∞
1 0
σ is the Pauli matrix , N is given by
3 0 1
(cid:18) − (cid:19)
N = 1 1 1 e−14πiσ3, (1.20)
√2 1 1
(cid:18)− (cid:19)
and
2 4k+3 2k−1 ( 1)jtj 2j+1
θ(ζ;t0,...,t2k−1)= ζ 2 2 − ζ 2 , (1.21)
4k+3 − 2j +1
j=0
X
wherethe fractional powers denote (as usualthroughoutthis paper)the principal
branches analytic for ζ C ( ,0] and positive for ζ > 0.
∈ \ −∞
(d) Φ is bounded near 0.
(2k) (2k)
ThefunctionsΦ = Φ andΦ =Φ appearingin(1.14)aretheanalyticextensions
1 1 2 2
ofthefunctionsΦ andΦ fromthesectorinbetween Γ andΓ totheentirecomplex
11 21 1 2
plane. (Oneverifies theanalyticity bymultiplyingthejumpmatrices: seeRemark 2.2.)
4
A solution Φ to this RH problem satisfies a linear system of the form
Φ′ = AΦ, Φ′ = B Φ, (1.22)
ζ tj j
where A is a polynomial in ζ of degree 2k +1, and B is a polynomial in ζ of degree
j
(2k)
j + 1. The compatibility conditions for the system (1.22) yield the P equation,
I
the Painlev´e I equation of order 4k. The most general form of the RH problem for
(2k)
P involves a contour consisting of 4k+4 rays instead of 4. Our RH problem is the
I
(2k)
one corresponding to one particular solution of P , which was introduced by Br´ezin,
I
Marinari, and Parisi in [8]. The precise structure of the Painlev´e I hierarchy and the
(2k)
relation of the functions Φ and Φ to the P equation is not important for us, but
1 2 I
(k)
we refer the interested reader to [35, 32, 33]. The Fredholm determinant det(I K )
s
−
is the object we want to study, and we only need to know the RH characterization of
Φ and Φ for that purpose.
1 2
Remark 1.3 In the case k = 1, the existence of a RH solution Φ was proved in [11,
Lemma 2.3] for all real values of t ,t . The proof of this fact is however valid for
0 1
arbitrary k and arbitrary real values of the parameters t . It is important to stress that
j
the proof does not hold for odd members of the Painlev´e I hierarchy, in other words for
the function Φ(2k+1). Moreover, the RH problem for Φ(1) is not solvable for every real
t , because this would imply [21, 11] the existence of a real-valued Painlev´e I solution
0
without poles on the real line, which contradicts [29]. However, as a consequence of
Remark1.1, onlythekernelsgenerated bytheeven membersofthePainlev´e Ihierarchy
appear in unitary random matrix ensembles of the form (1.1).
Remark 1.4 ItshouldbenotedthattheRHproblemforΦ(2k) isnotuniquelysolvable:
1 0
there is a family of solutions of the form Φ(2k)(ζ), where ω can depend on
ω 1
(cid:18) (cid:19)
t ,...,t butnotonζ. Indeeditisobvious thattheleftmultiplication by aconstant
0 2k−1
matrix (independent of ζ) does not modify the jump conditions. If the matrix is lower-
triangular with 1 on both diagonal entries, neither the asymptotic condition (1.19) is
violated. One verifies as in [28] that, up to the left multiplication by such a matrix,
the RH problem is uniquely solvable. However, the kernel (1.14) is independent of the
choice of a solution and thus is well-defined.
It follows from (1.14) and (1.19) that
4k+3 4k+3
K(k)(u,v;t ,...,t )= (e−cu 2 −cv 2 ), as u,v + , (1.23)
0 2k−1
O → ∞
(k)
for some constant c > 0. Consequently, the asymptotics of lndet(I K ) can be
s
−
obtained by a standard series expansion. In particular,
4k+3
lndet(I K(k))= (e−cs 2 ), as s + . (1.24)
− s O → ∞
(k)
The question of the asymptotic behavior of det(I K ) as s is much more
s
− → −∞
subtle and will be addressed below.
1.1 Statement of results
In the present paper, we will answer the following questions. What are the “large gap”
(k)
asymptotics for the Fredholm determinant det(I K ) as s ? Can we find an
s
− → −∞
(k)
identity for det(I K ) which generalizes the Tracy-Widom formula (1.9) to general
s
−
k? What is the analogue of the Hastings-McLeod solution of Painlev´e II for general k,
and what are its properties?
5
1.1.1 Large gap asymptotics
ThefirstgoalofthispaperistofindlargegapasymptoticsfortheFredholmdeterminant
(k)
det(I K ) as s . Those asymptotics are known for the Airy kernel, i.e. in
s
− → −∞
the case k = 0, namely
s 3 1
lndet(I K(0)) = | | ln s +χ+ (s −3/2), (1.25)
− s − 12 − 8 | | O | |
where
1
χ = ln2+ζ′( 1), (1.26)
24 −
and ζ(s)is the Riemann zeta-function. The firsttwo terms at the r.h.s. of (1.25) follow
easily from (1.9) and (1.12). The expansion of the derivative of (1.25) was obtained by
Tracy and Widom who also conjectured the value (1.26) for the constant χ [39]. A full
proof of (1.25), (1.26) was given recently in [17] and another proof followed shortly in
[2].
We obtain
Theorem 1.5 Let K(k) be the kernel defined in (1.14) for arbitrary k = 0,1,2,...,
(k)
depending on parameters t ,...,t , and write K for the trace-class operator with
0 2k−1 s
kernel K(k) acting on L2(s,+ ). The asymptotic expansion for the Fredholm determi-
∞
(k)
nant det(I K ) as s is given by the formula
s
− → −∞
d 1 3a (s)
ds lndet(I −Ks(k))= 4a20(s)|s|4k+2+ 16a1(s)s +O(|s|−4k2+5). (1.27)
0
| |
Here a (s,t ,...,t ) and a (s,t ,...,t ) are defined as follows:
0 0 2k−1 1 0 2k−1
1 Γ(2k+ 3)
a (s) = 2
j Γ(j + 3) Γ(2k+2 j)
2 −
2k+1−j Γ(2k+ 3 m)
+ t 2 − s −m , (1.28)
2k+1−m
Γ(2k+2 j m)| |
!
m=2 − −
X
where the sum vanishes if j =2k, j = 2k+1, and Γ(x) is Euler’s Γ-function.
Remark 1.6 Expanding the right-hand side of (1.27) in negative powers of s and
| |
(k)
integrating, one easily finds asymptotics for det(I K ) itself except for the constant
s
−
of integration χ(k). In particular, to the leading order
(4k+1)! 2 s 4k+3
lndet(I K(k))= | | [1+o(1)], (1.29)
− s − 24k+1(2k)!(2k+1)! 4k+3
(cid:18) (cid:19)
which was conjectured in [7, 9]. The constant χ(k) has a representation, as in the Airy
case k = 0, in terms of a solution of a Painlev´e equation and also in terms of a limit
of multiple integrals (see Section 6). For k = 0, the relevant multiple integral can be
reduced to an explicitly computable Selberg integral, which allowed the authors in [17]
to obtain the simple expression (1.26).
Example 1.7 For k = 0, we have a (s) = 1 and a (s) = 2, so that (1.27) takes the
0 1 3
form
d 1 1
ds lndet(I −Ks(0)) = 4|s|2+ 8s +O(|s|−25).
| |
6
Integrating this expression gives (1.25) without fixing the constant χ.
For k = 1,
5 5 2
a (s;t ,t ) = +t s −2+2t s −3, a (s;t ,t ) = + t s −2.
0 0 1 1 0 1 0 1 1
8 | | | | 4 3 | |
Theorem 1.5 then gives
d 52 5 5 1 3
lndet(I K(1)) = s 6+ t s 4+ t s 3+ t2 s 2+t t s +t2+ + (s −3),
ds − s 28| | 16 1| | 8 0| | 4 1| | 0 1| | 0 8s O | |
| |
which integrates to
52 1 5 1 1
lndet(I K(1))= s 7 t s 5 t s 4 t2 s 3 t t s 2 t2 s
− s −28 7| | − 16 1| | − 32 0| | − 12 1| | − 2 0 1| | − 0| |
·
3
+χ(1) ln s + (s −2), (1.30)
− 8 | | O | |
where χ(1) is the constant of integration. In Section 6, we discuss 2 representations for
this constant mentioned in Remark 1.6.
1.1.2 The Painlev´e II hierarchy
On our way to the higher order analogues of the Tracy-Widom formula, let us first de-
scribethePainlev´e IIhierarchy andinparticular onedistinguishedsolution toit, which
is a natural higher order analogue of the Hastings-McLeod solution to the Painlev´e II
equation.
(n)
The n-th member of the Painlev´e II hierarchy, the P equation, is the following
II
differential equation of order 2n for q = q(x;τ ,...,τ ) [38, 14, 30, 35]:
1 n−1
n−1
d d
+2q [q q2]+ τ +2q [q q2]= xq α, n 1, (1.31)
n x ℓ ℓ x
dx L − dx L − − ≥
(cid:18) (cid:19) ℓ=1 (cid:18) (cid:19)
X
where the operator is defined by the Lenard recursion relation
n
L
d d3 d 1
f = +4f +2f f, f = , 0= 0, j 1. (1.32)
dxLj+1 dx3 dx x Lj L0 2 Lj ≥
(cid:18) (cid:19)
(j)
In particular, the first members of the hierarchy (we write q for the j-th derivative
x
of q with respect to x) are given by the formulas:
P(1) : q 2q3 = xq α, (1.33)
II xx− −
P(2) : (q(4) 10qq2 10q2q +6q5)+τ (q 2q3) = xq α, (1.34)
II x − x− xx 1 xx− −
P(3) : (q(6) 14q2q(4) 56qq q(3) 70(q )2q 42q(q )2
II x − x − x x − x xx− xx
+70q4q +140q3q2 20q7)+τ (q(4) 10qq2 10q2q +6q5)
xx x− 2 x − x − xx
+τ (q 2q3) = xq α. (1.35)
1 xx
− −
We are interested in the odd members of the above hierarchy, with the value of the pa-
rameter α = 1. One distinguished solution of the P(n) equation attracts our attention.
2 II
It has the following properties.
7
Theorem 1.8 Fix τ ,...,τ R. For n odd and α = 1, there exists a real solution
1 n−1 ∈ 2
(n)
q = q(x;τ ,...,τ ) of the P equation (1.31) which has no poles for the real values
1 n−1 II
of x, and the following asymptotic behavior as x :
→ ±∞
1
q(x)= + (x−4n2n+1), as x + , (1.36)
2x O → ∞
1
n!2 2n
q(x)= x + (x−1), as x . (1.37)
(2n)!| | O | | → −∞
(cid:18) (cid:19)
Remark 1.9 For n = 1, (1.36) and (1.37) together fix the solution to be the Hastings-
McLeod solution of Painlev´e II [26]. Note that the function q differs from the function
u given by (1.10)-(1.12), which is the Hastings-McLeod solution corresponding to the
value α = 0, while q corresponds to α = 1. We will explain below why the case α = 1,
2 2
and not α = 0, is relevant for the Fredholm determinant if k 1. For general odd n,
≥
(n)
although the P equation is of order 2n, it is our belief that (1.36) and (1.37) together
II
still (as for n = 1) determine the real solution q uniquely. We do not have a proof of
this fact, but we will provide some supporting heuristic arguments later on.
Remark 1.10 Theasymptotics(1.36)at+ indicatethatthetermsattheright-hand
∞
side of (1.31) are dominant compared to those at the left-hand side, and that there is a
balance between the terms xq and α = 1. There is a whole family of solutions sharing
2
the local asymptotic behavior (1.36), see [30]. The asymptotics (1.37) at indicate
−∞
a balance of the q2n+1-term at the left-hand side with the term xq at the right-hand
side. Again the existence of solutions with this local behavior was shown in [30].
Remark 1.11 It is a well-known fact that there is a one-to-one map between the so-
lutions of a Painlev´e equation and a so-called monodromy surface [23]. The solution
considered in Theorem 1.8 is mapped to the point on the monodromy surface corre-
sponding to the Stokes multipliers
s = s = = s = 0,
2 3 2n
···
s = s = i.
1 2n+1
−
We will characterize this Painlev´e transcendent in Section 5 in terms of a RH problem.
1.1.3 Painlev´e expression for the Fredholm determinants
Let n = 2k + 1. Define b = b (s;t ,...,t ) for j = 0,...,2k slightly modified
j j 0 2k−1
compared to a (1.28) by
j
Γ(2k+ 3)
b (s) = 2 s2k+1−j
j Γ(j + 3)Γ(2k+2 j)
2 −
2k−1 Γ(ℓ+ 1)
( 1)ℓt 2 sℓ−j, (1.38)
− − ℓΓ(j + 3)Γ(ℓ j+1)
ℓ=j 2 −
X
and let x = x(s;t ,...,t ) and τ = τ (s;t ,...,t ) be polynomials in s defined
0 2k−1 j j 0 2k−1
by
4k+1
x(s)= 24k+3b0(s;t0,...,t2k−1), (1.39)
−
4(k−j)+1
τj(s)= (2j +1)2 4k+3 bj(s;t0,...,t2k−1), j = 1,...,2k. (1.40)
For k = 0 all t ’s, τ ’s, and the sum in (1.38) vanish.
j j
We prove
8
Theorem 1.12 Let K(k) be the kernel defined in (1.14) for arbitrary k = 0,1,..., and
write K(k) for the trace-class operator with kernel K(k) acting on L2(s,+ ). Then the
s
∞
following identities hold:
d
lndet(I K(k))= Q[x(s);τ (s),...,τ (s)], (1.41)
ds − s 1 2k
where x(s),τ (s),...,τ (s) are given by (1.39)-(1.40). Furthermore, for any fixed
1 2k
τ ,...,τ ,
1 2k
x
Q(x;τ ,...,τ ) = u(ξ;τ ,...,τ )2dξ (1.42)
1 2k 1 2k
Z−∞
with u(x) = u(x;τ ,...,τ ) given by
1 2k
u(x) = 2−44kk++13 exp +∞ q(ξ) 1 dξ exp xq(ξ)dξ , (1.43)
− − 2ξ ·
(cid:26) Z2 (cid:18) (cid:19) (cid:27) (cid:26)Z2 (cid:27)
(2k+1)
where q(x;τ ,...,τ ) is the solution of the P equation characterized by the Stokes
1 2k II
multipliers
s = s = = s = 0, s = s = i.
2 3 4k+2 1 4k+3
··· −
This solution satisfies the conditions of Theorem 1.8.
Remark 1.13 By (1.39), the limit x corresponds to s .
→ ±∞ → ∓∞
Remark 1.14 The function u(x) with fixed τ ,...,τ can also be characterized as
1 2k
the solution of the linear second order differential equation
u′′(x) = [q (x)+q(x)2]u(x), (1.44)
x
with boundary conditions given by
u(x) = 2−44kk++13 x(1+o(1)), as x + , (1.45)
2 → ∞
r
u(x) = o(1), as x . (1.46)
→ −∞
This allows us to check that our result reproduces the Tracy-Widom formula (1.9) for
k = 0. Inthiscasex = 21/3s,andfurthermore,thereexistsaBacklundtransformation
−
relating q (solving Painlev´e II with α = 1) with the usual Hastings-McLeod solution
2
q (corresponding to Painlev´e II (1.10) with α= 0). Indeed we have [4, 14]
0
2−4/3(x+2q2(x)+2q (x)) = q ( 2−1/3x)2. (1.47)
x 0
−
Equation (1.44) thus becomes
x
u′′(x) = [21/3q ( 2−1/3x)2 ]u(x). (1.48)
0
− − 2
The substitution u(x) = 2−1/6q ( 2−1/3x) = 2−1/6q (s) reduces this equation to the
0 0
−
Painlev´e II equation (1.10) for q (s) w.r.t. the variable s. Moreover, we conclude from
0
theboundaryconditions(1.45), (1.46)thatq (s)istheHastings-McLeod solution. Now
0
taking one more s-derivative of (1.41) gives
d2
lndet(I K(k)) = q (s)2,
ds2 − s − 0
which is equivalent to the Tracy-Widom formula (1.9).
9
Remark 1.15 The function q (x)+q(x)2 appearing in (1.44) is a solution of a differ-
x
ential equation which is also known as the (2k+1)-th member of the Painlev´e XXXIV
hierarchy, and which is equivalent to the (2k +1)-th member of the Painlev´e II hier-
archy, see [14] and also [4, 28]. Since the Painlev´e II hierarchy is more standard and
better known, we prefer to state our results in terms of it. For k = 0, the function
q +q2 can, as explained above, be expressed in terms of the Hastings-McLeod solution
x
q for α = 0, relying on the Backlund transformation (1.47) which relates Painlev´e II
0
for α = 1/2 with α = 0 via the Painlev´e XXXIV equation [14]. For k > 0, a transfor-
mation for the Painlev´e II hierarchy relating the values α = 1 and α = 0 is, to the best
2
of our knowledge, not present in the literature.
Outline for the rest of the paper
Theproofs of Theorems 1.5 and 1.12 are based on aRH representation for the logarith-
(k)
mic derivative of the Fredholm determinants det(I K ). We present this identity
s
−
in Section 2. In Section 3, we obtain asymptotics for d lndet(I K(k)) as s
ds − s → −∞
by applying the steepest descent method of Deift and Zhou to the associated RH prob-
lem. Section 4 is devoted to the proof of Theorem 1.8, which relies on the study of a
RH problem for the Painlev´e II hierarchy. In Section 5, we identify the RH problem
associated to the Fredholm determinant with the one related to the Painlev´e II hierar-
chy. This enables us to give higher order analogues of the Tracy-Widom formula and
thus to prove Theorem 1.12. In Section 6 we discuss the multiplicative constant in the
(k)
asymptotics of det(I K ) as s .
s
− → −∞
2 Fredholm determinant in terms of a RH solution
In this section, we give a differential identity for the Fredholm determinant det(I K )
s
−
in terms of the solution of a RH problem. This type of identity was obtained in a very
general framework in [27, 18, 4, 5]. For the convenience of the reader, we recall briefly
how to do it.
Recall first that K is an integral operator acting on L2(s,+ ) with kernel K =
s
∞
K(k) which we can write in the form
fT(u)h(v) Φ 1 Φ
K(u,v) = , f = 1 , h = − 2 , (2.1)
u v Φ 2πi Φ
2 1
− (cid:18) (cid:19) (cid:18) (cid:19)
whereΦ andΦ arerelatedtothePainlev´eIhierarchyasexplainedintheintroduction.
1 2
Note that
d dK
lndet(I K ) = tr (I K )−1 s = ((I K )−1K )(s,s)
s s s s
ds − − − ds −
(cid:18) (cid:19)
= ((I K )−1(K I +I))(s,s) = R (s,s), (2.2)
s s s
− −
where I +R is the resolvent of the operator K given by
s s
I +R = (I K )−1. (2.3)
s s
−
The operator R has a kernel of the form [18, Lemma 2.8]:
s
FT(u)H(v)
R (u,v) = , F = (I K )−1f, H = (I K )−1h. (2.4)
s s s
u v − −
−
10