The lowest eigenvalue of Jacobi random matrix ensembles and Painlev\'e VI

Abstract

We present two complementary methods, each applicable in a different range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble. The first method solves an associated Painleve VI nonlinear differential equation numerically, with suitable initial conditions that we determine. The second method proceeds via constructing the power-series expansion of the Painleve VI function. Our results are applied in a forthcoming paper in which we model the distribution of the first zero above the central point of elliptic curve L-function families of finite conductor and of conjecturally orthogonal symmetry. Comment: 30 pages, 2 figures

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arXiv:1005.1298v2 [math.CA] 2 Sep 2010
THE LOWEST EIGENVALUE OF JACOBI RANDOM MATRIX
ENSEMBLES AND PAINLEV´E VI
EDUARDO DUE˜NEZ, DUC KHIEM HUYNH, JON P. KEATING, STEVEN J. MILLER,
AND NINA C. SNAITH
Abstract. We present two complementary methods, each applicable in a different
range, to evaluate the distribution of the lowest eigenvalue of random matrices in a
Jacobi ensemble. The first method solves an associated Painlev´ e VI nonlinear differ-
ential equation numerically, with suitable initial conditions that we determine. The
second method proceeds via constructing the power-series expansion of the Painlev´ e VI
function. Our results are applied in a forthcoming paper in which we model the dis-
tribution of the first zero above the central point of elliptic curve L-function families
of finite conductor and of conjecturally orthogonal symmetry.
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Appendix A.
Appendix B.
References
Introduction
Jacobi ensembles and their first eigenvalue
First method: The auxiliary Hamiltonian and Painlev´ e VI
Taylor series expansion for E(α,β)
N
First method: Initial conditions for the Painlev´ e equation
Second method: Symbolic solution using power series
Comparison of the two methods
Summary
MATLAB Code
SAGE Code
1
3
5
7(φ)
17
18
21
23
24
27
29
1. Introduction
We present techniques for calculating numerically the distribution of the lowest eigen-
value (or synonymously, we say the ‘first eigenvalue’) of random matrices in Jacobi en-
sembles J(a,b)
N
of N × N random matrices. We relate the distribution of the lowest eigenvalue of ma-
trices in JNto the probability E(α,β)
N
(0;I) that a Jacobi ensemble has no levels in some
. We proceed as follows. We introduce the Jacobi ensemble JN = J(a,b)
N
Date: September 3, 2010.
2010 Mathematics Subject Classification. 34M55 (primary), 15B52, 33C45, 65F15 (secondary).
Key words and phrases. Jacobi ensembles, Painlev´ e VI, Selberg-Aomoto’s integral.
The second-named author was partially supported by EPSRC, a CRM postdoctoral fellowship and
NSF grant DMS-0757627. The fourth-named author was partially supported by NSF grants DMS-
0855257 and DMS-097006. The final author was supported by funding from EPSRC. The first, second
and fourth authors thank the University of Bristol for its hospitality and support during the preparation
of this manuscript.
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2 DUE˜NEZ, HUYNH, KEATING, MILLER, AND SNAITH
interval I = [t,1] for 0 ≤ t ≤ 1. We use two complementary methods to evaluate E(α,β)
relying on its interpretation as the Okamoto τ-function of a Painlev´ e VI system, along
with an auxiliary Hamiltonian function h(t) for which Forrester and Witte [5] have es-
tablished explicit differential equations of Painlev´ e VI type. Our first method uses the
Selberg-Aomoto integral to obtain explicit initial conditions for the Painlev´ e IV equa-
tion satisfied by h(t) which are valid close to the edge t = 1. With these in hand we
provide the MATLAB code (relying on its built-in ordinary differential equation solver)
to numerically evaluate h(t) alongside E(α,β)
N
equation with explicit initial conditions at the edge t = 0 together with power series
manipulations to recursively find the power series expansions of h(t) and E(α,β)
plement this algorithm on SAGE using its ability to perform power series manipulations
and symbolic algebra.
The use of these two complementary methods is essential in order to compute h(t)
and E(α,β)
N
(t) accurately over the whole range 0 ≤ t ≤ 1. The Painlev´ e VI equation and
its solutions have singularities at the edges t = 0 and t = 1. The first method uses a
solution found starting from an explicit initial condition at a point t0= 1−ε close to 1,
where ε > 0 is a small positive parameter we determine empirically. Such an explicit
initial condition is found in Sections 4 and 5; however, the initial condition is correct
only up to terms of size O(ε2). The errors introduced by such approximation and by the
numerical Runge-Kutta method result in a computed solution whose range of reliability
may not extend to t close to the singularity at t = 0. The second method, described in
Section 6, constructs a truncated but otherwise exact power series for h(t) about t = 0
(the singularity at t = 0 is handled indirectly) up to terms of order O(tD+1) where D
is the degree of the truncation. Such a solution is reliable over any interval [0,u] with
u < 1, provided D is large enough, though not necessarily over the entire interval [0,1]
in view of the singularity at t = 1. In Section 7 we analyze the range of parameters
a,b,N for which both methods are stable in the sense that both numerically computed
solutions agree in some subinterval [u,v] of (0,1), which implies that the numerical solver
is robust for this range of parameters.
It is important to note that the methods we use apply to non-integer values of N.
Painlev´ e differential equations have played a role in many problems in random matrix
theory, ranging from the distribution of the eigenvalues in the bulk to the largest and
smallest eigenvalues, and have been extensively studied; for our purposes, the most
relevant are the investigations of solutions to Painlev´ e VI. We briefly mention some of
the literature. We refer the reader to the special edition of the Journal of Physics A
(Volume 39, Number 39, 2006), which celebrates 100 years of Painlev´ e VI, especially
the historical introduction and survey [14] and the article by Forrester and Witte [6] on
connections with random matrix theory; see also the recent works by Dai and Zhang [2]
and Chen and Zhang [1] for determinantal formulas obtained from ladder operators.
The main contribution of this paper is the derivation of an algorithm to compute
numerically the distribution of the lowest eigenvalue in the Jacobi ensembles, and a
discussion of its implementation and accuracy. The motivation for this project comes
from attempts to understand the observations in [13] on the distribution of the first zero
above the central point in families of elliptic curve L-functions when the conductors
are small. The Katz-Sarnak conjectures [8, 9] predict that as the conductors of the
elliptic curves tend to infinity their zero statistics should agree with the N → ∞ scaling
limits of the corresponding statistics of the eigenvalues of matrices from a classical
compact group. For suitable test functions this was proved in [12,17]; however, for finite
N
. The second method uses the Painlev´ e VI
N
. We im-

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THE LOWEST EIGENVALUE IN JACOBI ENSEMBLES AND PAINLEV´E VI3
conductors the numerical data in [13] is in sharp disagreement with the limiting behavior
of these random matrix ensembles. In particular, the first zero above the central point
is repelled, with the repulsion decreasing as the conductors increase. In a forthcoming
paper we complete the study of the low lying zeros of elliptic curve L-functions, and
obtain a model which describes the behavior of these zeros for finite conductors. One
of the key ingredients in our model is the lowest eigenvalue of these Jacobi ensembles
of N × N matrices, often requiring non-integer values of N, which is the main result of
this paper.
2. Jacobi ensembles and their first eigenvalue
Let JN= J(a,b)
N
denote the Jacobi ensemble on N levels 0 ≤ xj≤ 1, j = 1,2,...,N,
with real parameters a,b > −1. Explicitly, the N-level (joint) probability density of
levels of JNon [0,1]Nwith respect to its Lebesgue measure dx1dx2···dxNis given by
(2.1)
?C(a,b)
N
N
?
j=1
W(xj)
?
1≤j<k≤N
(xk− xj)2
where the weight function W = W(a,b)on [0,1] is given by
(2.2)W(x) = xb(1 − x)a
and?C(a,b)
variables φjdefined by
N
is the ensemble’s normalization constant.
above correspond to suitable ensembles of self-dual random matrices via the angular
Jacobi ensembles as described
(2.3)xj=1 + cosφj
2
,0 ≤ φj≤ π.
Note that the edges x = 0, x = 1 correspond respectively to φ = π, φ = 0 under this
change of variables. We refer the reader to [3] and the forthcoming book [7] for details
regarding the matrix realizations of Jacobi ensembles, for which we will otherwise have
no direct use. In what follows we will go back and forth between the abscissæ xjand
the angular variables φj, but will in any case refer to the associated ensemble by the
Jacobi name and denote it by JN. In terms of the angular variables, and with respect
to Lebesgue measure on (0,π)N, the N-level (joint) probability density for JN is given
by
(2.4)C(α,β)
N
N
?
j=1
w(cosφj)
?
1≤j<k≤N
(cosφk− cosφj)2
with the weight function w = w(α,β)on (0,π),
(2.5)w(cosφ) = (1 − cosφ)α(1 + cosφ)β.
The parameters α,β > −1
and C(α,β)
N
?C(α−1/2,β−1/2)
of matrices SO(2N), SO(2N + 1) and USp(2N), when the latter are endowed with an
invariant (Haar) probability measure and regarded as random matrix ensembles. The
case α = β = 0 corresponds to SO(2N), α = 1 and β = 0 corresponds to SO(2N + 1),
and α = β = 1 to USp(2N). This is explained in detail in [3]. Below in (4.23) we give an
2are related to a,b above by α = a + 1/2, β = b + 1/2,
is the appropriate normalization constant, namely C(α,β)
for the constant?C(a,b)
N
= 2−(N+a+b+2)N
NN
of (2.1). For suitable choices for α and β we obtain
the joint probability density of the N independent eigenphases for the classical groups

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4 DUE˜NEZ, HUYNH, KEATING, MILLER, AND SNAITH
explicit expression for the normalization constant?C(a,b)
only the Jacobi joint probability density via for instance the Accept-Reject Algorithm
whose applicability is quite broad; see for instance [15].)
As remarked above, the Jacobi ensemble JN describes the eigenvalue statistics in
suitable ensembles of self-dual random matrices having N pairs of eigenvalues e±iφj,
j = 1,2,...,N; we call φ the eigenphase of the eigenvalue eiφ. Let E(α,β)
the probability that a random matrix A ∈ JN has exactly n eigenphases in the inter-
val I = [0,φ]. As shorthand notation we write E(α,β)
the probability of having no eigenphases in the interval [0,φ]. The probability density
function ν(α,β)
N
N
. (Jacobi-distributed pseudoran-
dom sequences of levels can be generated from a uniform pseudorandom sequence using
N
(n;I) denote
N
(φ) for E(α,β)
N
(0;[0,φ]), namely
(φ) of the distribution of the first eigenphase is related to E(α,β)
N
(φ) by
(2.6)ν(α,β)
N
(φ) = −d
dφE(α,β)
N
(φ).
We can deduce the relation (2.6) as follows: assume that the interval [0,φ] contains no
eigenvalues. Then a small increment ε > 0 of the interval to [0,φ + ε] has two possible
outcomes. Either the interval [0,φ+ε] contains no eigenvalues, or it contains some. The
probability of the first event is E(α,β)
N
is the probability that the interval [φ,φ + ε] contains at least one eigenvalue; as ε → 0
there can be only one, namely the first eigenphase in [φ,φ + ε]. Thus
(φ + ε). It follows that E(α,β)
N
(φ) − E(α,β)
N
(φ + ε)
(2.7) lim
ε→0
E(α,β)
N
(φ) − E(α,β)
N
(φ + ε)
ε
= −d
dφE(α,β)
N
(φ)
indeed yields the probability density function ν(α,β)
alternative way to prove (2.6) is to observe that 1 − E(α,β)
[0,φ] contains at least one eigenphase, hence that the first eigenphase φminis at most φ;
otherwise said 1−E(α,β)
N
(φ) is the cumulative distribution function of the first eigenphase
φmin, so its derivative is equal to the probability density function ν(α,β)
we shall need to scale the angular variable φ by a factor of N/π in order to consider
eigenphases of mean unit spacing on [0,N].
We have
N
(φ) of the first eigenphase.
(φ) is the probability that
An
N
N
(φ). In Section 5
E(α,β)
N
(φ) = C(α,β)
N
?π
?
φ
···
?π
φ
N
?
j=1
(1 − cosφj)α(1 + cosφj)β
×
1≤j<k≤N
(cosφj− cosφk)2dφ1···dφN
(2.8)
for fixed α,β > −1/2 and the normalization constant C(α,β)
method to evaluate the multiple integral in equation (2.8) exactly. E(α,β)
to a Painlev´ e VI transcendental function h(t), namely a certain solution to a second-
order nonlinear ordinary differential equation. In Proposition 4.4 we provide the first
few terms of a power-series expansion of E(α,β)
initial conditions for the differential equation we aim to solve. Our main reference for
the theory is the work of Forrester and Witte [5]. Their result is stated for the abscissal
N
of (2.4). There is no known
N
(φ) is related
N
(φ) for φ close to 0; these provide the

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THE LOWEST EIGENVALUE IN JACOBI ENSEMBLES AND PAINLEV´E VI5
counterpart to the function E(α,β)
N
(φ) of (2.8), namely the function?E(a,b)
xb
N
(t) defined by
(2.9)
?E(a,b)
N
(t) =?C(a,b)
N
?t
0
···
?t
0
N
?
of (2.1). The functions (2.9) and (2.8) are related
j=1
j(1 − xj)a
?
1≤j<k≤N
(xj− xk)2dx1···dxN
with the normalization constant?C(a,b)
(2.10)
N
by the change of variables
a = α − 1/2 andb = β − 1/2
along with
(2.11)t =1 + cosφ
2
andxj=1 + cosφj
2
where 0 ≤ t,xj≤ 1. Explicitly,
(2.12)E(α,β)
N
(φ) =? E(a,b)
N
?1 + cosφ
2
?
.
3. First method: The auxiliary Hamiltonian and Painlev´ e VI
Both of our mutually complementary methods rely on the interpretation of?E(a,b)
auxiliary Hamiltonian h(t); this Hamiltonian arises as the solution of a Painlev´ e VI
equation with the exact parameters determined by Forrester and Witte in Proposition
13 of [5] as follows.
N
as an Okamoto τ-function and ensuing relation to a Painlev´ e system with associated
Proposition 3.1. Let a,b > −1 and N be a positive integer. The auxiliary Hamiltonian
(3.1)h(t) = t · e′
2[b] −1
2e2[b] + t(t − 1)d
dtlog?E(a,b)
,−a + b
2
N
(t)
where
b = (b1,b2,b3,b4) =
?
N +a + b
2
,a − b
2
,−N −a + b
2
?
, (3.2)
e2[b] = b1b2+ b1b3+ b1b4+ b2b3+ b2b4+ b3b4,
e′
2[b] = b1b3+ b1b4+ b3b4,
(3.3)
(3.4)
satisfies the following Painlev´ e VI equation in Jimbo-Miwa-Okamoto σ-form
(3.5)h′(t)?t(1 − t)h′′(t)?2+?h′(t)[2h − (2t − 1)h′(t)] + b1b2b3b4
Furthermore, we have the boundary condition (as t → 0)
?
Note that besides simplifying the notation in Proposition 13 of [5] we also swap the
a’s and b’s therein. The parameter a is equal to the order of vanishing of the Jacobi level
density at the edge t = 1, whereas a+1/2 is the order of vanishing of eigenphase density
at the edge φ = 0. We remark that the apostrophe in the symbol e′
meaning and is merely used to visually distinguish it from e2(in a manner consistent
with the notation of reference [5]), whereas the apostrophe in h′and elsewhere in this
manuscript means differentiation: h′(t) =dh
?2=
4?
k=1
(h′(t) + b2
k).
(3.6)h(t) =
−1
2e2[b] − N(b + N)
?
+
?
e′
2[b] +N(N + b)(2N + a + b)
2N + b
?
t + O(t2).
2has no specific
dt, h′′(t) =d2h
dt2,?E(a,b)
N
′(t) =d
dt?E(a,b)
N
(t), etc.