The Distribution of the first Eigenvalue Spacing at the Hard Edge of the
  Laguerre Unitary Ensemble
THE DISTRIBUTION OF THE FIRST EIGENVALUE SPACING
AT THE HARD EDGE OF THE LAGUERRE UNITARY
ENSEMBLE
PETER J. FORRESTER AND NICHOLAS S. WITTE
Abstract. The distribution function for the first eigenvalue spacing in the
Laguerre unitary ensemble of finite rank random matrices is found in terms of
a Painlevé V system, and the solution of its associated linear isomonodromic
system. In particular it is characterised by the polynomial solutions to the
isomonodromic equations which are also orthogonal with respect to a defor-
mation of the Laguerre weight. In the scaling to the hard edge regime we
find an analogous situation where a certain Painlevé III′ system and its asso-
ciated linear isomonodromic system characterise the scaled distribution. We
undertake extensive analytical studies of this system and use this knowledge
to accurately compute the distribution and its moments for various values of
the parameter a. In particular choosing a = ±1/2 allows the first eigenvalue
spacing distribution for random real orthogonal matrices to be computed.
1. Introduction
The Laguerre unitary ensemble (LUEn,a) of random matrices is specified by the
eigenvalue probability density function (p.d.f.)
(1.1) p(λ1, . . . , λn)
n!cn,n+a
e−λjλaj
1≤j<k≤n
(λk − λj)2, λ1, . . . , λn ∈ [0,∞),
where
(1.2) cm,n :=
Γ(n− j + 1)Γ(m− j + 2).
The naming relates to the fact that (1.1) is the eigenvalue p.d.f. of complex Hermit-
ian matrices X with measure invariant under unitary conjugation X 7→ UXU−1,
proportional to the generalised Laguerre form
(1.3) (detX)
e−Tr(X).
2000 Mathematics Subject Classification. 15A52, 33C45, 33E17, 42C05, 60K35, 62E15.
Key words and phrases. random matrices, eigenvalue distribution, Wishart matrices, Painlevé
equations, isomonodromic deformations.
http://arxiv.org/abs/0704.1926v2
2 PETER J. FORRESTER AND NICHOLAS S. WITTE
In multivariate statistics (1.1) is realised as the eigenvalue p.d.f. for the complex
case of the so-called Wishart matrices X = Y †Y . Here Y is an N × n (N ≥ n)
rectangular matrix of i.i.d. entries with distribution N[0, 1]+iN[0, 1]. In this setting
a = N − n, and so a is naturally a non-negative integer. The spectrum of complex
Wishart matrices has found recent application in studies of wireless communication
systems [34], where the matrix Y consists of the complex amplitudes of various
channels of transmitted waves as received by the antennas.
The matrix structure Y †Y is relevant to the study of the eigenvalues of the
(n+N)× (n+N) Hermitian matrix
(1.4) X̃ :=
0N×N Y
Y † 0n×n
Thus one has that X̃ has in general N − n zero eigenvalues, with the remaining
eigenvalues given by ± the positive square roots of the eigenvalues of Y †Y (see e.g.
[9]). This matrix structure has application to the study of the Dirac equation in
the context of quantum chromodynamics [36]. There most interest is in the scaling
behaviour of the smallest eigenvalues.
In the study of matrix Lie algebras one encounters antisymmetric matrices
(XT = −X) with pure imaginary complex elements. Specifically, such matrices
are the Hermitian part of the matrix Lie algebra
(1.5) i× (so(n,C)) := {i times n× n skew symmetric complex matrices}.
If the independent imaginary complex elements are i.i.d with distribution iN[0, 1],
then the p.d.f. of the positive eigenvalues is proportional to
n = 2m even:
exp(−λ2j)
1≤j<k≤m
(λ2k − λ2j)2,(1.6)
n = 2m+ 1 odd:
λ2j exp(−λ2j )
1≤j<k≤m
(λ2k − λ2j )2.(1.7)
This ensemble will be denoted by AS(n). Under the change of variables λ2j 7→ λj
these reduce to the LUEn,a with parameters a = −1/2, 1/2 respectively.
Antisymmetric Hermitian matrices X can be used to parameterise real orthog-
onal matrices R with determinant +1 (and thus, by definition, members of the
classical group O+(n)). Thus we can write R according to a Cayley transformation
(1.8) R =
In + iX
In − iX
Note from this that the property that the eigenvalues of X come in ± pairs is
consistent with the property that the eigenvalues of R come in complex conjugate
pairs e±iθ. This can be used (see e.g. [9]) to show that with the matrix R chosen
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 3
with uniform (Haar) measure, the eigenvalue p.d.f for the eigenvalues with angles
0 ≤ θ ≤ π is proportional to
n = 2m even:
1≤j<k≤m
(cos θk − cos θj)2,(1.9)
n = 2m+ 1 odd:
(1− cos θj)
1≤j<k≤m
(cos θk − cos θj)2.(1.10)
Note that for θl → 0 these have the same leading behaviour as (1.6), (1.7) with
λl → 0. This is consistent with the fact that the m → ∞ scaled joint distribution
function for the k smallest eigenvalues, p(k) say, is the same for both ensembles,
(1.11)
AS(n)
, . . . ,
) = lim
O+(n)
, . . . ,
where n = 2m, 2m + 1. This scaling is such that the average spacing between
eigenvalues approaches unity as k → ∞. The distribution (1.11) is of primary
importance in the study of the spectral interpretation of L-functions [20],[30]. From
the remark below (1.7) we know that
(1.12)
· · ·
AS(n)
x1, . . . ,
xk) = p
LUEm,a
(x1, . . . , xk),
where for n = 2m, a = −1/2, while for n = 2m+ 1, a = 1/2. Consequently
(1.13) lim
O+(n)
, . . . ,
= lim
X1 . . . Xk p
LUEm,a
π2X21
, . . . ,
π2X2k
Thus knowledge of the distribution p
LUEm,a
for a = ±1/2 suffices to compute the
scaled limit of p
O+(n)
In the case k = 1 a number of different characterisations of p
LUEm,a
, which is the
distribution of the smallest eigenvalue in LUEm,a, are known. First, with n 7→ n+1
in (1.1) for convenience, the p.d.f. of the smallest eigenvalue is given by fixing one
of the coordinates at x1, and integrating the remaining over [x1,∞). Thus
(1.14) p
LUEn+1,a
(x1) =
n!cn+1,n+1+a
e−x1xa1
dλ1 . . .
e−λjλaj (λj − x1)2
1≤j<k≤n
(λk − λj)2.
One has that
(1.15) p
LUEn+1,a
(x1) = −
ELUEn+1,a(x1),
4 PETER J. FORRESTER AND NICHOLAS S. WITTE
where
(1.16) ELUEn+1,a(t) :=
dλ1 · · ·
dλn+1p(λ1, . . . , λn+1),
is the probability (gap probability) that no eigenvalues are in the interval (0, t).
For integer values of the parameter a in (1.1), ELUEn,a(t) was studied by orthog-
onal polynomial techniques in [11], where it was evaluated as an a×a determinant,
and by the method of Jack polynomials in [10], giving an a-dimensional integral
form. In [32] (see also [13]), for general Re(a) > −1, it was expressed in terms of a
fifth Painlevé transcendent. Explicitly, it was found that
(1.17) ELUEn,a(t) = exp
UV(s)
where UV(s) satisfies the Jumbo-Miwa-Okamoto σ-form of the Painlevé V equation
(1.18) (tσ′′)2 −
σ − tσ′ + 2(σ′)2 + (ν0 + ν1 + ν2 + ν3)σ′
+ 4(ν0 + σ
′)(ν1 + σ
′)(ν2 + σ
′)(ν3 + σ
′) = 0,
with parameters
(1.19) ν0 = ν1 = 0, ν2 = n+ a, ν3 = n.
Alternatively the conventional Painlevé V parameters are
(1.20) α =
a2, β = 0, γ = −2n− a− 1, δ = −1
and in terms of the Okamoto parameters they are
(1.21) v2 − v1 = 0, v3 − v1 = n+ a, v4 − v1 = n, v3 − v4 = a.
Because the eigenvalue density is strictly zero for λ < 0, the neighbourhood of the
smallest eigenvalue is referred to as the hard edge, and is denoted by HEa. As
is consistent with (1.13), a well defined limit of (1.17) is obtained by the scaling
t 7→ t/4n and n→ ∞. Thus [33]
(1.22) lim
ELUEn,a(t/4n) := EHEa(t) = exp
UIII′(s)
where UIII′(t) satisfies the Jimbo-Miwa-Okamoto σ-form of the Painlevé III
′ equation
(1.23) (tσ′′)2 − v1v2(σ′)2 + σ′(4σ′ − 1)(σ − tσ′)−
(v1 − v2)2 = 0,
with parameters
(1.24) v1 = v2 = a,
and subject to the boundary condition
(1.25) UIII′(t) ∼
22a+2Γ(a+ 1)Γ(a+ 2)
ta+1.
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 5
Our interest in this paper is the distribution p
LUEm,a
and its hard edge scaled
limit. Analogous to (1.14), we see from (1.1) that
(1.26) p
LUEn+2,a
(x1, x2) =
n!cn+2,n+2+a
e−x1−x2(x1 − x2)2(x1x2)a
dλ1 . . .
e−λjλaj (λj − x1)2(λj − x2)2
1≤j<k≤n
(λk − λj)2,
where x1 denotes the smallest eigenvalue and x2 the second smallest eigenvalue. In
[12], for a integer, this was expressed as an (a + 2) × (a + 2) determinant. In the
hard edge scaled limit this gave
(1.27) lim
LUEn+2,a
) =: pHEa
(x1, x2)
= 2−4
e−x2/4
× det
Ij+2−k(
j=1,...,a
k=1,...a+2
x2 − x1
)(k−j)/2
Ij+2−k(
x2 − x1)
j=1,2
k=1,...a+2
We seek a Painlevé type characterisation of (1.26) and its scaled limit, valid for
general Re(a) > −1.
One use of knowledge of p
LUEm,a
is the computation of the distribution of the
spacing between the smallest and the second smallest eigenvalues. Denoting this
distribution by An,a for LUEn+2,a, we have
(1.28) An,a(y) :=
dx1 p
LUEn+2,a
(x1, x1 + y), y ∈ R+.
Important to our subsequent working is a rewrite of (1.26) and (1.28) in terms of
an integral of the form
(1.29) Dn(x1, x2)[w(λ)]
dλ1 . . .
w(λl)
(λl − x1)(λl − x2)
1≤j<k≤n
(λk − λj)2,
where I denotes the support of the weight w(λ). We have
(1.30) p(2)(x, x+ y) =
cn+2,n+2+a
e−(n+1)(x+y)−xy2[x(x + y)]aDn(−y,−y)[λ2(λ+ x+ y)ae−λχ>0],
(1.31)
An,a(y) =
cn+2,n+2+a
dt ta(t− y)ae−(n+2)tDn(−y,−y)[λ2(λ+ t)ae−λχ>0].
6 PETER J. FORRESTER AND NICHOLAS S. WITTE
These equations exhibit the occurrence of a deformation of the Laguerre weight
(1.32) w(x; t) := x2(x+ t)ae−x, x ∈ R+.
This deformed weight actually interpolates between two Laguerre weights - when
t→ 0 then we have the general parameter a+ 2 case, whilst if t→ ∞ in the sector
−π < arg(t) ≤ π we have the special parameter situation with an exponent of 2. In
fact virtually all of our analysis can be carried over to the more general situation
where the exponent 2 is an arbitrary complex parameter suitably restricted.
We begin in Section 2 by revising appropriate results from orthogonal poly-
nomial system theory and apply this to the particular deformed Laguerre weight
(1.32). This allows us, in Section 3, to characterise the distributions (1.30) and
(1.31) by a solution of the fifth Painlevé equation and its associated linear isomon-
odromic system (see Proposition 3.2). Section 4 is devoted to the determinant
evaluations of those distributions for positive integer values of the parameter a. We
proceed in Section 5 to the study of the hard edge limits
(x1, x2) := lim
LUEn,a
),(1.33)
Aa(z) := lim
An,a(
).(1.34)
It is found that these scaled distributions can be characterised by the solution of a
certain Painlevé III′ equation and its associated linear isomonodromic system (see
Propositions 5.2, 5.5 and Remark 5.3). In Section 6 this characterisation is used to
obtain the high precision numerical values of statistical characteristics of Aa(z) for
various integer values of a and for the values a = ±1/2, the latter being relevant
to (1.6), (1.7) with the change of variable λ2j 7→ λj . Let p
(x1, x2) denote the
scaled distribution of the eigenvalues eiθ1 , eiθ2(θ1, θ2 > 0) closest to the origin in
O+(2n + 1) and O+(2n) respectively. With the scaling chosen so that the bulk
density is unity, it follows from (1.13) and (1.33) that
(1.35) p±
(x1, x2) = 4π
2x1x2p
HE±1/2
(π2x21, π
2x22).
Consequently
A±(y) :=
dx p±
(x, x+ y)
= 4π2
dxx(x + y)p
HE±1/2
(π2x2, π2(x+ y)2).(1.36)
We use our results for p
HE±1/2
to provide the high precision numerical values of
statistical characteristics of A±(y).
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 7
2. Orthogonal Polynomial System
2.1. Semi-classical Orthogonal Polynomials. Consider the general orthogonal
polynomial system {pn(x)}∞n=0 defined by the orthogonality relations
(2.1)
dx w(x)pn(x)x
0 0 ≤ m < n
hn m = n
with I denoting the support of the weight w(x). We give special notation for the
coefficients of xn and xn−1 in pn(x),
(2.2) pn(x) = γnx
n + γn,1x
n−1 + . . . .
The corresponding monic polynomials are then
(2.3) πn(x) =
pn(x).
It follows from (2.1) that
dx w(x)(pn(x))
2 = γnhn,
and thus for pn(x) to be normalised as well as orthogonal we set γnhn = 1. A
consequence of the orthogonality relation is the three term recurrence relation
(2.4) an+1pn+1(x) = (x− bn)pn(x) − anpn−1(x), n ≥ 1,
and we consider the set of orthogonal polynomials with initial values p−1 = 0
and p0 = γ0. The three term recurrence coefficients are related to the polynomial
coefficients by [31], [14]
(2.5) an =
, bn =
γn+1,1
, n ≥ 1,
along with
(2.6) b0 = −
, a0 = 0, γ0,1 = 0.
A well known consequence of (2.4) is the Christoffel-Darboux summation
(2.7)
pj(x)pj(y) = an
[pn(x)pn−1(y)− pn−1(x)pn(y)]
Central objects in our probabilistic model are the Hankel determinants
(2.8) ∆n := det[µj+k−2]j,k=1,...,n, n ≥ 1, ∆0 := 1,
(2.9) Σn := det
µ0 · · · µn−2 µn
µn−1 · · · µ2n−3 µ2n−1
, n ≥ 1, Σ0 := 0,
8 PETER J. FORRESTER AND NICHOLAS S. WITTE
defined in terms of the moments {µn}n=0,1,...,∞ of the weight,
(2.10) µn :=
dx w(x)xn.
We have integral representations for ∆n
(2.11) ∆n =
dx1 . . .
w(xl)
1≤j<k≤n
(xk − xj)2, n ≥ 1,
and Σn
(2.12) Σn =
dx1 . . .
w(xl)
1≤j<k≤n
(xk − xj)2, n ≥ 1.
The three-term recurrence coefficients are related to these determinants by standard
result in orthogonal polynomial theory [31], [14]
a2n =
∆n+1∆n−1
, n ≥ 1,(2.13)
, n ≥ 0,(2.14)
γ2n =
, n ≥ 0,(2.15)
with initial values
(2.16) a21 =
µ0µ2 − µ21
, b0 =
, µ0γ
0 = 1.
The orthogonal polynomials themselves also have a determinantal representation
(2.17)
∆n∆n+1pn(x) = det
µ0 · · · µn
µn−1 · · · µ2n−1
1 · · · xn
, n ≥ 1,
and the integral representation
(2.18)
∆n∆n+1pn(x) =
dx1 . . .
w(xl)(x−xl)
1≤j<k≤n
(xk−xj)2.
Another set of polynomial solutions to the three term recurrence relation are
the associated polynomials {p(1)n (x)}∞n=0, defined by
(2.19) p
n−1(x) :=
ds w(s)
pn(s)− pn(x)
, n ≥ 0.
In particular these polynomials satisfy
(2.20) an+1p
n (x) = (x − bn)p
n−1(x)− anp
n−2(x),
with the initial conditions p
−1(x) = 0, p
0 (x) = µ0γ1. Note the shift by one
decrement in comparison to the three-term recurrence (2.4) for the polynomials
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 9
{pn(x)}∞n=0. We also need the definition of the moment generating function or
Stieltjes function
f(x) : =
, x /∈ I,(2.21)
, x /∈ I, x→ ∞.(2.22)
We define non-polynomial associated functions {ǫn(x)}∞n=0 by
(2.23) ǫn(x) := f(x)pn(x) − p(1)n−1(x),
which also satisfy the three term recurrence relation (2.4), namely
(2.24) an+1ǫn+1(x) = (x− bn)ǫn(x) − anǫn−1(x),
subject to the initial values ǫ−1(x) = 0, ǫ0(x) = γ0f(x). The associated functions
have an integral representation analogous to (2.18)
(2.25)
∆n∆n+1ǫn(x)
(n+ 1)!
dx1 . . .
dxn+1
w(xl)
x− xl
1≤j<k≤n+1
(xk − xj)2, x /∈ I.
The polynomials and their associated functions satisfy the Casoratian relation
(2.26) pn(x)ǫn−1(x)− pn−1(x)ǫn(x) =
, n ≥ 1.
Extending (2.2) and (2.5) we have
(2.27) pn(x) = γn
0≤i<j<n
bibj −
xn−2 +O(xn−3)
valid for n ≥ 1, while for the associated functions
(2.28) ǫn(x) = γ
x−n−1 +
x−n−2
0≤i≤j≤n
bibj +
x−n−3 +O(x−n−4)
valid for n ≥ 0.
Proposition 2.1 ([6],[4],[24]). Let
(2.29)
w(x) =
2V (x)
W (x)
10 PETER J. FORRESTER AND NICHOLAS S. WITTE
for V,W irreducible. The orthogonal polynomials and associated functions satisfy
a system of coupled first order linear differential equations with respect to x ( ′ ≡
d/dx)
Wp′n = (Ωn − V )pn − anΘnpn−1, n ≥ 1,(2.30)
Wp′n−1 = anΘn−1pn − (Ωn + V )pn−1, n ≥ 0,(2.31)
for certain coefficient functions V (x),W (x),Θn(x),Ωn(x). The associated func-
tions ǫn, ǫn−1 satisfy precisely the same set of equations.
If we define the 2× 2 matrix variable
(2.32) Yn(x; t) =
pn(x)
ǫn(x)
pn−1(x)
ǫn−1(x)
then the above coupled system can be written as
(2.33)
Yn(x) =
W (x)
Ωn(x) − V (x) −anΘn(x)
anΘn−1(x) −Ωn(x)− V (x)
Yn(x)
It follows that the coefficient functions are specified by
Θn =W [ǫnp
n − ǫ′npn] + 2V ǫnpn, n ≥ 0, Θ−1 = 0,
(2.34)
Ωn = anW [ǫn−1p
n − ǫ′npn−1] + anV [ǫnpn−1 + ǫn−1pn] , n ≥ 1, Ω0 = 0.
(2.35)
Proposition 2.2 ([24]). The coefficient functions arising in Proposition 2.1 satisfy
the recurrence relations
(Ωn+1 − Ωn)(x − bn) =W + a2n+1Θn+1 − a2nΘn−1, n ≥ 0,(2.36)
Ωn+1 +Ωn = (x − bn)Θn, n ≥ 0(2.37)
We will find it necessary to study the zeros of the orthogonal polynomial pn(x)
which we denote {x1,n < . . . < xj,n < . . . < xn,n}. They have an electrostatic
interpretation as the equilibrium positions of the mobile unit charges, and there is
a set of equations governing these equilibrium positions known as the Bethe Ansatz
equations.
Proposition 2.3 ([16]). The zeros {xj,n}nj=1 of the polynomial pn(x) satisfy the
coupled functional equations
(2.38) 2
k 6=j
xj,n − xk,n
Θ′n(xj,n)
Θn(xj,n)
′(xj,n) + 2V (xj,n)
W (xj,n)
for all 1 ≤ j ≤ n.
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 11
One can also represent many useful quantities in terms of sums over the ze-
ros and we illustrate this with an example. Firstly the consecutive ratios of the
orthogonal polynomials have a partial fraction decomposition
(2.39)
pn−1(x)
pn(x)
= − 1
W (xj,n)
Θn(xj,n)
x− xj,n
along with
(2.40) a2n = −
W (xj,n)
Θn(xj,n)
Of particular relevance to our application are the semi-classical class of orthog-
onal polynomial systems [25] defined by the property that V (x) andW (x) in (2.29)
are polynomials in x. The zeros of W (x) define finite singularities of the system
of ordinary differential equations (2.30), (2.31) and will feature prominently in this
study. Let xr be such a point with r ≥ 1. Then at xr the relations (2.36) and
(2.37) can be combined and integrated to yield
(2.41) Ω2n(xr)− V 2(xr) = a2nΘn(xr)Θn−1(xr), n ≥ 1.
In fact, at a given finite singular point xr, we can deduce the following bi-linear
identities, that factorise the one above.
Corollary 2.1. The coefficient functions evaluated at a finite singular point xr are
related to evaluations of the orthogonal polynomials and associated functions by the
relations
Ωn(xr) + V (xr) = 2anV (xr)pn(xr)ǫn−1(xr), n ≥ 1,(2.42)
Ωn(xr)− V (xr) = 2anV (xr)pn−1(xr)ǫn(xr), n ≥ 1,(2.43)
Θn(xr) = 2V (xr)pn(xr)ǫn(xr). n ≥ 0,(2.44)
From the theory of Uvarov the following general result for (1.29) is known.
Proposition 2.4 ([35]). The quantity Dn(x, x)[w(λ)], defined by the equal argu-
ment form of (1.29), is evaluated in terms of the polynomials pn(x) orthogonal with
respect to w(x) and coefficients γn,∆n of this system as
(2.45) Dn(x, x)[w(λ)] =
γnγn+1
[pn(x)p
n+1(x)− pn+1(x)p′n(x)].
Proof. This is a specialisation of Uvarov’s general result to the case k = 0 and
l = 2 where the integral Dn(x1, x2)[w(λ)] is a Hankel determinant with respect to
the weight w0,2(x), defined by
(2.46) w0,2(x)dx = dρ0,2(x), ρ0,2(x) =
(s− x1)(s− x2)w(s)ds.
12 PETER J. FORRESTER AND NICHOLAS S. WITTE
The non-confluent form of the corresponding identity states that
(2.47) Dn(x1, x2)[w(λ)] = (∆n+2∆n)
1/2 det[pn+k−1(xj)]j,k=1,2
x2 − x1
and the result follows under the confluence x2 → x1. �
We see from (2.45) that our main task is to obtain appropriate characterisations
of the orthogonal polynomials and their derivatives associated with the weight
(1.32).
2.2. Deformed Laguerre Orthogonal Polynomials. As we noted in the intro-
duction we see the appearance of a deformed Laguerre weight (1.32) which is a
member of the semi-classical class with the polynomials V,W in (2.29) specified by
(2.48) 2V (x; t) = −x2 + (a+ 2− t)x+ 2t, W (x; t) = x(x+ t),
and has finite singularities at x = 0,−t. The moments have the simple evaluation
(2.49) µn(t) = t
a+n+3Γ(n+ 3)U(n+ 3, a+ n+ 4; t), n ≥ 0, |arg(t)| < π,
where U(α, γ; z) is the confluent hypergeometric that is not analytic at z = 0. The
moments can be written as a sum of two parts one of which is analytic and the
other non-analytic about t = 0
(2.50) µn(t) = Γ(a+ n+ 3)1F1(−a;−a− n− 2; t)
+ (−1)n+3Γ(a+ 1)Γ(n+ 3)
Γ(a+ n+ 4)
ta+n+31F1(n+ 3; a+ n+ 4; t),
where we have to exclude the cases a ∈ Z≥0.
Within the semi-classical class the coefficient functions Θn(x),Ωn(x) are poly-
nomials with degree fixed independently of the index n. In particular we can relate
these polynomials to the coefficients of the orthogonal polynomials themselves.
Proposition 2.5. The coefficient functions are
Θn(x) = 2n+ a+ 3− t− bn − x, n ≥ 0,(2.51)
Ωn(x) = −
(2n+ a+ 2− t)x + (n+ 1)t− a2n −
, n ≥ 1.(2.52)
Proof. From the theory of [24] we note that the degrees of the coefficient functions
are degΘn ≤ max{degW − 2, degV − 1} = 1 and degΩn ≤ max{degW − 1, degV −
1, degΘn−1, degU+1} = 2. To obtain explicit forms for these we use the definitions
(2.34) and (2.35) and the large x→ ∞ expansions of the polynomials and associated
functions given in (2.27) and (2.28). The first equalities in (2.51) and (2.52) then
follow. �
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 13
We will also find it convenient to make the following definitions motivated by
the above result,
θn := 2n+ a+ 3− t− bn,(2.53)
κn := (n+ 1)t− a2n −
.(2.54)
Proposition 2.6. The spectral derivatives of the polynomials pn(x) are (
∂/∂x)
x(x+ t)p′n = (nx+ κn − t)pn − an(θn − x)pn−1(2.55)
x(x+ t)p′n−1 = an(θn−1 − x)pn − (−x2 + (n+ a+ 2− t)x + t+ κn)pn−1(2.56)
Proof. This follows from the general form of the spectral derivatives (2.30) and
(2.30), along with the explicit particular forms (2.51) and (2.52). �
Proposition 2.7. The deformation derivatives of the orthogonal polynomials are
( ˙ ≡ ∂/∂t)
t(x+ t)ṗn =
(n+ 1)t− κn −
(x+ t)(θn + t)
pn + an(θn + t)pn−1
(2.57)
t(x+ t)ṗn−1 = −an(θn−1 + t)pn +
(x+ t)(θn−1 + t) + κn − (n+ a+ 1)t
(2.58)
Proof. We will opt to establish this relation directly from the orthonormality con-
ditions on the polynomials
dx w(x)pn(x)pn−i(x) = δi,0, 0 ≤ i ≤ n.
Differentiating this with respect to t leaves us with the relation
(2.59) 0 = a
dx w(x)
pnpn−i
dx w(x)ṗnpn−i +
dx w(x)pnṗn−i,
where use of the logarithmic derivative of w(x) has been made. Now we employ
ṗn−i =
γ̇n−i
pn−i +Πn−i−1,
to write the last term of (2.59) as
dx w(x)pnṗn−i =
γ̇n−i
δi,0 =
dx w(x)pnpn−i.
14 PETER J. FORRESTER AND NICHOLAS S. WITTE
Considering the first term of (2.59) we note
dx w(x)
pn(x)pn−i(x)
dx w(x)pn(x)
pn−i(x)− pn−i(−t)
+ pn−i(−t)
dx w(x)
pn(x)
x + t
= pn−i(−t)
dx w(x)
pn(x)
x + t
=− pn−i(−t)ǫn(−t).
But we can recast pn−i(−t) as
pn−i(−t) =
pn−j(−t)δi,j ,
pn−j(−t)
dx w(x)pn−j(x)pn−i(x),
dx w(x)pn−i(x)
pn−j(−t)pn−j(x),
dx w(x)pn−i(x)
pj(−t)pj(x).
Combining these we deduce that
dx w(x)pn−i(x)
ṗn(x) +
pn(x) − aǫn(−t)
pj(−t)pj(x)
for i = 0, . . . , n. The factor in curly brackets in the integrand must be a polynomial
in x with degree less than or equal to n, and yet is orthogonal to all polynomials
pj for j = 0, . . . , n, and thus must be identically zero. This gives us our first form
for the deformation derivative of pn,
(2.60) ṗn(x) = −
pn(x) + aǫn(−t)
pj(−t)pj(x).
Equating coefficients of pn(x) in this relation we find
(2.61) 2
= apn(−t)ǫn(−t).
Furthermore using the Christoffel-Darboux formula (2.7), and the above equation,
we can express this derivative solely as a linear combination of pn and pn−1
(2.62) ṗn(x) = aǫn(−t)
pn(−t) + an
pn−1(−t)
pn(x)
− aanǫn(−t)
pn(−t)
pn−1(x).
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 15
Using the bilinear product relations (2.43) and (2.44) we arrive at the result (2.57).
The second of the two relations can be found by shifting n 7→ n− 1 in (2.62) and
using the three term recurrence relation. �
In the matrix formulation the spectral derivative take the particular form
(2.63) ∂xYn(x; t) =
Yn(x; t).
The residue matrices are explicitly given by
κn − t −anθn
anθn−1 −κn − t
, χ0 = 0,−2(2.64)
(n+ 1)t− κn an(θn + t)
−an(θn−1 + t) κn − (n+ a+ 1)t
, χt = 0,−a(2.65)
(2.66)
Our linear system has two regular singularities at x = 0,−t and an irregular sin-
gularity at x = ∞ with Poincaré index 1. In this formulation the deformation
derivative is
(2.67) ∂tYn(x; t) =
B + At
Yn(x; t)
(2.68) B = 1
−θn − t 0
0 θn−1 + t
As we will see in Section 3.2 (2.63) and (2.67) form the monodromy preserving
system corresponding to the fifth Painlevé equation with a form equivalent to that
discussed by Jimbo [18], in contrast to other forms studied in [19], [7], [8] or [17].
Corollary 2.2. The polynomial coefficients satisfy the following coupled, first order
mixed deformation derivative and difference equations
= 2 + bn−1 − bn, n ≥ 1,(2.69)
tḃn = a
n − a2n+1 + bn, n ≥ 0,(2.70)
with the initial t = 0 values for bn and a
n given by (2.87) and (2.88) respectively.
This system of differential equations is equivalent to the Schlesinger equations.
Proof. There are several methods of proof available here. The first is using the
general result of [24], expressing the deformation derivatives of the polynomial
coefficients in terms of a sum of the coefficient functions over the movable finite
16 PETER J. FORRESTER AND NICHOLAS S. WITTE
singular points
Θn(xr)−Θn−1(xr)
W ′(xr)
ẋr , n ≥ 1,(2.71)
ḃn =
Ωn+1(xr)− Ωn(xr)
W ′(xr)
ẋr n ≥ 1.(2.72)
The only finite singular point contributing here is x = −t.
Alternatively one can find these derivatives from the polynomial derivatives by
examining selected coefficients. For example by considering the coefficients of xn−1
in (2.60) we have
(2.73) γ̇n,1 =
aǫn(−t)pn(−t)γn,1 + aǫn(−t)pn−1(−t)γn−1,
and therefore
(2.74)
˙(γn,1
= aǫn(−t)pn−1(−t).
Consequently, using (2.5), we find that
a [ǫn−1(−t)pn−1(−t)− ǫn(−t)pn(−t)] ,(2.75)
ḃn = a [anǫn(−t)pn−1(−t)− an+1ǫn+1(−t)pn(−t)] ,(2.76)
which are identical to (2.69) and (2.70) respectively. �
Corollary 2.3. The recurrences for the polynomial coefficients are
(2.77) a2n+2 − a2n
= 2t+ bn+1 [2n+ a+ 6− t− bn+1]− bn [2n+ a+ 2− t− bn] , n ≥ 1.
(2.78) a2n+1 [2n+ a+ 5− t− bn − bn+1]− a2n [2n+ a+ 1− t− bn−1 − bn]
= −bn(bn + t), n ≥ 1.
The initial data b0(t) and a
1(t) are given by (2.16) with the evaluation of the mo-
ments (2.49).
Proof. The first relation (2.77) follows by substituting the explicit forms for the co-
efficient functions, (2.51) and (2.52), into the relation (2.37) and requiring equality
of the polynomials in x. Equality is trivial for x2 and x1, whilst the non-trivial
equality for x0 gives (2.77). The second relation (2.78) follows from the same pro-
cedure applied to the recurrence relation (2.36) and again the only nontrivial result
occurs for the x0 part. �
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 17
This result can also be recovered from a specialisation of work in [5]. In their sys-
tem of monic orthogonal polynomials the three term recurrence coefficients βn, γn
(not to be confused with our use of these symbols subsequently) are related to ours
(2.79) βn = bn, γn = a
Equation (39) of this work implies
(2.80) γn+1 + γn = (2n+ 3)t+ (2n+ a+ 4− t)βn + 2
βk − β2n,
and differencing this once leads directly to (2.77). In addition their equation (40)
implies
(2.81) γn+1βn+1 = (2n+ a+ 5− t− βn)γn+1 + 2
βk(βk + t).
Again differencing this once one finds precisely (2.78).
A check of the above results can be made when t → 0 whilst all parameters
are kept fixed since this, as noted before, corresponds to the Laguerre weight with
parameter a+ 2. Therefore we have
a2n(0) = n(n+ a+ 2),(2.82)
bn(0) = 2n+ a+ 3,(2.83)
∆n(0) =
j=1 j!Γ(j + a+ 2)
.(2.84)
In our normalisation we have
(2.85) pn(x; 0) = (−1)n
Γ(n+ a+ 3)
L(a+2)n (x),
where L
n (x) are the standard associated Laguerre polynomials of degree n and
index α. We see that the spectral derivative equations (2.55) and (2.56) reduce
to the standard expressions for the derivative of the Laguerre polynomials, the
coefficients in the right-hand sides of the deformation derivatives (2.57) and (2.58)
vanish, the recurrence relations (2.77) and (2.78) are identically satisfied, and the
right-hand sides of (2.69) and (2.70) are zero.
In fact we will need to develop expansions about t = 0 in order to charac-
terise our quantities as particular solutions of difference and differential equations
in Section 3. To this end we have the following result.
18 PETER J. FORRESTER AND NICHOLAS S. WITTE
Proposition 2.8. For fixed n, a /∈ Z≥0 the Hankel determinant (2.8) with the
weight (1.32) has the expansion about t = 0
(2.86) ∆n(t) = ∆n(0)
(a− 1)(n+ 1)
(a+ 1)(n− 1)
nt2 +O(t3)
− 2Γ(a+ 1)
Γ(a+ 3)Γ2(a+ 4)
Γ(a+ n+ 3)
ta+3 (1 + O(t)) + O(t2a+6)
with |arg(t)| < π. Consequently, under the same conditions, the three-term recur-
rence coefficients have the expansions about t = 0
(2.87) bn(t) = 2n+ a+ 3−
2a(2n+ a+ 3)
(a+ 3)(a+ 2)2(a+ 1)
t2 +O(t3)
(a+ 2)(a+ 1)Γ2(a+ 3)
Γ(a+ n+ 3)
Γ(n+ 1)
ta+3 (1 + O(t)) + O(t2a+6),
(2.88) a2n(t) = n(n+ a+ 2)
1− 2a
(a+ 3)(a+ 2)2(a+ 1)
t2 +O(t3)
(a+ 1)Γ(a+ 3)Γ(a+ 4)
Γ(a+ n+ 2)
Γ(n+ 1)
ta+3 (1 + O(t)) + O(t2a+6)
Proof. We adopt the method of expanding the Hankel determinant by expanding
the moments to leading order
(2.89) µn(t) = Γ(a+ n+ 3) + aΓ(a+ n+ 2)t+
a(a− 1)Γ(a+ n+ 1)t2 +O(t3)
+ (−1)n+1Γ(a+ 1)Γ(n+ 3)
Γ(a+ n+ 4)
tn+a+3 +O(tn+a+4),
as t→ 0 using (2.50). The determinant can be expanded to leading orders in t, ta+3
and the resulting determinants evaluated using the identity [26]
(2.90) det(Γ(zk + j))j,k=0,...,n−1 =
Γ(zj)
0≤j<k≤n−1
(zk − zj),
where {z0, . . . , zn−1} is an arbitrary sequence not necessarily in arithmetic progres-
sion. �
We conclude this section by noting some identities relating the polynomial coeffi-
cients and the zeros of the polynomials. Firstly we give the Bethe Ansatz equations
for the zeros of the deformed Laguerre orthogonal polynomials which can be directly
deduced from Proposition 2.3.
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 19
Corollary 2.4. The zeros xj,n of the deformed Laguerre orthogonal polynomials
pn(x) satisfy the functional equations
(2.91)
xj,n + t
xj,n − θn
k 6=j
xj,n − xk,n
= 1, 1 ≤ j ≤ n.
According to the electrostatic interpretation, the terms of (2.91) can be inter-
preted in the following way - the first is the interaction of the mobile unit charge
at xj,n with the fixed charge of size 3 at the singularity x = 0, the second with the
fixed charge of size a + 1 at the singularity x = −t, the third with a fixed charge
of size −1 at the apparent singularity x = θn, the fourth the mutual repulsion
with the other mobile charges and the term on the right-hand side is the linear
confining potential. From the partial fraction decomposition (2.39) specialised to
the arguments x = 0,−t we have the summation identities.
Proposition 2.9. The following summations over the zeros have the explicit eval-
uations
κn − (n+ 1)t
θn + t
θn − xj,n
,(2.92)
κn − t
t+ xj,n
θn − xj,n
,(2.93)
θn + t
κn − t
θn − xj,n
,(2.94)
a2n =
xj,n(xj,n + t)
xj,n − θn
.(2.95)
In addition we can characterise the motion of the zeros with respect to the
deformation variable.
Proposition 2.10. The zeros xj,n(t) satisfy the differential equation with respect
(2.96) tẋj,n =
θn + t
θn − xj,n
xj,n.
Proof. This follows by equating
(2.97)
ṗn(x)
pn(x)
ẋj,n
x− xj,n
and (2.57), and then employing (2.39) along with (2.48), (2.51). �
3. Difference and Differential Equations
3.1. Difference Equations. In the first subsection we derive an alternative dif-
ference system in terms of the new variables θn(t), κn(t) as specified by (2.53),
(2.54).
20 PETER J. FORRESTER AND NICHOLAS S. WITTE
Proposition 3.1. The auxiliary functions θn(t), κn(t) satisfy a system of coupled
first order recurrence relations
κn+1 + κn = θn(θn + t− 2n− a− 3), n ≥ 0,(3.1)
θn + t
θn−1 + t
(κn − t)(κn + t)
[κn − (n+ a+ 1)t][κn − (n+ 1)t]
, n ≥ 1.(3.2)
The initial values θ0 and κ0 are given by
(3.3) θ0(t) = −2t
dx e−xx(t+ x)a
dx e−xx2(t+ x)a
, κ0 = t.
Proof. The first of the recurrence relations (2.77) can be exactly summed and the
result is
(3.4) a2n+1 + a
n = (2n+ 3)t+ (2n+ a+ 4− t)bn + 2
bi − b2n.
Recalling the second relation of (2.5) the summation appearing here can done by
recasting the equation in terms of the new variables and yields
(3.5) κn+1 + κn = −θnbn,
which is (3.1). The second member of the coupled set is most easily found from
the general relation (2.41) evaluated at the finite singular points x = 0,−t and
employing the new variables. These two key identities are
(κn + t)(κn − t) = a2nθnθn−1,(3.6)
[κn − (n+ a+ 1)t][κn − (n+ 1)t] = a2n(θn + t)(θn−1 + t).(3.7)
The ratio of these two identities yields the relation (3.2). �
There are other recurrence relations which will be used subsequently, and the
first is
(3.8) a2n(θn + θn−1 + t) = −(2n+ a+ 2)κn + [n2 + (n+ 1)(a+ 2)]t.
This follows from the subtraction of (3.6) from (3.7). The second relation
(3.9) a2n+1 − a2n − bn − t = 2κn + bnθn,
is derived by writing the definition of κn+1 − κn in terms of the old variables and
then employing (3.1). The last relation
(3.10) a2n+1θn+1 − a2nθn−1 = bn(2κn + bnθn),
is a consequence of (2.78) along with the use of (3.9).
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 21
As a consequence of relations (3.6) from (3.7) we have
θn + t
κn − t
κn − t
κn − (n+ 1)t
θn + t
κn + t
− a2n
θn−1 + t
κn − (n+ a+ 1)t
κn − t
κn − (n+ a+ 1)t
n+ a+ 2 +
κn + t
,(3.11)
(3.12) n+
κn − t
n+ a+ 2 +
κn + t
n7→n+1
= θn + t.
3.2. Reduction to Painlevé V. Here we will identify the fifth Painlevé system
as the solution to our system of equations characterising the deformed Laguerre or-
thogonal polynomial system. This is most simply seen in terms of the new variables
θn, κn rather than the basic orthogonal polynomial variables an, bn.
Proposition 3.2. The auxiliary quantities θn(t), κn(t) satisfy the coupled first or-
der ordinary differential equations
(3.13) tθ̇n = 2κn + θn(2n+ a+ 3− t− θn),
(3.14) tκ̇n =
θn + t
κ2n +
2n+ a+ 3− (2n+ a+ 2) t
θn + t
− [n2 + (n+ 1)(a+ 2)]t− t
+ (n+ 1)(n+ a+ 1)
θn + t
Equations (3.13) and (3.14) can be solved in terms of the fifth Painlevé system
(3.15) θn = t
, κn = t(1 + qp),
where q, p are the Hamiltonian variables of the Okamoto PV [27] system with the
parameters
(3.16) α =
, β = −2, γ = −(2n+ a+ 3), δ = −
(3.17) v2 − v1 = −2, v3 − v1 = n+ a, v4 − v1 = n, v3 − v4 = a.
The solutions satisfy the boundary value data at t = 0
(3.18) θn(t) =
2a(2n+ a+ 3)
(a+ 3)(a+ 2)2(a+ 1)
t2 +O(t3)
(a+ 2)(a+ 1)Γ2(a+ 3)
Γ(a+ n+ 3)
Γ(n+ 1)
ta+3 (1 + O(t)) + O(t2a+6),
22 PETER J. FORRESTER AND NICHOLAS S. WITTE
(3.19) κn(t) =
2n+ a+ 2
4n(n+ a+ 2)a
(a+ 3)(a+ 2)2(a+ 1)
t2 +O(t3)
(a+ 2)(a+ 1)Γ2(a+ 3)
Γ(a+ n+ 3)
ta+3 (1 + O(t)) + O(t2a+6),
provided a /∈ Z≥0 and |arg(t)| < π.
Proof. If we employ (3.9) in (2.70) and then substitute for bn in terms of θn then
the result is (3.13). Let us define the shorthand notation
(3.20) Γn :=
Furthermore when the deformation derivative (2.70) is summed on the free index
the result is
(3.21) tΓ̇n = a
n + Γn.
Now if compute the deformation derivative of κn and use (2.69) along with (3.21)
we arrive at
(3.22) tκ̇n = κn − a2n(θn − θn−1).
Now the idea is to eliminate a2n and θn−1, which appear in (3.22), through use
of the recurrence relations. Equation (3.2) is a linear equation for θn−1 in terms
of the unshifted variables and the solution can be substituted into (3.8) yielding
a linear relation for a2n in terms of unshifted variables. Then both solutions can
be substituted into (3.22) and the result is (3.14). One can easily verify that the
transformation to the Hamiltonian variables q, p (3.15) with the parameters (3.16)
yields the Hamilton equations of motion for the Hamiltonian in [27]. �
Remark 3.1. A few remarks can now be made regarding the identification of the
recurrences (3.1) and (3.2). This is different in appearance from the discrete in-
tegrable equations that arose in the study of the Laguerre unitary ensemble [13]
which were explicitly identified with the system in the Sakai scheme, with the ra-
tional surface D
5 → E
6 , and has a continuous limit of Painlevé IV . In fact we
find that the variables of the latter system can be expressed in terms of our own
(3.23) xn =
κn + θn(n+ a+ 1− t− θn)
θn + t
, yn = −
and it is clear that one cannot transform (3.1) and (3.2) into this system using
such a transformation. Also the two systems arise as different Schlesinger-type
transformations - in our case as a sequence where α0 7→ α0 + 1, α2 7→ α2 − 1
whereas in the other case as α0 7→ α0 + 1, α3 7→ α3 − 1.
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 23
Remark 3.2. In [13] two fundamental quantities were studied - the τ -function τ [n](t)
and its logarithmic derivative the σ-function Vn(t; a, µ). Their relation to the ob-
jects of the present work are
(3.24) τ [n] = c(n, a)n!e−ntt−n
2−n(a+4)∆n(t),
where c(n, a) is an unspecified constant and
(3.25) Vn(t; a, 2) = −nt− 4n+ t
log∆n(t).
We also note that
(3.26) Vn(t; a, 2) = Γn + n(n+ a− 2− t)
and consequently
Γn = −n(n+ a+ 2) + t
log∆n(t)(3.27)
bn = 2n+ a+ 3 + t
∆n(t)
∆n+1(t)
(3.28)
Remark 3.3. The new variable Γn(t) possess an expansion as t → 0 which can be
directly found from (2.86) and (2.87)
(3.29) n(n+ a+ 2) + Γn(t) =
nt− 2n(n+ a+ 2)a
(a+ 3)(a+ 2)2(a+ 1)
t2 +O(t3)
(a+ 2)(a+ 1)Γ(a+ 3)Γ(a+ 4)
Γ(a+ n+ 3)
ta+3 (1 + O(t)) + O(t2a+6),
again provided a /∈ Z≥0.
Remark 3.4. The use of Proposition 3.2 is in the computation of the orthogonal
polynomials in (2.4) corresponding to the weight (1.32). For this we note from
(2.53) that
(3.30) bn = 2n+ a+ 3− t− θn,
while (2.54) together with (2.5), (2.6) show
a2n = (n+ 1)t−
− κn(3.31)
= (n+ 1)t+
bj − κn,(3.32)
(the quantity an is positive, so the positive square root of this equation is to be
taken). Further, it follows from (2.13) and (2.5) that
(3.33)
1 · · · γ2n−1
= ∆n,
where each γj is the coefficient of x
j in pj(x) as specified by (2.2). All terms in the
equation (2.45) for Dn(x, x) are then known, and the task is then to compute the
integral as required by (1.31).
24 PETER J. FORRESTER AND NICHOLAS S. WITTE
An alternative system of coupled first order ordinary differential equations which
will be used for scaling to the hard edge is given in the following proposition.
Proposition 3.3. The variables θn(t),Γn(t) satisfy the coupled first order ordinary
differential equations
(3.34) tθ̇n = θn −
θ4n − 2(2n+ a+ 2− t)θ3n
+ [4Γn + (2n+ a+ 2− t)2 − 4(n+ 1)t]θ2n
+ 4t[Γn + n(n+ a+ 2− t) + a+ 2− t]θn + 4t2
(3.35) tΓ̇n = (n+ 1)t+
θn(2n+ a+ 2− t− θn) +
θ4n − 2(2n+ a+ 2− t)θ3n
+ [4Γn + (2n+ a+ 2− t)2 − 4(n+ 1)t]θ2n
+ 4t[Γn + n(n+ a+ 2− t) + a+ 2− t]θn + 4t2
Proof. We proceed in a series of steps. Firstly we use (3.6) to solve for θn−1 in
terms of θn, κn and Γn. In the second step we substitute this solution for θn−1 into
(3.7) and solve the following quadratic equation for κn in terms of θn and Γn,
(3.36)
κ2n+θn(2n+a+2−t−θn)κn+[(n+1)t−Γn]θn(θn+t)−[n2+(n+1)(a+2)]tθn−t2 = 0.
The choice of the sign of the square-root branch follows from the expansions (3.18)
and (3.29) on one hand, and on the other hand noting that as t→ 0
(3.37)
θ4n − 2(2n+ a+ 2− t)θ3n
+ [4Γn + (2n+ a+ 2− t)2 − 4(n+ 1)t]θ2n
+ 4t[Γn + n(n+ a+ 2− t) + a+ 2− t]θn + 4t2
2a(2n+ a+ 3)
(a+ 3)(a+ 2)2(a+ 1)
t2 +O(t3).
In the final step we use these solutions for θn−1 and κn in (3.22) and (3.21). �
In preparation for the hard edge scaling limit we need to make evaluations of
the polynomials at the finite singular points. Firstly considering x = 0 we note
(3.38)
πn(0)
πn−1(0)
pn(0)
pn−1(0)
κn + t
= a2n
κn − t
as follows immediately from (2.55). Furthermore we also have
(3.39) t ˙(log πn(0)) = t
˙(log pn(0)) +
(θn + t) = n+
κn − t
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 25
which follows from (2.57) and the above result. The corresponding result for the
polynomial ratio at x = −t is
(3.40)
pn(−t)
pn−1(−t)
κn − (n+ a+ 1)t
θn−1 + t
θn + t
κn − (n+ 1)t
After [1] we define the orthogonal polynomial ratios
(3.41) Qn(x; t) :=
pn(x; t)
pn(0; t)
because we are interested in the scaling properties of the orthogonal polynomial
system at the edge of their interval of orthogonality, x = 0. Then (2.4) and Propo-
sitions 2.6 and 2.7 can be translated into the following three corollaries.
Corollary 3.1. The three-term recurrence for {Qn}n=0,1,... system is
(3.42) bn(Qn+1 +Qn−1 − 2Qn) + (bn + 2
κn − t
)(Qn+1 −Qn−1) + 2xQn = 0.
Corollary 3.2. The spectral derivatives of Qn, Qn−1 are
x(x+ t)Q′n = nxQn + (κn − t)
Qn −Qn−1 +
(3.43)
x(x+ t)Q′n−1 = x[x − (n+ a+ 2− t)]Qn−1 + (κn + t)
Qn −Qn−1 −
(3.44)
Corollary 3.3. The deformation derivatives of Qn, Qn−1 are
t(x+ t)Q̇n = −x(n+
κn − t
)Qn + (κn − t)
θn + t
[Qn−1 −Qn] ,(3.45)
t(x+ t)Q̇n−1 = x[n+ a+ 2 +
κn + t
]Qn−1 + (κn + t)
θn−1 + t
[Qn−1 −Qn] .
(3.46)
As we noted earlier the polynomial ratio Qn(x; t) has a product representation
(3.47) Qn(x; t) =
where again xj,n is the j-th zero of the polynomial. We can use this fact to compute
sums of the inverse powers of the zeros from the above differential equations.
Proposition 3.4. The increment of the sum of the reciprocals of the zeros going
from n− 1 to n is given by
(3.48) 3t
xj,n−1
κn − t
+ n+ a+ 2 +
κn + t
26 PETER J. FORRESTER AND NICHOLAS S. WITTE
Proof. The required increment of the sum of the reciprocals of the zeros is the
order x term in the expansion of ψn(x; t) := Qn(x; t)/Qn−1(x; t) about x = 0, and
this can evaluated from the same expansion of the differential equation for ψn with
respect to x. This latter differential equation is easily found from (3.43) and (3.44)
and is
(3.49)
x(x+t)ψ′n =
x− θn
(κn−t)+
2κn+x[−x+2n+a+2−t]
x− θn−1
(κn+t)ψ
At the other finite singular point, x = −t, we have as a consequence of these
corollaries
(3.50)
Qn(−t; t)
Qn−1(−t; t)
κn − t
κn − (n+ 1)t
θn + t
κn − (n+ a+ 1)t
κn + t
θn−1 + t
(3.51)
Qn(−t; t) =
t− κn − nθn
θn(θn + t)
Qn(−t; t).
Note that the derivative with respect to t in the latter equation is a total derivative.
To complete our preparations for the hard edge scaling we need to identify two
polynomial variables that will scale to independent variables in the scaling limit.
The first is the orthogonal polynomial ratio Qn, and for the second a number of
choices could be made but a simple choice is
(3.52) Rn := Qn −Qn−1.
Corollary 3.4. The spectral derivatives of Qn, Rn are
x(x + t)Q′n = x
κn − t
Qn + (κn − t)
θn − x
Rn,(3.53)
x(x+ t)R′n = x
κn − t
+ n+ a+ 2 +
κn + t
− x− t
Qn(3.54)
κn − t
+ x(x + t)− (a+ 2)x− 2t
Proof. This follows from Corollary 3.2. �
Corollary 3.5. The deformation derivatives of Qn, Rn are
t(x + t)Q̇n = −x
κn − t
Qn − (κn − t)
θn + t
(3.55)
t(x+ t)Ṙn = −x
κn − t
+ n+ a+ 2 +
κn + t
(3.56)
n+ a+ 2 +
κn + t
+ (κn + t)
θn−1 + t
− (κn − t)
θn + t
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 27
Proof. This follows from Corollary 3.3. �
3.3. Inequalities and Bounds. A key step in proving our hard edge scaling limits
will be bounds on the variables θn, κn and some auxiliary quantities. The first step
is the following result.
Lemma 3.1. The variables θn(t), κn(t) satisfy the inequalities
θn + t
κn − (n+ 1)t
κn − t
< 0,(3.57)
κn + t
θn−1 + t
κn − (n+ a+ 1)t
< 0,(3.58)
for all positive, real and bounded t and n ≥ 1.
Proof. That the ratios given in (3.57) and (3.58) are negative is a consequence of
the fact that the polynomial pn(x) evaluated on the negative real axis, i.e. exterior
to the interval of orthogonality, has a fixed sign. Specifically (−1)npn(−y) > 0
for real, positive y. Using the ratio relations (3.38) and (3.40) we have the upper
bounds. From the Christoffel-Darboux formula (2.7) at equal arguments we note
(3.59) pn−1(x)p
n(x)− p′n−1(x)pn(x) > 0,
and from the above pn−1(x)pn(x) < 0 for x ∈ −R+ we conclude
(3.60)
p′n(x)
pn(x)
p′n−1(x)
pn−1(x)
under the conditions on x. Integrating this inequality from 0 to −y ∈ −R+ we
arrive at
(3.61)
pn(−y)
pn−1(−y)
pn(0)
pn−1(0)
Then identifying these ratios with (3.38) and (3.40) in the case y = t leads to the
relative inequalities. �
The above set of inequalities must all apply simultaneously and we see in fact
that it implies restrictions on the variables θn, κn.
Lemma 3.2. For bounded t ∈ R+ and n ≥ 0 the variables θn, κn satisfy inequalities
which place them in one of three cases, as illustrated in Figure 1 -
Case I:
0 < θn,(3.62)
θn(θn + t− 2n− a− 3) + t < κn < t− nθn,(3.63)
Case II:
−t ≤ θn ≤ 0,(3.64)
t− nθn ≤ κn ≤ t+min{nt, θn(θn + t− 2n− a− 3)} ≤ (n+ 1)t,(3.65)
28 PETER J. FORRESTER AND NICHOLAS S. WITTE
(n+1)t
t−nθn
(θn+t)(θn−2n−a−3)+(n+1)t
θn(θn+t−n)+t
θn(θn+t−2n−a−3)+t
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Figure 1. A pictorial form of the inequalities taking the example
of a = 1/2, n = 2 and t = 5/3, which illustrates the generic
situation for n+ a+ 3 > t.
Case III:
θn < −t,(3.66)
(n+ 1)t < κn < t− nθn.(3.67)
Proof. From the three inequalities implied by (3.57) we see that θn ≷ 0 according
as κn ≶ t, θn + t ≷ 0 according as κn ≶ (n+ 1)t, and θn(θn + t) ≷ 0 according as
κn ≶ t − nθn. In (3.58) we make the replacement n 7→ n + 1 and employ (3.1) to
eliminate κn+1. This inequality now reads
(3.68) − t− κn − (n+ 1)t
θn + t
< −κn − t
< bn = 2n+ a+ 3− t− θn.
Consequently these three inequalities imply θn ≷ 0 according as κn − t ≷ θn(θn +
t− 2n− a− 3), θn + t ≷ 0 according as κn − (n+ 1)t ≷ (θn + t)(θn − 2n− a− 3),
and θn(θn + t) ≷ 0 according as κn ≶ θn(θn + t− n) + t. Combining these sets of
inequalities leads to the three cases above. �
We see that Case II applies in our situation.
Lemma 3.3. For all n and t ∈ R+ we have −t ≤ θn ≤ 0 and t − nθn ≤ κn ≤
(n+ 1)t.
Proof. From the residue formula (2.44) we recall that Θn(0) = θn = 2tpn(0)ǫn(0)
and Θn(−t) = θn + t = −atpn(−t)ǫn(−t). As we noted in the proof of Lemma
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 29
(3.1) it is immediate from the integral representations of the polynomials and their
associated functions, (2.18) and (2.25), that (−1)npn(−t) ≥ 0 and (−1)n+1ǫn(−t) ≥
0 for all real t ≥ 0. This places θn in the range applying to Case II. �
We can also draw some conclusions concerning the zeros of the orthogonal poly-
nomials which will be important subsequently.
Corollary 3.6. Each zero xj,n(t) is a monotonically decreasing function of t and
interpolates between the Laguerre zero with t = 0 and exponent a + 2 and the
Laguerre zero with t = ∞ and exponent 2,
(3.69) xj,n(0) > xj,n(t) > xj,n(∞) > 0,
for all 1 ≤ j ≤ n and bounded t > 0.
Corollary 3.7. The following bounds on the reciprocal sums over the zeros hold
xj,n(t)
,(3.70)
xj,n(t) + t
.(3.71)
Proof. From the two-sided bound on θn we can deduce
(3.72)
xj,n(t) + t
xj,n(t)− θn
xj,n(t)
Employing this in the Bethe Ansatz (2.91) summed over j we arrive at the above
bounds. �
4. Special Case a ∈ Z≥0
Our evaluation of the distribution function in terms of the fifth Painlevé sys-
tem is with all three free parameters variable in some sense - one is fixed in this
application at a positive integer, one is the index n ∈ Z and the remaining one is
a ∈ C. Up to this point we have studied in some depth the recurrence relations
with respect to n while a has been left arbitrary other than being restricted because
of the existence considerations. From the point of view of the Painlevé theory it is
quite natural that the transcendental objects become classical when a ∈ Z for either
positive or negative subsets of the integers. In particular it is expected that the τ
functions in the theory will have Hankel determinantal forms of classical function
entries with a rank dependent on a. It is these cases which have been studied in the
past [12],[11] using methods which transform the integral into the determinantal
representations and then employ confluent Vandermonde identities.
30 PETER J. FORRESTER AND NICHOLAS S. WITTE
Proposition 4.1. When a ∈ Z>0 we have the evaluation for the Hankel determi-
(4.1) ∆n(t) =
cn+1,n+1+a
(n+ a)!
det[L
(j+k+1−a)
n+a+1−j−k(−t)]j,k=1,...,a,
and for a = 0
(4.2) ∆n(t) =
cn+1,n+1
Proof. In [12] Eq. (3.18) (after correcting) states
(4.3) ∆n(t) =
cn+1,n+1+a
(n+ a)!
(−1)a(a−1)/2 det[Dj+k−2x L
−(a−3)
n+a−1 (x)|x=−t]j,k=1,...,a,
where Dx := d/dx. Using the Laguerre polynomial identity
(4.4) Dmx L
n (x) = (−1)mL
(α+m)
n−m (x), m ∈ Z≥0,
with the proviso L
n (x) = 0 for n < 0, we arrive at (4.1). �
As a consequence of the relations (3.28) and (3.27) the variables θn(t),Γn(t) will
have a× a determinant forms, and in particular for a = 0
(4.5) θn(t) = −t, κn(t) = (n+ 1)t, Γn(t) = −n(n+ 2).
The orthogonal polynomials also have determinantal forms of the following type.
Proposition 4.2. When a ∈ Z>0 the orthogonal polynomials are given by
(4.6)
∆n∆n+1pn(x; t) = (−1)n+a+⌊
⌋a! . . . (n+ a)!1! . . . (n+ 1)!
× (x+ t)−a det
(k+1−a)
n+a+1−k(x)
k=1,...a+1
(j+k−a)
n+a+2−j−k(−t)
j=1,...,a
k=1,...a+1
and for a = 0
(4.7) pn(x; t) = (−1)n
(n+ 2)(n+ 1)
L(2)n (x).
If a > n+1 we note that L
(a+1)
n+1−a(−t) = 0. Consequently, under the same condition,
the polynomial ratio is given by
(4.8) Qn(x; t) =
(k+1−a)
n+a+1−k(x)
k=1,...a+1
(j+k−a)
n+a+2−j−k(−t)
j=1,...,a
k=1,...a+1
n+a+1−k
k=1,...a+1
(j+k−a)
n+a+2−j−k(−t)
j=1,...,a
k=1,...a+1
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 31
Proof. Starting with the integral representation (2.18) we follow the procedures
used in [12]. Taking one factor of the squared product of differences we write it like
(4.9)
1≤j<k≤n
(xk − xj) =
l=0 cl
det[L
k−1(xj)]j,k=1,...,n,
using the Vandermonde identity and where cn is the leading coefficient of L
n (x)
and α is a parameter to be fixed later. Of the remaining factors in the integrand
we write
(4.10)
(xj + t)
a(x− xj)
1≤j<k≤n
(xk − xj) =
(−1)a(n+1)
l=0 l!
l=0 cl
(x+ t)−a
× det
k−1(xj)
j=1,...,n
k=1,...,N
k−1(x)
k=1,...,N
Dj−1y L
k−1(y)|y=−t
j=1,...,a
k=1,...,N
where the confluent Vandermonde identity has been used and N = n+ a+ 1.
Reassembling the integral with these two factors, then expanding the determi-
nant in (4.9) we multiply each of n factors into the determinant of (4.10). Making
use of the antisymmetry of the row ordering in the first n rows of the determinant
we can perform the n integrals as long as we choose α = 2. Then
(4.11)
∆n∆n+1pn(x; t) =
(−1)a(n+1)
l=0 l!
l=0 cl
l=0 cl
(j + 2)!
(x+ t)−a
× det
n+k−1(x)
k=1,...,a+1
Dj−1y L
n+k−1(y)|y=−t
j=1,...,a
k=1,...,a+1
Using the identities
(4.12) L(α−1)n (x) = L
n (x)− L
n−1(x),
along with elementary column operations, and then identity (4.4) we are lead to
(4.6). The evaluation for the polynomial ratio (4.8) is a simple consequence of the
first evaluation along with
(4.13) L
(−a+1+k)
n+a+1−k (0) =
n+ a+ 1− k
32 PETER J. FORRESTER AND NICHOLAS S. WITTE
Proposition 4.3. When a ∈ Z≥0 the deformed Hankel determinant is given by
(4.14) Dn(x, x) = cn+2,n+2+a(−1)
(a+1)(a+2)(x+ t)−2a
× det
(j+k−1−a)
n+a+3−j−k(−t)
j=1,...,a
k=1,...a+2
(j+k−1−a)
n+a+3−j−k(x)
j=1,2
k=1,...a+2
Proof. The quantity Dn(x, x) was essentially computed in [12] in Eq. (3.20), which
can be recast as
(4.15) Dn(x, x) = cn+2,n+2+a(−1)
(a+1)(a+2)(x+ t)−2a
× det
Dj+k−2u L
−(a−1)
n+a+1 (u)|u=−t
j=1,...,a
k=1,...a+2
Dj+k−2u L
−(a−1)
n+a+1 (u)|u=x
j=1,2
k=1,...a+2
Then (4.14) follows by application of the identity (4.4). �
As a consequence the distribution of the first eigenvalue spacing is
(4.16) An,a(y) = (−1)
(a+1)(a+2)y2ey
dt ta(t− y)−ae−(n+2)t
× det
(j+k−a−1)
n+a+3−j−k(−t)
j=1,...,a
k=1,...a+2
(j+k−a−1)
n+a+3−j−k(−y)
j=1,2
k=1,...a+2
for a = 1, 2, 3, . . . and
(4.17) An,0(y) =
y2e−(n+1)y
n+1(−y)L
n−1(−y)−
L(2)n (−y)
for a = 0.
5. Hard Edge Scaling
5.1. General Case. We define new scaled spectral variables s, z by
(5.1) t =
, x = −
in the triangular domain s > z > 0 and study the scaling of the finite distribu-
tion (1.31) as the polynomial degree n → ∞. What is required here is not just
the asymptotic scaling of the orthogonal polynomial coefficients but also of the
polynomials themselves in the neighbourhood of an endpoint of the interval of or-
thogonality. For the deformed Laguerre polynomials this would be a generalisation
of the asymptotics of Hilb’s type for the Laguerre polynomials as found in Szegö
(5.2) ex/2xµ/2L(µ)n (−x) =
Γ(µ+ n+ 1)
M)µ/2
Mx) + O(n
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 33
with M = 4n + 2µ + 2 as n → ∞ and Iµ(z) the standard modified Bessel func-
tion. Despite a resurgence in activity around these questions, especially the use of
Riemann-Hilbert techniques on these problems, there are no results available for
our particular problem. A general review of the asymptotics of orthogonal polyno-
mials can be found in [23], and an introduction to the Riemann-Hilbert approach
to the asymptotics is the chapter in [22]. However to establish the nature of and
the existence of limits for our variables we do not require such techniques.
Lemma 5.1. Under the scaling of the (5.1) 4nθn(t) and κn(t) are bounded for all
real positive t and n ≥ 1.
Proof. From the result of Lemma 3.3 we see that
(5.3) − s ≤ 4nθn(s/4n) ≤ 0, s/4n− 4nθn(s/4n) ≤ κn(s/4n) ≤ (n+ 1)s/4n,
and the assertion follows. �
Corollary 5.1. Under the above conditions
(5.4) n+
κn(t)− t
θn(t)
t=s/4n
= O(1), as n→ ∞.
Proof. The above lemma states that
(5.5) θn(s/4n) = O(1), as n→ ∞,
and we find as a consequence that also
(5.6) ε := n(n+ a+ 2)− s/4 + Γn(s/4n) = O(1), as n→ ∞.
The discriminantD appearing in the workings of Proposition 3.3 can then be written
(5.7)
D2 := θ4n−2(2n+a+2−t)θ3n+[4(ε−t)+(a+2−t)2−4nt]θ2n+4t[ε+a+2−t]θn+4t2,
and therefore
(5.8)
θn(t)
t=s/4n
= O(1), as n→ ∞.
From the formula for κn in Proposition 3.3 we note that
(5.9) n+
κn − t
= −a+ 2
(θn + t)−
and the result then follows. �
Proposition 5.1. For bounded s ∈ R+ under the scaling (5.1) the variables θn(t)
and Γn(t) converge to limits in the following manner
4nθn(t)|t=s/4n = µ(s),(5.10)
n(n+ a+ 2) + Γn(t)|t=s/4n = ν(s).(5.11)
34 PETER J. FORRESTER AND NICHOLAS S. WITTE
The variable κn(t) converges like
(5.12) lim
κn(t)|t=s/4n = −
µ(s).
Proof. Firstly we note that
η := a2n(t)− n(n+ a+ 2)
t=s/4n
= O(1), as n→ ∞,
as follows from (5.6). Starting with (3.8) we see that
θn − θn−1 = 2θn + t+
(2n+ a+ 2)κn − [n2 + (n+ 1)(a+ 2)]t
(2n+ a+ 2 + η)t+ 2(n+ a+ 2)θn
κn − t
,(5.13)
and thus
nθn−(n−1)θn−1 = θn−1+
(2n+ a+ 2 + η)t+ 2(n+ a+ 2)θn
κn − t
Thus we have shown
nθn(t)− (n− 1)θn−1(t)|t=s/4n = O(n
−1), as n→ ∞.
Now we can write the quantity of interest
nθn(t)|t=s/4n − (n− 1)θn−1(t)|t=s/4(n−1)
= [nθn(t)− (n− 1)θn−1(t)]|t=s/4n
+ (n− 1)θn−1(t)|t=s/4n − (n− 1)θn−1(t)|t=s/4(n−1) ,
so we require bounds on the difference of the last two terms on the right-hand side.
Let t = s/4n and t> = s/4(n− 1). Because θn(t) is continuously differentiable and
its derivative is given by (3.13)
(n− 1) |θn−1(t)− θn−1(t>)|
≤ (n− 1)(t> − t) max
u∈(t,t>)
|θ̇n−1(u)|,
u∈(t,t>)
|θ̇n−1(u)|,
u∈(t,t>)
u−1|2κn−1(u) + θn−1(u)(2n+ a+ 1− u− θn−1(u))|,
u∈(t,t>)
u−1|θn−1(u)|
n− 1 + κn−1(u)− u
θn−1(u)
+ a+ 3− u− θn−1(u) +
θn−1(u)
≤ t> max
u∈(t,t>)
n− 1 + κn−1(u)− u
θn−1(u)
+ a+ 3 + u+ |θn−1(u)|+
|θn−1(u)|
= O(n−1), as n→ ∞.
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 35
Thus the limit shown in (5.10) exists. Turning our attention to (5.12) we can use
(3.1) to compute that
κn+1 − κn = −2κn − bnθn,
= −2κn − (2n+ a+ 3− t− θn)θn,
= −2θn
κn − t
− 2t− (a+ 3)θn + θn(θn + t),
and therefore
[κn+1(t)− κn(t)]|t=s/4n = O(n
−1), as n→ ∞.
Now in this case the quantity we require is
κn+1(t)|t=s/4(n+1) − κn(t)|t=s/4n
= κn+1(t)|t=s/4(n+1) − κn+1(t)|t=s/4n + [κn+1(t)− κn(t)]|t=s/4n ,
and therefore we need to bound the difference of first two terms on the right-
hand side. Let us denote t< = s/4(n + 1). Again because κn(t) is continuously
differentiable with derivative (3.22) we have
|κn+1(t<)− κn+1(t)|
≤ (t− t<) max
u∈(t<,t)
|κ̇n+1(u)|,
4n(n+ 1)
u∈(t<,t)
|κ̇n+1(u)|,
4n(n+ 1)
u∈(t<,t)
u−1|κn+1(u)− a2n+1(u)(θn+1(u)− θn(u))|,
u∈(t<,t)
|κn+1(u)|+ (2n+ a+ 4 + |η(u)|)u
+ 2(n+ a+ 3)|θn+1(u)|
n+ 1 +
κn+1(u)− u
θn+1(u)
= O(n−1), as n→ ∞.
In the last two steps we have used (5.13) and the subsequent estimates. Thus the
limit in (5.12) follows. The fact that the limit of κn(t) under the hard edge scaling
is related to the limit given in (5.10) follows from the relation (3.2). The limit
(5.11) is a consequence of the limits in the primary variables. �
In addition the following combinations of variables possess scaling limits which
will subsequently be useful.
36 PETER J. FORRESTER AND NICHOLAS S. WITTE
Corollary 5.2. For bounded s ∈ R+ the following limits as n→ ∞ exist
2κn(t) + θn(t)bn(t)
θn(t)
t=s/4n
,(5.14)
κn(t)− t
θn(t)
t=s/4n
C(s),(5.15)
n+ a+ 2 +
κn(t) + t
θn−1(t)
t=s/4n
−C(s),(5.16)
κn(t)− t
θn(t)
+ n+ a+ 2 +
κn(t) + t
θn−1(t)
t=s/4n
ξ(s).(5.17)
Proof. The limit in (5.15) is a consequence of (5.4) in a previous corollary. The
scaling limit of (5.16) follows from that of (5.15) and the identity (3.11). The limit
(5.17) can be derived from (3.12) and as a result one can deduce that
κn − t
+ n+ a+ 2 +
κn + t
is of order O(n−1). �
We note other relations amongst the scaling limit functions, namely
(5.18) 2µC(s) = −[(a+ 2)µ+ 2s]
[(a+ 2)µ+ 2s]2 + 4µ(µ+ s)ν − µ(µ+ s)2
(5.19) ξ(s) = −
sC(C + a)
(5.20) 2C + a+ 3 = s
µ̇− 2
Proposition 5.2. The scaled variables µ(s), ν(s) are characterised by solutions to
the PIII′ system with parameters v1 = a+ 2, v2 = a− 2. In particular
(5.21) ν(s) = −σIII′(s) +
s− a− 2,
where σIII′(s) satisfies the Jimbo-Miwa-Okamoto σ-form for PIII′ with above param-
eters. The boundary conditions to uniquely specify the solution ν(s) are
(5.22) ν(s) =
8(a+ 3)(a+ 2)2(a+ 1)
s2 +O(s3)
4a+3(a+ 2)(a+ 1)Γ(a+ 3)Γ(a+ 4)
sa+3 (1 + O(s)) + O(s2a+6),
assuming a /∈ Z≥0 and |arg(s)| < π.
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 37
Proof. Introducing the scaling ansatzes (5.10) and (5.11) formally into the differen-
tial equation (3.34) under the scaling (5.1) we find that the highest order nontrivial
relation
(5.23) µ(s) + s = 4sν̇(s),
at order n−1. Proceeding in the same manner with the differential equation (3.35)
we find the highest order relation is
(5.24) sµ̇(s) = µ+
[(a+ 2)µ+ 2s]2 + 4µ(µ+ s)ν − µ(µ+ s)2
which occurs at order n−1 as well. Eliminating µ(s) using (5.23) we find that (5.24)
yields
(5.25) s2(ν̈)2 − (a+ 2)2(ν̇)2 + ν̇(4ν̇ − 1)(sν̇ − ν) +
a(a+ 2)ν̇ −
a2 = 0,
which is almost the Jimbo-Miwa-Okamoto σ-form for PIII′ [28]. The boundary
conditions follow from the application of the scaling limit (5.11) to the expansion
about t = 0, Equation (3.29). �
In addition we find the scaling behaviour of the Hankel determinants and poly-
nomial evaluations to be given by the following propositions.
Proposition 5.3. As n→ ∞ under the hard edge scaling the Hankel determinants
scale as
(5.26) ∆n(t)|t=s/4n ∼
1! . . . (n− 1)!Γ(a+3) . . .Γ(n+ a+2) exp
and the monic polynomials evaluated at x = 0 scale as
(5.27) πn(0; t)|t=s/4n ∼
(−1)n(a+ 3)n exp
Proof. The relation (5.26) arises from integrating (3.27) with respect to t and then
employing the scaling form for the integrand as given by (5.11). The second relation
is derived by integrating (3.39) and using (5.15) for the scaling of the resulting
integrand. The factorial and Pochhammer prefactors arise from the normalisations
at t = 0. �
We find that the polynomial ratios have well defined scaling behaviour rather
than the polynomial themselves.
Proposition 5.4. The polynomial ratios Qn(x; t), Rn(x; t) scale as
Qn(x; t)|x=−z/4n,t=s/4n = q(z; s),(5.28)
nRn(x; t)|x=−z/4n,t=s/4n = p(z; s),(5.29)
where q(z; s), p(z; s) are entire functions of z.
38 PETER J. FORRESTER AND NICHOLAS S. WITTE
Proof. We start with the product form of the scaled polynomial
(5.30) Qn(−
4nxj,n(s/4n)
and seek bounds for the logarithm of the ratio of the n-th scaled polynomial to the
(n− 1)st. When 0 < s, z < ∞ we can write the general bound as the sum of four
contributions
(5.31)
Qn(−z/4n; s/4n)
Qn−1(−z/4(n− 1); s/4(n− 1))
4nxn,n(t)
4nxj,n(t)
− log
4(n− 1)xj,n(t)
4(n− 1)xj,n(t)
− log
4(n− 1)xj,n(t>)
4(n− 1)xj,n(t>)
− log
4(n− 1)xj,n−1(t>)
where t = s/4n and t> = s/4(n− 1). Using the inequality
< A−B
for A > B > 0 we can find a simpler bound
(5.32) LHS of (5.31) ≤ z
4nxn,n(t)
4nxj,n(t)
4(n− 1)xj,n(t)
4(n− 1)xj,n(t)
4(n− 1)xj,n(t>)
4(n− 1)xj,n(t>)
4(n− 1)xj,n−1(t>)
Considering the first sum of (5.32) we see that this is
4nxj,n(t)
4(n− 1)xj,n(t)
4n(n− 1)
xj,n(t)
4n(n− 1)
xj,n(t)
4n(n− 1)
= zO(n−1), as n→ ∞,
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 39
for all bounded s. The second term of (5.32) is
4(n− 1)xj,n(t)
4(n− 1)xj,n(t>)
4(n− 1)
xj,n(t)
xj,n(t>)
The summand appearing here can be bounded in the following way
xj,n(t)
xj,n(t>)
|xj,n(t>)− xj,n(t)|
xj,n(t>)xj,n(t)
xj,n(t>)xj,n(t)
(t> − t) max
u∈(t,t>)
|ẋj,n(u)|.(5.33)
Now from (2.96) we note that
|ẋj,n(u)| = u−1
θn(u) + u
−θn(u) + xj,n(u)
xj,n(u),
and so furnishes a bound on (5.33)
xj,n(t)
xj,n(t>)
xj,n(t>)xj,n(t)
4n(n− 1)
u∈(t,t>)
θn(u) + u
−θn(u) + xj,n(u)
xj,n(u),
xj,n(t>)xj,n(t)
xj,n(t) max
u∈(t,t>)
θn(u) + u
−θn(u) + xj,n(u)
xj,n(t>)
u∈(t,t>)
θn(u) + u
−θn(u) + xj,n(u)
This means that
xj,n(t)
xj,n(t>)
u∈(t,t>)
θn(u) + u
−θn(u) + xj,n(u)
xj,n(t>)
u∈(t,t>)
θn(u) + u
−θn(u) + xj,n(u)
xj,n(t>)
u∈(t,t>)
θn(u) + u
−θn(u) + xj,n(u)
xj,n(t>)
2(n− 1)
u∈(t,t>)
θn(u) + u
−θn(u) + xj,n(u)
2(n− 1)
u∈(t,t>)
θn(u) + u
−θn(u) + xj,n(u)
2(n− 1) maxu∈(t,t>)
κn(u)− u
θn(u)
where we have used (2.94) in the last step. The total contribution of the second
term is therefore bounded by
8(n− 1)2
O(1) = zO(n−1).
40 PETER J. FORRESTER AND NICHOLAS S. WITTE
The third sum in (5.32) is
4(n− 1)
xj,n(t>)
xj,n−1(t>)
4(n− 1)
xj,n(t>)
xj,n−1(t>)
4(n− 1)
xj,n(t>)
xj,n−1(t>)
From the identity (3.48) and the scaling of the variables involved as given in (5.17)
we conclude that the contribution of this sum is bounded by
4(n− 1)
O(1) = zO(n−1).
Finally we note that the isolated term in (5.32) is of order zO(n−2) as a leading
estimate of the largest zero xn,n is of order O(n). This establishes that Qn under
the hard edge scaling of the independent variables converges to a limit as n → ∞,
for all real, positive and bounded z, s. �
The spectral and deformation derivatives of the Qn, Rn system scale to the
corresponding derivatives of the q, p system as in the following result.
Proposition 5.5. Specify scaled quantities as in Proposition 5.1, Corollary 5.2
and Proposition 5.2. The spectral derivatives of the q, p system are
(s− z)z∂zq = −zCq − (µ+ z)p,(5.34)
(s− z)z∂zp = −z
(z − s)
q + [−2s+ z(C + a+ 2)]p,(5.35)
and their deformation derivatives are
(s− z)s∂sq = zCq + (µ+ s)p,(5.36)
(s− z)s∂sp = zξq − [s(2C + a)− zC]p,(5.37)
The boundary conditions satisfied by the solutions q(z; s) and p(z; s) of the above
system on the domain s ≥ z ≥ 0 along z = 0 are
q(0; s) = 1,(5.38)
p(0; s) = 0,(5.39)
for all s > 0.
Proof. The first spectral derivative (5.34) follows from the scaling of (3.53) and
employing (5.10), (5.12) and (5.15). The second member (5.35) is derived from
the scaling of (3.54) and using (5.15) and (5.17). The first deformation derivative
(5.36) follows from the scaling of (3.55) and utilising (5.10), (5.12) and (5.15). The
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 41
second deformation derivative (5.37) arises from the scaling of (3.56), employing
(5.16) and (5.17) and noting that
(κn + t)
θn−1 + t
− (κn − t)
θn + t
t=s/4n
−s(2C + a)
The boundary conditions (5.38) and (5.39) follow from the definitions (3.41) and
(3.52) respectively and the scalings in Proposition 5.4. �
Remark 5.1. Both the spectral derivative (3.43) and the deformation derivative
(3.45) scale to
(5.40) (µ+ s)z∂zq + (µ+ z)s∂sq + zC(s)q = 0,
and this mixed derivative equation can be easily found from Proposition 5.5 by
eliminating the variable p between (5.34) and (5.36), i.e. (µ+ s) times (5.34) plus
(µ+ z) times (5.36). The three-term recurrence relation (3.42) scales to
(5.41) z2∂2zq + 2sz∂z∂sq + s
2∂2sq + s
µ̇− 2
[z∂zq + s∂sq]−
zq = 0.
This can also be recovered from the equations of Proposition 5.5. If we eliminate q
between (5.34) and (5.36) by adding them we find that
(5.42) p = z∂zq + s∂sq.
Employing this we find that (5.41) is equivalent to
(5.43) z∂zp+ s∂sp+ (−1 + s
µ̇− 2
)p− 1
zq = 0,
which can be found by adding (5.35) and (5.37) and noting the relation (5.20).
Remark 5.2. In addition to the boundary conditions at z = 0 given by (5.38) and
(5.39) there are also relations along s = z
(5.44)
q(s; s) = − C
q(s; s),
and further
(5.45)
p(s; s)
q(s; s)
C + a
= − sC
However these are a consequence of the spectral and deformation derivatives and
so do not constitute independent boundary conditions.
Remark 5.3. The compatibility of the two sets of derivatives, (5.34) and (5.35) on
the one hand, and (5.36) and (5.37) on the other hand, affords a check on our
results. We find that compatibility of (5.34) and (5.36) leads us to conclude that
(5.46) ξ = sĊ +
(µ+ s),
42 PETER J. FORRESTER AND NICHOLAS S. WITTE
and we also recover (5.20). Similar considerations applied to (5.35) and (5.37) imply
(5.47) sξ̇ = −(2C + a+ 2)ξ + 1
s(2C + a),
and we get (5.46) again. Using (5.19) to eliminate µ we arrive at a coupled pair of
first order ordinary differential equations
sĊ = ξ +
C(C + a),(5.48)
sξ̇ = −(2C + a+ 2)ξ + 1
s(2C + a).(5.49)
Using the latter equation to eliminate C we obtain a second order ordinary differ-
ential equation for ξ, which by means of the transformation
(5.50) ξ(s) =
sy(s)
y(s)− 1
is transformed into the standard equation for the fifth Painlevé transcendent. This
is a degenerate case of PV which reduces to the third Painlevé transcendent be-
cause the parameters are αV = 9/2, βV = −a2/2, γV = 1/2, δV = 0. Making an
independent variable transformation s 7→ 2s so that γV = 1 we determine the
PIII parameters to be αIII = 2(2 − a), βIII = 2(a + 3), γIII = 1, δIII = −1 which is
consistent with those in Proposition 5.2.
The matrix form of the spectral derivatives (5.34,5.35) and deformation deriva-
tives (5.36,5.37) yield the Lax pairs
∂zΨ =
z − s
Ψ,(5.51)
∂sΨ =
B − As
z − s
Ψ,(5.52)
in the matrix variable
(5.53) Ψ(z; s) =
q(z; s)
p(z; s)
The system has two regular singularities at z = 0, s and an irregular one at z = ∞
with a Poincaré index of 1. This system is essentially equivalent to the isomon-
odromic system of the fifth Painlevé equation but is the degenerate case. The
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 43
residue matrices are
0 −µ(s)s
,(5.54)
µ(s) + s
ξ(s) −C(s)− a
,(5.55)
,(5.56)
−C(s) 0
−ξ(s) −C(s)
.(5.57)
Local convergent expansions about the regular singularities take the form for z = 0
q(z; s) =
r0m(s)z
χ0+m,(5.58)
p(z; s) =
u0m(s)z
χ0+m,(5.59)
for |z| < s and the initial relations amongst the coefficients are found to be
(5.60) sχ0r
0 = −µu00, sχ0u00 = −2su00.
This implies that χ0 = −2 or χ0 = 0 and u00 = 0. The latter case applies here as
both q, p are analytic at z = 0, and in addition we also require p = 0 on z = 0. In
addition r00 = 1. The recurrence relations for general m are
s(m+ 2)u0m = (C + a+m+ 1)u
m−1 +
r0m−1 −
r0m−2,(5.61)
smr0m = −µu0m + (m− 1− C)r0m−1 − u0m−1.(5.62)
For z = s we have the convergent expansion
q(z; s) =
rsm(s)(s− z)χs+m,(5.63)
p(z; s) =
usm(s)(s− z)χs+m,(5.64)
for |z − s| < s and where in this case the initial relations are
(5.65) s(−χs + C)rs0 = −(µ+ s)us0, (C + a+ χs)us0 = ξrs0.
Combining these we get a relation which is identical to the second equality in (5.45)
only if χs = 0,−a. The former case is the one we must choose as q, p are well-defined
44 PETER J. FORRESTER AND NICHOLAS S. WITTE
and finite on z = s. For general m the recurrence relations are
s(C −m)rsm + (µ+ s)usm = (C −m+ 1)rsm−1 + usm−1,
(5.66)
−ξrsm + s(C + a+m)usm = −
rsm−1 + (C + a+m+ 1)u
m−1 +
rsm−2.
(5.67)
This system has a unique solution only if s2m(a + m) 6= 0 for s > 0 and m ≥ 1
which in turn means that a 6= −N. We also note that the two sets of coefficients
are related by
(5.68) rsm(s) = (−1)m
snr0m+n(s),
and an identical relation for usm(s).
Proposition 5.6. The determinant Dn(x, x) scales as
(5.69) Dn(x, x)|x=−z/4n,t=s/4n ∼
−4∆nπn(0)πn+1(0) [q∂zp− p∂zq] .
Proof. We can employ the polynomials Qn, Rn in the evaluation (2.45) and find
Dn(x, x) = ∆nπn(0)πn+1(0)[Qn+1R
n+1 − Rn+1Q′n+1].
Applying the scaling of Proposition 5.4 to this expression we arrive at (5.69). �
Proposition 5.7. The distribution An,a(y) scales to
(5.70)
An,a(y)|y=z/4n ∼
Aa(z).
Proof. This is the only possible scaling consistent with the scaling of the indepen-
dent variable to the hard edge, y = z/4n. �
Proposition 5.8. The distribution of the first eigenvalue spacing at the hard edge
is given by
(5.71) Aa(z) =
42a+3Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3)
ds sa(s− z)a exp
[ν(v) + 2C(v)]
[q∂zp− p∂zq] .
Proof. We apply the scaling (5.70) to Eq. (1.31) and utilise the previous relation
for the scaling of the integrand (5.69). For the first three factors on the right-hand
side of (5.69) we can use the scaling results of (5.26) and (5.27), yielding the above
integral representation. �
We note that the factor of the integrand of (5.71) can be written as
(5.72) q∂zp− p∂zq =
q2 − 2
p2 − 1
(ξq − Cp)ξq − (C + a)p
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 45
The corresponding distribution A±(z) defined in (1.35) and (1.36) for the special
cases a = ±1/2 is given by
(5.73) A±(z) =
42a+2Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3)
ds s2a+1(s− z)2a+1(2s− z)2 exp
[ν(v) + 2C(v)]
× [q∂zp− p∂zq]| s7→s2
z 7→z(2s−z)
5.2. Special Case a ∈ Z≥0.
Proposition 5.9. In the special case a ∈ Z≥0 we have
(5.74)
∆n(t)|t=s/4n ∼
cn+1,n+1+a
(n+ a)!
(−1)⌊a/2⌋(2n)as−a det
Ij+2−k(
j,k=1,...,a
(5.75) ν(s) = s
log s−a det
Ij+2−k(
j,k=1,...,a
whilst for a = 0 we find ν(s) = 0.
Proof. The first relation (5.74) follows from an application of the Hilb type asymp-
totic formula (5.2) to (4.1) and the second follows by using this result in (3.27). �
Another consequence of the Hilb formula is the scaling of the orthogonal poly-
nomial ratio as given by (4.8).
Proposition 5.10. The scaled orthogonal polynomial ratio q(z; s) is
(5.76) q(z; s) = z−3/2
(z/s)k/2I3−k(
k=1,...a+1
[Ij+2−k(
s)] j=1,...,a
k=1,...a+1
s, 1/2s, 1/s3/2, 0, . . . , 0
[Ij+2−k(
s)] j=1,...,a
k=1,...a+1
, a ≥ 1
and for a = 0 is
(5.77) q(z; s) =
Finally the eigenvalue spacing distribution (4.16) takes the following form in the
scaling limit.
46 PETER J. FORRESTER AND NICHOLAS S. WITTE
Proposition 5.11. The distribution of the first eigenvalue spacing at the hard edge
for a ∈ Z>0 is
(5.78) Aa(z) = 2
−4z1/2
e−s/4
× det
[Ij+2−k(
s)] j=1,...,a
k=1,...a+2
(s/z)(2−k)/2Ij+2−k(
j=1,2
k=1,...a+2
for a ≥ 1 and for a = 0 is
(5.79) A0(z) =
e−z/4
I22 (
z)− I1(
z)I3(
Proof. We apply the Hilb asymptotic formula (5.2) to (4.16). As 1
(a+1)(a+2)+
1 + ⌊a/2⌋ is always even for a ∈ Z we have (5.78). �
6. Analytical Studies at the Hard Edge
In this section of our study we intend to develop the analytical and non-formal
theory of the solutions to the defining ordinary and partial differential equations
described in the previous sections. This is because we wish to compute precision
numerical data characterising the distribution function of the first eigenvalue spac-
ing at the hard edge Aa(z) for arbitrary parameter a. For this purpose it is not
sufficient to employ a single local expansion of the σ-function, about s = 0 say,
because it has a finite convergence domain and one cannot use this to evaluate the
s-integrals on the interval [0,∞). For this reason we construct a patchwork of over-
lapping local expansions including Taylor series expansions about regular points
s0 for positive and real values. A similar approach was undertaken by Prähofer
and Spohn in their study [29] of the exact scaling functions for one-dimensional
stationary KPZ growth.
6.1. The σ-function expansion about s = 0. The terms given in the expansion
of the Painlevé III′ σ-function (5.22) are the minimum required to specify the full
non-analytic Puiseux-type expansion of the particular solution for ν(s) about s = 0
in the sector −π ≤ arg(s) < π. This is the primary data specifying our particular
solution and we need to use this in various ways in order to compute the distribution
Aa(z).
Proposition 6.1 ([18],[21]). The Painlevé III′ σ-function ν(s) has a Puiseux-type
expansion about the fixed regular singular point s = 0 of the form
(6.1) ν(s) =
ck,js
j+ka, a ∈ C, 0 ≤ Re(a) < 1,
with |arg(s)| < π and is convergent in a finite domain s ∈ {z ∈ C : |z| < R, |za| <
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 47
The coefficients ck,j are determined by recurrences which follow from the sub-
stitution of expansion (6.1) into the relation (5.25). We will assume that a is not
rational for simplicity. Considering terms in the resulting equation with sp and
p ∈ Z≥0 then the p = 0 case implies that if c0,0 = 0 then this choice fixes
(6.2) c0,1 =
4(a+ 2)
The p = 1 is automatically zero but for the p = 2 case one has the two options
(6.3) c0,2 = −
8(a+ 3)(a+ 2)2(a+ 1)
, or 0.
The former case applies here by comparison with (5.22). For p ≥ 3 the recurrence
(6.4) −
a(p+ a+ 1)(p− a− 3)
(a+ 3)(a+ 2)2(a+ 1)
c0,p −
j(p− j)c0,jc0,p+1−j
+ 8c0,2
(j − 1)(p− j)c0,j−1c0,p+1−j
j(p+ 2− j)[(j − 1)(p+ 1− j)− (a+ 2)2]c0,jc0,p+2−j
p+2−j
jm(p+ 1− j −m)c0,jc0,mc0,p+2−j−m = 0,
which allows for c0,p to be recursively found, valid for a /∈ Z. Terms with s−2+qa for
q = 2k ≥ 2 imply that ck,0 = 0 given that c0,0 = 0 and a 6= 3/(k − 1),−1/(k + 1).
Consequently all terms with s−1+qa vanish also. The generic recurrence relation
following from examining the sp+qa term is
(6.5)
a(a+ 2)(p+ 1 + qa)cq,p+1 =
(j + ka)[p− j + (q − k)a]ck,jcq−k,p+1−j
(j+ka)[p+2−j+(q−k)a][(j−1+ka)[p+1−j+(q−k)a]−(a+2)2]ck,jcq−k,p+2−j
p+2−j
(j+ka)(m+la)[p+1−j−m+(q−k−l)a]ck,jcl,mcq−k−l,p+2−j−m,
for q, p ≥ 0. The convention is taken that sums with upper limits less than their
lower limits are zero, or equivalently coefficients with negative indices are zero.
Contrary to appearances the highest coefficient in (6.5) will turn out to be cq,p as
cq,p+2 occurs with c0,0 as a factor and cq,p+1 has a factor of
(6.6)
a(a+ 2) + c0,0 − 2(a+ 2)2c0,1 − 8c0,0c0,1,
48 PETER J. FORRESTER AND NICHOLAS S. WITTE
which is zero as a consequence of c0,0 = 0 and (6.2). We find that the coefficient
cq,p occurs as a linear term and has a factor of
(6.7) (p−1+qa)c0,1(4c0,1−1)+4(p+qa)(p−1+qa−(a+2)2)c0,2−16(p+qa)c0,0c0,2.
With the above evaluations (6.2) and (6.3) this is
(6.8) − a(p+ 1 + (q + 1)a)(p− 3 + (q − 1)a)
2(a+ 3)(a+ 2)2(a+ 1)
This vanishes when q = 1, p = 3 and we find that c1,3 is undetermined and therefore
is a free parameter. With these choices a triangular subset of the coefficients vanish
(6.9) ck,j = 0, j = 0, . . . , 3k − 1 for k = 1, . . . ,∞,
so that the initial non-zero term in the j−sum is ck,3k. All other terms are fixed
by c1,3 and a.
For the variable µ(s) we can deduce the following Puiseux-type expansion from
the above work
(6.10) µ(s) ∼
(a+ 3)(a+ 2)2(a+ 1)
4a+2(a+ 2)(a+ 1)Γ2(a+ 3)
sa+3.
In regard to the quantity 2C(s) we require as much detail about this as for ν(s).
This variable is a σ-function for an identical problem, where the fixed exponent 2
in the weight (1.32) is replaced by 3. If we define the expansion coefficients for this
object
(6.11) 2C(s) =
ak,js
j+ka, a ∈ C, 0 ≤ Re(a) < 1,
then the defining recurrences for these using (5.20) are
(6.12) a0,0 = 0, a0,1 = 4(a+ 2)(a+ 1)c0,2,
and for j ≥ 2
(6.13) a0,j = 2(a+ 2)
− (j + 1)(j − 2− a)c0,j+1 +
la0,j+1−lc0,l
For the case k ≥ 1 we have ak,j = 0 for j = 0, .., 3k− 1 and the remaining non-zero
terms are given by
(6.14) a1,j = 2(a+ 2)
− (j + 1 + a)(j − 2)c1,j+1
(l + a)a0,j+1−lc1,l +
la1,j+1−lc0,l
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 49
for k = 1 and the general case k ≥ 2 by
(6.15) ak,j = 2(a+ 2)
− (j + 1 + ka)(j − 2 + (k − 1)a)ck,j+1
(l + ka)a0,j+1−lck,l +
lak,j+1−lc0,l
(l +ma)ak−m,j+1−lcm,l
The first few terms are
(6.16) 2C(s) ∼
2(a+ 3)(a+ 2)
s− a(a
2 − 5a− 18)
8(a+ 4)(a+ 3)2(a+ 2)2(a+ 1)
4a+2(a+ 2)(a+ 1)Γ(a+ 3)Γ(a+ 4)
sa+3.
(6.17) ξ(s) ∼
4(a+ 3)
s− 3a
8(a+ 4)(a+ 3)2(a+ 2)
4a+7(a+ 3)(a+ 2)(a+ 1)Γ2(a+ 4)
sa+4.
6.2. The q, p expansion about s = 0. In this subsection we seek local expansions
about s = 0 for the coefficients functions r0m(s), u
m(s), r
m(s), u
m(s) appearing in
(5.58), (5.59), (5.63) and (5.64). In parallel with the transcendent quantities these
will have Puisuex-type expansions of the form
r0m(s) =
k,j≥0
r0m,k,js
j+ka,(6.18)
rsm(s) =
k,j≥0
rsm,k,js
j+ka,(6.19)
with analogous expansions for the remaining two coefficients. This is immediately
clear because the recurrence relations for these imply that they are polynomial
functions of the variables µ(s), C(s), ξ(s). These recurrence relations imply the
following ones for the z = 0 coefficients r0m,k,j , u
m,k,j
(6.20) (a+m+ 2)u0m,k,0 =
r0m−1,k,0, mr
m,k,0 = u
m,k,0,
for m ≥ 1, k ≥ 0. For the general case j ≥ 1, k ≥ 0 we have
(6.21) (m+ 2)u0m,k,j−1 = (a+m+ 1)u
m−1,k,j +
ak−q,j−pu
m−1,q,p
r0m−1,k,j−1−
[j−p+(k−q)a]
ck−q,j−p +
ak−q,j−p
r0m−1,q,p−
r0m−2,k,j ,
50 PETER J. FORRESTER AND NICHOLAS S. WITTE
(6.22) mr0m,k,j−1 = (m− 1)r0m−1,k,j −
ak−q,j−pr
m−1,q,p
+ u0m,k,j−1 − 4
[j − p+ (k − q)a]ck−q,j−pu0m,q,p − u0m−1,k,j,
These equations can be solved for successive values of m starting with the m = 0
values for all k, j
(6.23) u00,k,j = 0, r
0,k,j = 0, except for k = j = 0 where r
0,0,0 = 1,
which follow from r00(s) = 1, u
0(s) = 0. The next few coefficients can be read off
(6.24) r01(s) ∼
4(a+ 3)
16(a+ 4)(a+ 3)2(a+ 2)
128.4a(a+ 3)(a+ 2)(a+ 1)Γ(a+ 5)Γ(a+ 4)
sa+3,
(6.25) u01(s) ∼
4(a+ 3)
8(a+ 4)(a+ 3)2(a+ 2)
128.4a(a+ 3)(a+ 2)(a+ 1)Γ2(a+ 4)
sa+3,
(6.26) r02(s) ∼
32(a+ 4)(a+ 3)
64(a+ 5)(a+ 4)(a+ 3)2(a+ 2)
512.4a(a+ 3)(a+ 2)(a+ 1)Γ(a+ 6)Γ(a+ 4)
sa+3,
(6.27) u02(s) ∼
16(a+ 4)(a+ 3)
64(a+ 5)(a+ 4)(a+ 3)2(a+ 2)
512.4a(a+ 3)(a+ 2)(a+ 1)Γ(a+ 5)Γ(a+ 4)
sa+3.
The analogous results for rsm,k,j , u
m,k,j are
(6.28) (a+m+ 2)usm,k,0 = −
rsm−1,k,0, mr
m,k,0 = u
m,k,0,
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 51
for m ≥ 1, k ≥ 0. Again for the general case j ≥ 1, k ≥ 0 we have
(6.29) (a+m)usm,k,j−1 +
ak−q,j−1−pu
m,q,p
= (a+m+ 1)usm−1,k,j +
ak−q,j−pu
m−1,q,p
[j − 1− p+ (k − q)a]
ck−q,j−1−p +
ak−q,j−1−p
rsm,q,p
rsm−1,k,j−1 −
[j − p+ (k − q)a]
ck−q,j−p +
ak−q,j−p
rsm−1,q,p
r0m−2,k,j ,
(6.30) mrsm,k,j−1 −
ak−q,j−1−pr
m,q,p
= (m− 1)rsm−1,k,j −
ak−q,j−pr
m−1,q,p
− usm−1,k,j + 4
[j − p+ (k − q)a]ck−q,j−pusm,q,p,
Again these can be solved for successive values of m starting with the initial values
given by the relations
(6.31) rs0,k,j =
r0n,k,j−n,
along with an identical formula for us0,k,j.
6.3. The σ-function expansion about s = ∞. The nature of the expansion of
ν(s) about s = ∞ is rather different because this fixed singular point is irregular
in the case of the third Painlevé transcendent.
Proposition 6.2. The formal asymptotic expansion of ν(s) about s = ∞ has the
(6.32) ν(s) ∼
−j/2,
where d−1 = ± 12a.
Proof. We start with the general ansatz of
(6.33) ν(s) = dskα +O(s(k−1)α),
52 PETER J. FORRESTER AND NICHOLAS S. WITTE
where k ∈ N and α ∈ C with 0 < Re(α) < 1. Using (5.25) we find that the only
terms which can balance the O(1) term are the O(s2kα−1) and O(s3kα−2) terms. If
kα < 1 then the former choice applies and we find kα = 1/2. If we assume kα > 1
then the latter case must be chosen but this leads to kα = 2/3 in contradiction to
the hypothesis. With the correct choice of kα we find the above relation for leading
coefficient, d−1. Considering the sub-leading terms we note that it is only possible
for the terms of O(s−α) to balance those of O(s−1/2) so that in fact α = 1/2
and k = 1. This is entirely consistent with the expansion for the Painlevé III
transcendent qIII(t) about t = ∞ as found in [15]. �
The choice of the sign of the leading coefficient is positive in our application.
Consequently we find the first few terms of the asymptotic expansions of the σ-
function
(6.34) ν(s) =
as1/2 − 1
a(a+ 4) +
as−1/2 +
a2s−1
a(16a2 − 7)s−3/2 +O(s−2),
and the auxiliary variables
(6.35) µ(s) + s =
as1/2 − 15
as−1/2 − 15
a2s−1
a(16a2 − 7)s−3/2 +O(s−2),
(6.36) 2C(s) + a =
as−1/2 +
a2s−1 +
a(8a2 − 21)s−3/2 +O(s−2),
(6.37) ξ(s) =
as1/2 − 35
as−1/2 − 35
a2s−1 +O(s−3/2).
The large s-regime also implies a simplification of spectral derivative (5.51)
which becomes
(6.38) ∂zΨ =
2(z − s)
2(z − s) −
I+O(s−1/2)
Using the substitution Ψ 7→ exp(−a ln(s−z)I/2)Ψ this decouples and can be solved
in terms of the modified Bessel functions. An application of the boundary condition
(5.38) implies that
q(z; s) ∼
z),(6.39)
p(z; s) ∼
.(6.40)
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 53
6.4. The σ-function expansion about a regular point. In this subsection we
seek Taylor series expansions for the sigma function and derived variables about
regular points s0, taken to be positive and real without any loss of generality. Let
us write
(6.41) ν(s) =
dj(s− s0)j ,
and using (5.25) we find the following recurrence relations for the coefficients dj
(6.42) 4s20d
a(a+2)d1+ d0d1 − [s0+(a+2)2]d21 +4(s0d1 − d0)d21 = 0,
and for n ≥ 1, d2 6= 0
(6.43) 4s20(n+ 2)(n+ 1)d2dn+2 = −
a(a+ 2)(n+ 1)dn+1
+ [s0 + (a+ 2)
(j + 1)(n− j + 1)dj+1dn−j+1
(j + 1)(n− j − 1)dj+1dn−j
(j + 1)j(n− j + 1)(n− j)dj+1dn−j+1
− 2s0
(j + 2)(j + 1)(n− j + 1)(n− j)dj+2dn−j+1
− s20
(j + 2)(j + 1)(n− j + 2)(n− j + 1)dj+2dn−j+2
(i+ 1)(j + 1)(n− i− j − 1)di+1dj+1dn−i−j
− 4s0
(i+ 1)(j + 1)(n− i− j + 1)di+1dj+1dn−i−j+1.
These recurrences are solved subject to the initial values of
ck,js
0 ,(6.44)
(j + ka)ck,js
j−1+ka
0 ,(6.45)
54 PETER J. FORRESTER AND NICHOLAS S. WITTE
which in turn can be found from the solutions to the recurrences (6.4) and (6.5).
In addition we define
µ(s) =
fj(s− s0)j ,(6.46)
C(s) =
gj(s− s0)j ,(6.47)
ξ(s) =
hj(s− s0)j .(6.48)
The coefficients appearing here are computed using the recurrences
(6.49)
f0 = s0(4d1 − 1), f1 = 4d1 − 1 + 8s0d2, fj = 4[jdj + s0(j + 1)dj+1], j ≥ 2,
and subject to f0 6= 0
2f0g0 = −2s0 − (a+ 3)f0 + s0f1,(6.50)
2f0g1 = −2− (a+ 2)f1 + 2s0f2 − 2f1g0,(6.51)
2f0gj = (j − a− 3)fj + (j + 1)s0fj+1 − 2
fj−kgk, j ≥ 2,(6.52)
and provided s0 + f0 6= 0
(s0 + f0)h0 = −s0g0(a+ g0),(6.53)
(s0 + f0)h1 = −as0g1 − 2s0g0g1 − (a+ g0)g0 − (1 + f1)h0,(6.54)
(s0 + f0)hj = −as0gj − agj−1 − s0
gj−kgk −
gj−kgk−1(6.55)
− (1 + f1)hj−1 −
fj−khk, j ≥ 2.
6.5. The q, p expansion about a regular point. We will seek a Taylor series ap-
proximation for the scaled polynomial and associated function q(z; s), p(z; s) about
the regular point (z0, s0) with 0 < z0 < s0. Let us write
q(z; s) =
j,k≥0
rj,k(z − z0)j(s− s0)k,(6.56)
p(z; s) =
j,k≥0
uj,k(z − z0)j(s− s0)k,(6.57)
Using the first spectral derivative (5.34) we obtain the recurrence relation
(6.58)
z0(s0−z0)(j+1)rj+1,k+(s0−2z0)jrj,k−(j−1)rj−1,k+z0(j+1)rj+1,k−1+jrj,k−1
= −z0uj,k − uj−1,k −
[fk−luj,l + gk−lrj−1,l + z0gk−lrj,l] ,
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 55
for j, k ≥ 1. When k = 0 and j ≥ 1 we have the specialisation
(6.59) z0(s0 − z0)(j + 1)rj+1,0 + [(s0 − 2z0)j + z0g0]rj,0 + [g0 − j + 1]rj−1,0
= −(f0 + z0)uj,0 − uj−1,0.
The second spectral derivative (5.35) yields the recurrence relation
(6.60)
z0(s0 − z0)(j + 1)uj+1,k + [(s0 − 2z0)j + 2s0 − (a+ 2)z0]uj,k − (a+ j + 1)uj−1,k
+ z0(j + 1)uj+1,k−1 + (j + 2)uj,k−1
z0(s0 − z0)rj,k +
(s0 − 2z0)rj−1,k −
rj−2,k +
z0rj,k−1 +
rj−1,k−1
hk−l(rj−1,l + z0rj,l) +
gk−l(uj−1,l + z0uj,l),
for j, k ≥ 1. When k = 0 and j ≥ 1 we have the specialisation
(6.61)
z0(s0−z0)(j+1)uj+1,0+[(s0−2z0)j+2s0−(a+2)z0−z0g0]uj,0−[a+1+g0+j]uj−1,0
z0(s0 − z0)− h0z0]rj,0 + [
(s0 − 2z0)− h0]rj−1,0 −
rj−2,0.
Using the first deformation derivative (5.36) we obtain the recurrence relation
(6.62)
s0(s0−z0)(k+1)rj,k+1−s0(k+1)rj−1,k+1+(2s0−z0)krj,k−krj−1,k+(k−1)rj,k−1
= s0uj,k + uj,k−1 +
[fk−luj,l + gk−lrj−1,l + z0gk−lrj,l] ,
for j, k ≥ 1. When j = 0 and k ≥ 1 we have the specialisation
(6.63) s0(s0 − z0)(k + 1)r0,k+1 + (2s0 − z0)kr0,k + (k − 1)r0,k−1
= s0u0,k + u0,k−1 +
[fk−lu0,l + z0gk−lr0,l].
The second deformation derivative (5.37) in turn gives us the recurrence relation
(6.64) s0(s0 − z0)(k + 1)uj,k+1 − s0(k + 1)uj−1,k+1
+ [(2s0 − z0)k + as0]uj,k − kuj−1,k + (a+ k − 1)uj,k−1
hk−l(rj−1,l+z0rj,l)−(2s0−z0)
gk−luj,l+
gk−luj−1,l−2
gk−1−luj,l,
56 PETER J. FORRESTER AND NICHOLAS S. WITTE
for j, k ≥ 1. When j = 0 and k ≥ 1 we have the specialisation
(6.65) s0(s0 − z0)(k + 1)u0,k+1 + [(2s0 − z0)k + as0]u0,k + [a+ k − 1]u0,k−1
hk−lr0,l − (2s0 − z0)
gk−lu0,l − 2
gk−1−lu0,l.
These recurrences can be solved in the following way. First (6.59) and (6.61) are
solved for rj,0, uj,0 for j ≥ 1 in terms of r0,0, u0,0. Then these solutions can be
substituted into the boundary conditions
rj,0(−z0)j = 1,(6.66)
uj,0(−z0)j = 0,(6.67)
and this allows for r0,0, u0,0 to be found. Next the sequence r0,k, u0,k can be found
for k ≥ 1 using (6.63) and (6.65). Finally the two general systems (6.58),(6.60) and
(6.62),(6.64) can be employed to compute rj,k, uj,k for j, k ≥ 1.
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 57
6.6. Numerical studies at the Hard Edge. Using the integral formula for the
distribution Aa(z) as given by (5.71) it is possible to compute values of this and
the examples of a = 0, 1, 2 are plotted in Figure 2. However we wish to characterise
it in a precise quantitative way and evaluate the moments of this distribution
(6.68) mk :=
dz zkAa(z), k ∈ Z≥0.
These are easily seen to be
(6.69) mk =
42a+3Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3)
ds sk+2a+3e−s/4eF (s)
du uk+2(1− u)aG(us; s),
where
(6.70) F (s) :=
[ν(v) + 2C(v)],
(6.71) G(z; s) := q∂zp− p∂zq.
We note that by employing the large s asymptotic form of q(z; s), p(z; s) as given in
(6.39,6.40) we can deduce the asymptotic form of the spacing distribution is given
(6.72) Aa(z) ∼
Ce−z/4+(a+2)
zz−1/2−a
where C is a constant which cannot be found from our methods. For small z we
find that
(6.73) Aa(z) ∼
42a+3Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3)
ds s2a
− ξ(s)
[ν(v) + 2C(v)]
where we have used (5.58,5.59) and the fact that u01(s) = 1/12−ξ(s)/3s. Therefore
we can conclude that the moments exist for Re(k) > −3, Re(a) > −1 and Re(k +
2a) > −4. An instance where exact evaluation of the moments can be made is the
case a = 0 and the first four of these are
m1 = 4e
2 [I0(2)− I1(2)] , m2 = 32e2I0(2),(6.74)
m3 = 384e
2 [2I0(2) + I1(2)] , m4 = 2048e
2 [13I0(2) + 9I1(2)] .(6.75)
We investigated the distributions Aa(z) and A
±(z) for the two special cases of
a = ±1/2 in some detail because of the motivations provided by (1.6) and (1.7).
58 PETER J. FORRESTER AND NICHOLAS S. WITTE
The analogue of (6.69) for A±(z) is
(6.76) m±k =
24a+5Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3)
ds sk/2+2a+3e−s/4eF (s)
du uk+2(1 − u)2a+1(2− u)2G(u(2− u)s; s),
for a = ± 1
. The statistical data for Aa(z) for the cases a = −1/2, 0, 1/2, 1, 2 are
given in Table 1 and the data for A±(z) is given in Table 2.
Our strategy is that by employing local Puiseux-type and Taylor expansions for
the two factors in the integrand, namely eF (s) and G(z; s), within a given finite
member of the patchwork of local expansion domains the above integrals restricted
to this domain can be exactly evaluated. This is essential as numerical quadrature
algorithms implemented in either computer algebra software or compiled language
packages (e.g. QUADPACK) have minimal attained error tolerances which cannot
be reduced below a fixed bound. For the compiled language option with a floating
point representation of 64 bits the best one could expect is a relative error of around
10−15 but often it is far worse and around 10−8 − 10−9. To illustrate this we have
computed the statistical data for the a = 1, 2 cases using QUADPACK routines
and the results are displayed in Table 1. In the case of the Puiseux-type expansions
the integrals are
(6.77)
ds sk+3+l+n+(m+2)ae−s/4,
(6.78)
∫ 1/2
du uk+2+n(1− u)a or
du uk+2(1 − u)a+n,
which for k, l,m, n ∈ Z≥0 can be evaluated in terms of radicals, the Gamma func-
tion at integer arguments, a rational function of a and the Whittaker function
M(α, β;S/4) or its specialisations depending on a. For example in the case a = 1/2
the s-integral reduces to the error and exponential functions. For the Taylor ex-
pansion case we have the double integral
(6.79)
ds sk+3+2a(s− s0)me−s/4
du uk+2(1− u)a(us− z0)l,
which for k, l,m ∈ Z≥0 and s0 ∈ (s1, s2) is evaluated in terms of a rational function
of a, a polynomial function of s0, z0 and the Whittaker functions M(α, β; s1,2/4).
Therefore the only sources of error are from the truncation of the expansions and the
finite number of intervals, both of which can be adjusted to reduce the contributing
errors.
The computations were performed, in most cases, using the computer algebra
system Maple with a sufficiently large number of decimal digits and found that
250 digits was more than adequate. In addition we found that we had to tailor
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 59
the numerical parameters for each case of a = ±1/2 differently as the errors varied
quite strongly with a (this was especially pronounced as a approached −1). We
discuss the case of a = 1/2 first. In regard to the truncations about the singular
point s = 0, we found that 1890 terms in the expansion of the transcendent vari-
able (6.1) with k ≤ 30, j + k ≤ 120 yielded an error for the second derivative of
ν(s) at s = 2 which was estimated to be 4.6 × 10−98. At most 100 terms were re-
tained in each of the expansions of the transcendent variables about regular points
(6.41),(6.46),(6.47) and (6.48) because much fewer, of the order of j, k ≤ 20, were
required in the corresponding expansions of the linear variables. The number of
intervals in the s-direction was taken to be 19 with the sequence of s0 values being
{0, 2, 5/2, 3, 4, 6, 9, 13, 19, 25, 30, 38, 54, 72, 90, 115, 150, 200, 300, 500}. The bound-
aries of the s-interval with node s0 were taken to be located at the midpoints
of s0 and its neighbouring nodes. This sequence of nodes was chosen to be close to
an optimal situation yielding the largest separation of each node from its preceding
node, yet close enough so to ensure that the error in ν′′(s) at the node was less
than 9.8 × 10−37. For each s-interval with node s0 two expansion points in the
z-direction were chosen because a single expansion point could never ensure that
all of the integration region would fall within the domain of convergence about
that point. The two points that together yielded the largest convergence domain
were found to be located at z0 = 0 and z0 = s0. Subdividing the z-interval into
three sub-intervals was found to contribute a variation of less than 3.2 × 10−19
to the normalisation. Another criteria that the sequence of s0 nodes had to sat-
isfy was that each (s, z) integration region fell completely within the union of the
convergence domains about (s0, 0) and (s0, s0). For the expansions of the linear
variables about the lines z = 0, s and about the singular point s = 0, as defined
in (6.18,6.19) along with (5.58,5.59,5.63,5.64), we chose the cut-off in the sum to
be 20. The expansions of linear variables about the lines z = 0, s and about a
regular point s = s0 > 0 as defined in (6.56,6.57) were cutoff at 25. An overall es-
timate of the accuracy is provided by the normalisation, which was unity to within
1.6×10−18. The second case with a = −1/2 was more demanding computationally.
We needed 5150 terms in the expansion of the transcendent variable (6.1) with
k ≤ 50, j + k ≤ 200 and computed these with compiled code using the multiple-
precision library MPFUN [3],[2]. The error for ν′′(s) at s = 2 was estimated to be
around 7.9×10−59. Again only 100 terms were retained in each of the expansions of
the transcendent variables about regular points. A larger number of intervals in the
s-direction were employed, namely 24, and the sequence of s0 values was taken to be
{0, 2, 5/2, 3, 7/2, 4, 5, 6, 7, 9, 11, 14, 18, 22, 28, 36, 46, 58, 72, 90, 114, 144, 180, 220, 300}.
This time the sequence of nodes was chosen to ensure that the error in ν′′(s) at
each node was less than 3.6×10−60. And again for each s-interval with node s0 two
expansion points in the z-direction were chosen at z0 = 0, s0. In the expansions of
60 PETER J. FORRESTER AND NICHOLAS S. WITTE
the linear variables about the lines z = 0, s and about the singular point s = 0 the
cut-off in the sum was chosen to be 20 as before. The expansions of these variables
about the lines z = 0, s and about a regular point s = s0 > 0 was terminated at
the cutoff of 25 also. The estimate of the accuracy provided by the normalisation
was 4.6× 10−20.
Because the raw moments grow rapidly with order we have computed some
standard statistical quantities instead using the definitions of the variance σ2, the
skewness γ1 and the kurtosis excess γ2
(6.80) σ2 = µ2, γ1 =
, γ2 =
in terms of the central moments
µ2 = m2 −m21,(6.81)
µ3 = m3 − 3m1m2 + 2m31,(6.82)
µ4 = m4 − 4m1m3 + 6m21m2 − 3m41.(6.83)
DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 61
Figure 2. The distribution of the first eigenvalue spacing at the
hard edge of random hermitian matrices Aa(z) for integral values
of a = 0, 1, 2.
Acknowledgements
This work was supported by the Australian Research Council.
62 PETER J. FORRESTER AND NICHOLAS S. WITTE
Table 1. Low order statistics of the distribution Aa(z) for various
values of the parameter a.
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Department of Mathematics and Statistics, University of Melbourne,Victoria 3010,
Australia
E-mail address: p.forrester@ms.unimelb.edu.au
E-mail address: n.witte@ms.unimelb.edu.au
	1. Introduction
	2. Orthogonal Polynomial System
	2.1. Semi-classical Orthogonal Polynomials
	2.2. Deformed Laguerre Orthogonal Polynomials
	3. Difference and Differential Equations
	3.1. Difference Equations
	3.2. Reduction to Painlevé V
	3.3. Inequalities and Bounds
	4. Special Case  a Z0 
	5. Hard Edge Scaling
	5.1. General Case
	5.2. Special Case  a Z0 
	6. Analytical Studies at the Hard Edge
	6.1. The -function expansion about  s=0 
	6.2. The  q, p  expansion about  s=0 
	6.3. The -function expansion about  s=
	6.4. The -function expansion about a regular point
	6.5. The  q, p  expansion about a regular point
	6.6. Numerical studies at the Hard Edge
	Acknowledgements
	References
