arXiv:1701.07402v2 [math-ph] 29 Jul 2017

Smallest eigenvalue density for regular or
fixed-trace complex Wishart-Laguerre ensemble and
entanglement in coupled kicked tops
Santosh Kumar & Bharath Sambasivam & Shashank Anand
Department of Physics, Shiv Nadar University, Gautam Buddha Nagar, Uttar
Pradesh – 201314, India
E-mail: skumar.physics@gmail.com

Abstract. The statistical behaviour of the smallest eigenvalue has important
implications for systems which can be modeled using a Wishart-Laguerre ensemble, the
regular one or the fixed trace one. For example, the density of the smallest eigenvalue of
the Wishart-Laguerre ensemble plays a crucial role in characterizing multiple channel
telecommunication systems. Similarly, in the quantum entanglement problem, the
smallest eigenvalue of the fixed trace ensemble carries information regarding the nature
of entanglement.
For real Wishart-Laguerre matrices, there exists an elegant recurrence scheme
suggested by Edelman to directly obtain the exact expression for the smallest
eigenvalue density. In the case of complex Wishart-Laguerre matrices, for finding exact
and explicit expressions for the smallest eigenvalue density, existing results based on
determinants become impractical when the determinants involve large-size matrices.
In this work, we derive a recurrence scheme for the complex case which is analogous to
that of Edelman’s for the real case. This is used to obtain exact results for the smallest
eigenvalue density for both the regular, and the fixed trace complex Wishart-Laguerre
ensembles. We validate our analytical results using Monte Carlo simulations. We also
study scaled Wishart-Laguerre ensemble and investigate its efficacy in approximating
the fixed-trace ensemble. Eventually, we apply our result for the fixed-trace ensemble
to investigate the behaviour of the smallest eigenvalue in the paradigmatic system of
coupled kicked tops.

1. Introduction
Wishart-Laguerre ensembles constitute an important class of random matrix
ensembles [1, 2] and have found diverse applications in the field of multivariate
statistics [3–5], problems related to time series [6–8], low energy quantum
chromodynamics [9, 10], multiple-channel telecommunication [11–13], quantum
entanglement [14–20], etc. Often the smallest eigenvalue distribution plays a crucial role
in investigating the behaviour of the system under study. For instance, in the context
of multiple input multiple output (MIMO) communication, the smallest eigenvalue of
Wishart-Laguerre ensemble determines the minimum distance between the received

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

2

vectors [21], and also the lower bound on the channel capacity that eventually is used
in antenna selection techniques [22]. Similarly, the smallest eigenvalue density of fixed
trace Wishart-Laguerre ensemble serves as an important metric for characterizing the
entanglement in bipartite systems [23, 24].
Matrices governed by Wishart distribution are parametrized by their size (n) and
the degree of freedom (m) [3–5]; see section 2. In this paper we use the term regular
to mean unrestricted trace Wishart matrices with m ≥ n. The smallest eigenvalue
of Wishart matrices have been studied extensively, both for regular, and fixed trace
scenarios. Moreover, finite-dimension, as well as large-dimension asymptotic cases have
been explored. Our focus here is on the finite-dimensional scenario with the primary
objective to obtain explicit expressions for the smallest eigenvalue density.
In the case of regular Wishart-Laguerre ensemble, for real matrices at finite
n, m, Edelman, among other things, has provided a recursion-based scheme to obtain
the smallest eigenvalue density [25, 26]. For complex matrices, the result for the
cumulative distribution of the smallest eigenvalue goes back to Khatri, who worked
out a determinant-based expression [27]. Forrester and Hughes have given closed
expressions for the density of the smallest and second-smallest eigenvalues in terms
of Wronskian and Toeplitz determinants [28]. Further generalizations applicable to a
wider symmetry class of Wishart matrices have been considered in [29–31]. Damgaard
and Nishigaki have derived the probability distribution of the kth smallest eigenvalue of
Dirac operator in the microscopic scaling limit for real, complex as well as quaternion
cases and demonstrated the universality of the results [32]. These eigenvalues have
direct relationship with those of the Wishart-Laguerre ensemble. In [33] Akemann et
al. have further explored the smallest eigenvalue distribution of real Wishart-Laguerre
matrices and validated the universality in the microscopic limit for the correlated case
also. Moreover, in a very recent work by Edelman, Guionnet, and Péché [34], the
behaviour of the smallest eigenvalue coming from finite random matrices (including
Wishart) based on non-Gaussian entries has been investigated.
For the fixed trace case, Chen, Liu and Zhou [35] have derived the smallest
eigenvalue density in terms of sum of Jack polynomials. Moreover, for the complex case
this expression has been simplified to inverse Laplace transform of a certain determinant.
In [36], for the real Wishart matrices, Edelman’s recursive approach for the regular
Wishart-Laguerre ensemble has been used by Akemann and Vivo to obtain the smallest
eigenvalue density for the fixed trace Wishart-Laguerre ensemble.
For the complex case, the exact result for the smallest eigenvalue density is
available in terms of determinants involving n-dimensional [21, 27, 37] or α-dimensional
matrices [28, 35], where α = m − n is the rectangularity. These results have been used
for asymptotic analysis when n → ∞ for α fixed and these correspond to eigenvalue
distributions comprising Bessel kernel [28,32,38]. On the other hand, if both n, α → ∞,
an analysis involving Fredholm determinant with Airy kernel is possible and that leads
to the celebrated Tracy-Widom (TW) distribution [39–41]. In [42] the transition regime
between the Bessel and Airy densities has also been investigated. While these asymptotic

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

3

results give a wealth of information regarding the universal aspects, if one desires
explicit expressions for the smallest eigenvalue density for large but finite n, α then
these determinant based results turn out to be impractical, even with the availability
of modern computational packages. We should remark that the smallest eigenvalue
density has also been worked out for correlated Wishart-Laguerre ensembles, both for
real and complex cases [37, 43, 44]. However, the exact results are, again, in terms of
determinants or Pfaffians.
The iterative scheme provided by Edelman [25, 26] is quite an effective and
convenient way to calculate the density for the case of real matrices, and one can handle
large values of dimensionality n and rectangularity α. For the complex Wishart-Laguerre
ensemble, Forrester and Hughes have worked out an iterative scheme for the cumulative
distribution of the smallest eigenvalue. However, to the best of our knowledge, an
iterative scheme analogous to that of Edelman’s, for direct evaluation of the probability
density of the smallest eigenvalue has hitherto remained unavailable. In the present
work, we derive the recurrence scheme that applies to the complex Wishart-Laguerre
ensemble. These results involve an ‘exponential times polynomial’ structure. Since the
fixed trace ensemble is related to the regular Wishart-Laguerre ensemble via a Laplace
transform, the structure of the smallest eigenvalue density in the latter leads to a very
convenient calculation of density in the former case as well [36]. Moreover, arbitrary
moments of the smallest eigenvalue are also readily obtained. In addition, for the regular
Wishart-Laguerre ensemble we also indicate a relation between the recurrence relation
and the determinantal results of Forrester and Hughes [28], and explicitly demonstrate
the equivalence between the two results for rectangularity α = 0, 1. Similarly, for
the fixed-trace scenario we prove the equivalence of the recursion-based expression and
the result of Chen, Liu and Zhou [35] based on the inverse Laplace transform of a
determinant, again for α = 0, 1.
Finally, we use the smallest eigenvalue density of the fixed trace ensemble to study
entanglement formation in the paradigmatic system of coupled kicked tops. We should
note that although the Floquet operator involved in this system belongs to the circular
orthogonal ensemble (COE) [1,2], the dynamically generated states are not random real
vectors in the Schmidt basis [45]. Rather, they are complex, and hence, the results for
the complex fixed trace Wishart-Laguerre ensemble are applicable.
2. Wishart-Laguerre ensemble
Consider complex matrices A of dimensions n × m taken from the density PA (A) ∝

exp − tr AA† . We assume without loss of generality that n ≤ m. Then, the
non-negative definite matrices W = AA† constitute the (regular) Wishart-Laguerre
ensemble with the probability density
PW (W) ∝ (det W)m−n exp (− tr W) .

(1)

The joint probability density of unordered eigenvalues (λj ≥ 0, j = 1, ..., n) of W is

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

4

given by [1, 2] ‡
P (λ1 , ..., λn ) =

Cn,α ∆2n ({λ})

n
Y

λαj e−λj ,

(2)

j=1

with α = m − n, and
−1
Cn,α

=

n
Y

Γ(j + 1)Γ(j + α).

(3)

j=1

Q

Also, ∆n ({λ}) = 1≤k<j≤n (λj − λk ) is the Vandermonde determinant. For this classical
ensemble, exact expression for correlation functions of all orders are known [1, 2]. For
example, the first order marginal density (one-level density),
Z ∞
Z ∞
p(λ) =
dλ2 · · ·
dλn P (λ, λ2 , ..., λn ),
(4)
0

0

is given by
n−1
1 −λ α X Γ(j + 1)  (α) 2
Lj (λ)
p(λ) = e λ
n
Γ(j + α + 1)
j=0

=

Γ(n) −λ α (α)
(α+1)
e λ [Ln−1 (λ)L(α+1)
(λ) − L(α)
n
n (λ)Ln−1 (λ)].
Γ(m)

(5)

(γ)

Here Li (λ) represents the associated Laguerre polynomials [46].
We now focus on the statistics of the smallest eigenvalue of W. The probability
density for the smallest eigenvalue can be calculated using the joint probability
density (2) as [25, 26, 28]
Z ∞
Z ∞
dλn P (x, λ2 , ..., λn ).
(6)
dλ2 · · ·
f (x) = n
x

x

As shown in Appendix A, this can be brought down to the form
f (x) = cn,m e−nx xα gn,m (x).

(7)

Here the normalization factor cn,m is given by
n−1

cn,m

Y Γ(i + 2)
1
=
,
Γ(n)Γ(m) i=1 Γ(i + α)

(8)

and gn,m (x) is obtained using the following recurrence scheme:
I. Set S0 = gn,m−1 (x), S−1 = 0
II. Iterate the following for i = 1 to n − 1 :
x dSi−1
Si = (x + m − i + 1)Si−1 −
n − i dx
m−i
+ x (i − 1)
Si−2
n−i
III. Obtain gn,m (x) = Sn−1
‡ We should note that m × m–dimensional matrices A† A share the eigenvalues λ1 , ..., λn of AA† . The
other m − n eigenvalues are all zeros. The joint density in this case can be written by introducing deltafunctions for these zero-eigenvalues and implementing proper symmetrization among all eigenvalues.

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

5

The initial case (m = n) is given by gn,n (x) = 1. Thus, starting from the square case,
for which the result is simple (f (x) = ne−nx ), we can go up to any desired rectangularity
by repeated application of the above recurrence scheme. We note that (7) is of the form
f (x) =

αn+1
X

hj xj−1 e−nx ,

(9)

j=α+1

where hj are n, m dependent rational coefficients. The lower and upper limits of the
summation in (9) are α + 1 and αn + 1, respectively. This is because the recurrence
scheme applied for rectangularity α gives gn,m (x) as a polynomial of degree α(n − 1),
and there is the factor xα in (7). The coefficients hj can be extracted once the above
recursion has been applied.
The above simple structure for the probability density gives easy access to the η–th
moment of the smallest eigenvalue of the Wishart-Laguerre ensemble. We obtain
Z ∞
αn+1
X hj
η
η
Γ(j + η).
(10)
hx i =
x f (x) dx =
nj+η
0
j=α+1
We would like to remark that this relationship holds not only for non-negative integer
values of η, but also for any complex η such that Re(η) > −α − 1.
In Appendix B we provide simple Mathematica [47] codes that produce exact results
for the density as well as the η–th moment for any desired value of n, m by implementing
the above results.
In figure 1 we consider the smallest eigenvalue density and compare the analytical
results with Monte Carlo simulations using 105 matrices for n = 8, 15, and several α
values. We find excellent agreement in all cases.
8

1.4

(a)

1.2

α=0

6
� (�)

α=1
4

α=2
α=3

2

(b)

1.0

α=5

0.8

α = 10

0.6

α = 15

0.4
0.2

0
0.0

0.5

1.0

1.5

2.0

0.0
0
�

1

2

3

4

5

6

7

Probability density of the smallest eigenvalue for the WishartLaguerre ensemble with (a) n = 8, (b) n = 15, and α values as indicated. The
solid lines are analytical predictions based on (7), while the symbols (filledcircles, squares, triangles) represent results of Monte Carlo simulations.
Figure 1.

Smallest eigenvalue density for complex Wishart-Laguerre ensemble
Forrester and Hughes’ result for the smallest eigenvalue density reads [28]
i
Γ(n + 1) −nx α h α+j−k−1 (3−α)
e x det Dt
Lm−2 (t)|t=−x
,
f (x) = (−1)α(α−1)/2
Γ(m)
j,k=1,...,α
where Dt ≡ d/dt. Comparing this result with (7), we find that
h
i
α+j−k−1 (3−α)
α(α−1)/2 Γ(n + 1)
det Dt
Lm−2 (t)|t=−x
.
gn,m (x) = (−1)
Γ(m) cn,m
j,k=1,...,α

6

(11)

(12)

Therefore, the recurrence scheme essentially leads to the evaluation of the above
determinant, which otherwise becomes difficult to evaluate directly for large α values.
Demonstrating the equality of the two sides in (12) directly seems challenging for
arbitrary n, m, if at all feasible. However, as shown below, for α = 1, we find that gn,m (x)
does lead to the associated Laguerre polynomial as evaluated by the determinantal
expression. Before analyzing the results of α = 1, we also consider α = 0, which
corresponds to the square case.
2.1. Results for α = 0
In this case gnm = 1 and in the expression (9), there is just one term in the sum, viz.
j = 1. The corresponding value of the coefficient hj is n. Thus, the smallest eigenvalue
density reads
f (x) = ne−nx .

(13)

Also, the moment-expression is given by
Γ(η + 1)
.
(14)
nη
These expressions agree with those derived in [25,28], as they should. We note that (11)
leads to the correct density, as the determinant part has to be taken as 1 for α = 0.
hxη i =

2.2. Results for α = 1
This is a nontrivial case. As shown in Appendix C, in this case, Si in the
(x). Consequently, gn,n+1 (x) =
recurrence relation can be identified with Γ(i + 1)Ln−i+1
i
(2)
Γ(n)Ln−1 (−x). Also cn,n+1 = 1/Γ(n), which leads to the smallest eigenvalue expression
(2)

f (x) = e−nx xLn−1 (−x).

(15)

This agrees with (11) when evaluated for α = 1. The use of the expansion of the
Laguerre polynomial [46] leads to the coefficient hj in (9) as
hj =

Γ(n + 2)
,
Γ(n − j + 2)Γ(j + 1)Γ(j − 1)

j = 2, 3, ..., n + 1.

(16)

The η–th moment follows as
η

hx i = Γ(n + 2)

n+1
X
j=2

nj+η Γ(n

Γ(j + η)
.
− j + 2)Γ(j + 1)Γ(j − 1)

(17)

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

7

3. Fixed trace Wishart-Laguerre ensemble
Fixed trace ensembles constitute a special class of random matrices and can take care
of system dependent constraints. For the Wishart-Laguerre case, the corresponding
fixed trace ensemble arises naturally in the quantum entanglement problem in bipartite
systems [15, 17, 18]. With the trace value fixed at unity, it models the reduced density
matrix; see section 5. Using the Wishart matrices W from the preceding section, the
fixed trace ensemble can be realized by considering the matrices F = W/ tr W [18, 20].
The corresponding probability density is
PF (F) ∝ (det F)α δ(tr F − 1).

(18)

The joint density of unordered eigenvalues (0 ≤ µj ≤ 1; j = 1, ..., n) of F is obtained
as [15, 17, 18]
!
n
n
X
Y
F
2
PF (µ1 , ..., µn ) = Cn,α δ
µi − 1 ∆n ({µ})
µαj ,
(19)
i=1

j=1

F
Cn,α

= Γ(nm) Cn,α [48]. The corresponding marginal density has been derived
where
in [49] as a single sum over hypergeometric 5 F4 , and as a double sum over polynomials
in [50]. In [48] it has been given as a single sum over the Gauss hypergeometric function
(2 F1 ):
n−1
X
pF (µ) =
Ki µi+α (1 − µ)−i+nm−α−2
×

i=0
1−n,i−nm+α+1
n Fα+1


−n,i−nm+α+1
− (n − i − 1)Fα+1
.

(20)

µ
)/Γ(c). Also, the coefficient Ki is given
Here we used the notation Fca,b := 2 F1 (a, b; c; 1−µ
by
(−1)i Γ(m + 1)Γ(nm)
Ki =
.
(21)
nΓ(i + 1)Γ(n − i)Γ(i + α + 2)Γ(nm − α − i − 1)
Figure 2 shows the comparison between analytical and Monte Carlo results for the
marginal density of the fixed trace ensemble. We find excellent agreement.
Using Selberg’s integral and its properties [1, 2], it can be shown that both the
average and the variance of the trace for the regular Wishart-Laguerre ensemble is mn.
Therefore, if we consider the ensemble of matrices W/mn, the corresponding eigenvalues
are 1/mn times the eigenvalues of W. Moreover, while individually these scaled matrices
may not have trace one, on average, it is one. In addition, the variance of trace for this
scaled ensemble is 1/mn, which becomes negligible for large n, m. Therefore, it is
expected that this scaled ensemble will approximate the behaviour of the fixed trace
ensemble. For instance, the marginal density for this scaled ensemble,

pe(µ) = mn p(mnµ),

(22)

should serve as an approximation to pF (µ). We can also use Marčenko-Pastur
density [51] to write down an expression for pe(µ) valid for large n, m:
p
p
(1 ± n/m)
m (µ+ − µ)(µ − µ− )
peMP (µ) =
; µ± =
.
(23)
2π
µ
n

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

8

Marginal density for fixed trace Wishart-Laguerre ensemble
using (20) with n = 8, and α values as indicated. The solid lines are analytical
predictions based on (20), and the symbols correspond to Monte Carlo results.
Figure 2.

This relation of the fixed trace ensemble with the scaled ensemble has been used
in [19, 52–54]. In figure 3 we plot the exact one-eigenvalue density (20) for the fixed
trace ensemble, as well as the densities (22), (23) based on the scaled ensemble. We
find that while the density obtained using the scaled ensemble is not able to capture the
oscillations, it does capture the overall shape of the density quite well.
25



�� ( μ )� � ( μ )� � ��( μ )

(a)

(b)

25

20
20
15
10
5

20

(c)

15

15
10

10

5

5

0
0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00

0.05

0.10

0.15

0
0.00 0.02 0.04 0.06 0.08 0.10

�

Figure 3. Marginal density of eigenvalues for the fixed trace Wishart-Laguerre

ensemble: Comparison between exact (solid black), scaled (dashed red), scaled
Marčenko-Pastur (dotted blue) as given by (20), (22) and (23), respectively.
(a) n = m = 15, (b) n = 20, m = 30, (c) n = 25, m = 75.

The exact result for the density of the smallest eigenvalue for the fixed trace
ensemble can be obtained using (7), (9), and the Laplace-inversion result
L−1 {s−a e−nsx }(t = 1) = (1 − nx)a−1 Θ(1 − nx)/Γ(a),
with Θ(z) being the Heaviside-theta function. We have
fF (x) = Γ(nm)L−1 {s1−nm f (sx)}(t = 1)

(24)

Smallest eigenvalue density for complex Wishart-Laguerre ensemble
500

9

400

(a)
400

α=0

300
α=5

α=1

300
� � (�)

(b)

α=2
200

α = 10

200

α = 15

α=3
100

100

0
0
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.000
�

0.005

0.010

0.015

0.020

Figure 4. Probability density of the smallest eigenvalue for the fixed trace

Wishart-Laguerre ensemble with (a) n = 8, (b) n = 15, and α values as
indicated. The solid lines are analytical predictions based on (25), while the
symbols (filled- circles, squares, triangles) represent results of Monte Carlo
simulations.

= Γ(nm)

αn+1
X
j=α+1

hj

(1 − nx)nm−j−1 xj−1
Θ(1 − nx).
Γ(nm − j)

(25)

F
/Cn,α . In [36] a
The prefactor Γ(nm) comes from the ratio of normalizations, viz. Cn,α
similar strategy has been used for the real case. In figure 4 we show the comparison
between the analytical prediction and the numerical simulation for the smallest
eigenvalue density. They are in excellent agreement.
The idea of using scaled Wishart-Laguerre ensemble, as discussed above, can be
applied here as well. Therefore, an approximate density for the smallest eigenvalue can
be written using (7) as

fe(x) = mnf (mnx).

(26)

In figure 5 we compare this approximation with the exact result given by (25). The
approximation works pretty well. This approximate relation between the two densities
is also the reason behind the very similar shapes of the curves in figures 1 and 4,
respectively.
We also find that, using the first equality in (25), it follows that the η–th moment
of the smallest eigenvalue for the fixed trace ensemble is related to that of the regular
Wishart-Laguerre ensemble as
hxη iF =

Γ(nm)
hxη i.
Γ(nm + η)

(27)

This, similar to (10), holds for Re(η) > −α − 1.
Mathematica [47] codes to obtain explicit results for the above smallest eigenvalue
density of the fixed trace ensemble, as well as the moments are given in Appendix B.

Smallest eigenvalue density for complex Wishart-Laguerre ensemble
120

(a)


� � (�)� � (�)

100

80

(b)

(c)

400

60

80

300

40

60

500

10

200

40

20

20

100

0
0
0
0.00 0.01 0.02 0.03 0.04 0.05 0.00 0.01 0.02 0.03 0.04 0.05 0.000 0.002 0.004 0.006 0.008 0.010

�

Figure 5. Comparison between the exact (fF (x): solid black) and approximate

(fe(x): dashed red) probability densities for the smallest eigenvalue of the fixed
trace Wishart-Laguerre ensemble, as given by (25) and (26), respectively. The
parameter values used are (a) n = m = 5, (b) n = 8, m = 13, and (c)
n = 20, m = 30.

Similar to the unrestricted trace case, we discuss below the cases α = 0, 1 for the
fixed trace scenario.
3.1. Results for α = 0
For α = 0 we just have one term in the series (25), and h1 = n. Therefore, we arrive at
2 −2

fF (x) = n(n2 − 1)(1 − nx)n

Θ(1 − nx).

(28)

Also, the expression for the η-th moment is given by
Γ(η + 1)Γ(n2 )
hx iF = η
.
n Γ(n2 + η)
η

(29)

These expressions are in agreement with those obtained in [23, 24].
3.2. Results for α = 1
In this case use of the result (16) for hj in (25) leads to the smallest eigenvalue density
expression
2

fF (x) = Γ(n +n)Γ(n+2)

n+1
X
j=2

2

(1 − nx)n +n−j−1 xj−1
Θ(1−nx).(30)
Γ(n − j + 2)Γ(j + 1)Γ(j − 1)Γ(n2 + n − j)

Also, the η–th moment follows as
n+1

Γ(n2 + n)Γ(n + 2) X
Γ(j + η)
hx iF =
.
2
j+η
Γ(n + n + η) j=2 n Γ(n − j + 2)Γ(j + 1)Γ(j − 1)
η

(31)

Chen, Liu and Zhou have provided the result for the cumulative distribution § of
the smallest eigenvalue for the complex case in terms of an inverse-Laplace transform
§ More appropriately, the survival function or the reliability function.

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

11

involving a determinant [35]:
n
o  1 − nx 
1
(k)
mn−1 −1
−mn
Q(x) = Γ(mn)x
L
s
det[Ln+j−k (−s)]j,k=0,...,α−1
; 0 < x ≤ .(32)
x
n
We set α = 1 in this expression and use the expansion for associated Laguerre
polynomial [46], later on. The inverse Laplace transform can then be explicitly
performed leading us to
2
n
X
xj (1 − nx)n +n−j−1
2
.
(33)
Q(x) = Γ(n + 1)Γ(n + n)
Γ2 (j + 1)Γ(n − j + 1)Γ(n2 + n − j)
j=0
The probability density of the smallest eigenvalue follows upon using fF (x) =
−dQ(x)/dx as
2
n
X
nxj (1 − nx)n +n−j−2
2
fF (x) = Γ(n + 1)Γ(n + n)
Γ2 (j + 1)Γ(n − j + 1)Γ(n2 + n − j − 1)
j=0
2

−Γ(n + 1)Γ(n + n)

n+1
X
j=1

2

xj−1 (1 − nx)n +n−j−1
.
Γ(j)Γ(j + 1)Γ(n − j + 1)Γ(n2 + n − j)

(34)

In the second term we start the sum from j = 1 as j = 0 term is zero because of the
diverging gamma function Γ(j) in the denominator. Moreover, we have added a term
j = n + 1 which, again, is zero because of the diverging Γ(n − j + 1) in the denominator.
Next, we consider j → j − 1 in the summand of the first term, and hence the sum runs
from j = 1 to n + 1. The two terms can then be combined to yield (30) by noticing that
the n = 1 term is zero. We note that (32) also produces the correct result for α = 0 if
the determinant value in this case is interpreted as 1.
4. Large n, α evaluations and comparison with Tracy-Widom density
The universality aspects of the regular Wishart-Laguerre ensemble have been explored in
several notable works [29–34,38–41,55–60]. For the fixed trace case, the local statistical
properties of the eigenvalues have been studied in [35, 61]. In particular, it has been
shown that the fixed trace and the regular Wishart-Laguerre ensembles share identical
universal behaviour for large n at the hard edge, in the bulk and at the soft edge for α
fixed [61].
For the complex Wishart-Laguerre ensemble, in the square case (α = 0), the
smallest eigenvalue scaled by n gives rise to an exponential density [23, 25, 26].
Interestingly, this is true for all n in this case, as evident from (13). For large n it
has been shown that this result holds even if the matrices W are constructed from nonGaussian A [55] (cf. section 2) with certain considerations. For the fixed trace case,
in view of its connection to the scaled Wishart-Laguerre ensemble (25), as discussed
in section 3, the smallest eigenvalue has to be scaled by n3 to obtain the exponential
density [23, 24]. This can be easily verified to be true from (28). Furthermore, very
recently, 1/n corrections to the scaled smallest eigenvalue has been worked our for close
to square cases [34, 59, 60].

Smallest eigenvalue density for complex Wishart-Laguerre ensemble
0.5

0.5

0.5

����������� �������

(a)

(b)

(c)

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.0
-6

-4

-2

0

2

0.5

0.0
-6

-4

-2

0

2

0.5

0.0
-6

(d)

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0

0

2

0.0
-6

2

(f)
0.4

-2

-2

(e)
0.4

-4

-4

0.5

0.4

0.0
-6

12

-4

-2

�

0

2

0.0
-6

-4

-2

0

2

Figure 6. Comparison of Tracy-Widom density (solid black) with densities

−σf (σx + η) (dashed red) and −(σ/mn)fF ((σx + η)/mn) (dotted blue) for
(a) n = 25, m = 125, (b) n = 25, m = 225, (c) n = 25, m = 425, (d)
n = 50, m = 150, (e) n = 50, m = 250, and (f) n = 50, m = 450. It should be
noted that the rectangularity α varies as 100 for (a), (d); 200 for (b), (e); and
400 for (c), (f). Also, the aspect ratio n/m is 1/5 for (a), (e), and 1/9 for (b),
(f), respectively.

For the rectangular case, Feldheim and Sodin [56] have shown that, in the limit m →
∞, n → ∞ with n/m bounded away from 1, the shifted and scaled smallest eigenvalue,
(λmin − η)/σ, leads to the Tracy-Widom density [39, 40]. Here η = (n1/2 − m1/2 )2 and
σ = (n1/2 − m1/2 )(n−1/2 − m−1/2 )1/3 < 0. The convergence, however, is slower when
α = m − n is close to 0. This can be attributed to the fact that the hard-edge behaviour
is prevalent unless α is large [58]. We should also mention that the Tracy-Widom
density captures the largest typical fluctuations of the smallest eigenvalue, while the
larger atypical fluctuations are described by large deviation results, as derived in [57]
by Katzav and Castillo.
As a consequence of identical universal behaviour of spectra of the regular and
fixed-trace ensembles [61], the Tracy-Widom density is also expected in the case of fixed
trace ensemble. The proper scaling in this case can be inferred from the connection
with the scaled Wishart-Laguerre ensemble, as discussed in Sec. 3. This implies that
the density of (mnµmin − η)/σ will converge to the Tracy-Widom result.
The recursion scheme given in section 2 enables us to work out the exact results
for the smallest eigenvalue density for large values of n and α and hence to explore
the above limit. In view of the scaling and shift indicated above, −σf (σx + η)

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

13

and −(σ/mn)fF ((σx + η)/mn) should coincide with the Tracy-Widom density of
the unitarily-invariant class. We examine this in figure 6. We can see that as the
rectangularity α increases the agreement improves for both n = 25 and n = 50. This
is because the hard-edge effect is diminished with increasing α. We also find that for a
given aspect ratio n/m < 1, as expected, the agreement is better for larger n.
5. Entanglement in bipartite systems
Consider a bipartite partition of an N1 N2 –dimensional Hilbert space H(N1 N2 ) consisting
(N )
(N )
of subsystems A and B, which belong to Hilbert spaces HA 1 and HB 2 , respectively,
(N )
(N )
such that H(N1 N2 ) = HA 1 ⊗ HB 2 . A general state |ψi of H(N1 N2 ) is given in terms of
(N )
(N )
the orthonormal states |iA i of HA 1 , and |αB i of HB 2 as
|ψi =

N1 X
N2
X

xi,α |iA i ⊗ |αB i,

(35)

i=1 α=1

PN1 PN2
2
where xi,α are complex coefficients, such that hψ|ψi =
i=1
α=1 |xi,α | = 1. The
density matrix for the composite system, considering a pure state scenario, is given by
ρ = |ψihψ| =

N1 X
N2
X

xi,α x∗j,β |iA ihj A | ⊗ |αB ihβ B |,

(36)

i,j=1 α,β=1

with tr[ρ] = 1. The reduced density matrix for subsystem, say A, can be obtained by
tracing out the subsystem B as
ρA =

N2
X

0

0

hα |ρ|β i =

α0 ,β 0 =1

N1
X

Fi,j |iA ihj A |,

(37)

i,j=1

P 2
∗
where Fi,j = N
α=1 xi,α xj,α can be viewed as the matrix elements of some N1 × N1 –
dimensional matrix F = XX† . Here X is a rectangular matrix of dimension N1 × N2
that has xi,α as its elements. In the eigenbasis of F, (37) can be written as
ρA =

N1
X

µi |µAi ihµAi |.

(38)

i=1

The eigenvalues µi of F are referred to as the Schmidt eigenvalues or Schmidt numbers.
Due to the trace condition, they satisfy
N1
X

µi = tr XX† = tr F = 1.

(39)

i=1

Suppose N1 ≤ N2 . Now, if we sample all normalized density matrices with equal
probabilities, i.e., if we choose the coefficients xi,α randomly using the Hilbert-Schmidt
density PX (X) ∝ δ(tr XX† − 1), then F defined here is statistically equivalent to the
F defined in (18), and the statistics of the Schmidt eigenvalues are described exactly
by the joint eigenvalue density of the fixed trace Wishart-Laguerre ensemble (19), with
N1 = n, N2 = m [19, 20]. It should be noted that the reduced density matrix for

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

14

the subsystem B will correspond to the matrix X† X, which will share the eigenvalues
µ1 , ..., µn , and will have the rest of its m − n eigenvalues as zero. As such, it carries the
same amount of information as F.
The Schmidt eigenvalues can be used to study various entanglement measures
such as von-Neumann entropy, Renyi entropies, concurrence, purity etc. As a
consequence, fixed trace Wishart-Laguerre ensemble has been extensively used to model
the reduced density matrices arising in the study of entanglement formation in bipartite
systems [14–16, 20, 23, 24, 35, 36, 48–50, 52–54, 61–67]. These works have explored several
aspects such as moments and distributions of Schmidt eigenvalues and entanglement
measures.
The density of the minimum eigenvalue in the present context not only sheds light
on the nature of the entanglement, but also provides important information about the
degree to which the effective dimension of the Hilbert space of the smaller subsystem
can be reduced [23, 24]. The smallest eigenvalue assumes values from 0 to 1/n. In
P
the extreme case of 1/n, it follows from the trace constraint ni=1 µi = 1, that all the
eigenvalues must have the same value 1/n. Consequently, the von-Neumann entropy,
P
− ni=1 µi ln µi , assumes its maximum value ln n, thereby making the corresponding
state maximally entangled. In the other extreme of the smallest eigenvalue being 0 (or
very close to 0), while it does not provide information regarding entanglement, from
the Schmidt decomposition it follows that the effective Hilbert space dimension of the
subsystem gets reduced by one.
In the next section we consider a system of coupled kicked tops and explore to what
extent the behaviour of the smallest Schmidt eigenvalue is described by the fixed trace
Wishart-Laguerre ensemble.
6. Coupled kicked tops
The kicked top system has been a paradigm for studying chaos, both classically and
quantum mechanically [68, 69]. Remarkably, it has also been realized experimentally
using an ensemble of Caesium atoms [70]. In the study of entanglement formation in
bipartite systems, a coupled system of two kicked tops has turned out to be of great
importance and has been investigated by a number of researchers [45, 53, 54, 65, 71, 72].
The full Hamiltonian of the coupled kicked top system is
H = H1 ⊗ 1N2 + 1N1 ⊗ H2 + H12 .

(40)

∞
kr 2 X
π
J
δ(t − ν), r = 1, 2,
Hr = Jyr +
2
2jr zr ν=−∞

(41)

Here,

represent the Hamiltonians for the individual tops, and
∞
X

H12 = √
(Jz ⊗ Jz2 )
δ(t − ν)
j1 j2 1
ν=−∞

(42)

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

15

is the interaction term. Also, 1Nr represents identity operator that acts on Nr dimensional Hilbert space H(Nr ) . The Hamiltonians H1 and H2 correspond respectively
to N1 (= 2j1 + 1)-dimensional, and N2 (= 2j2 + 1)-dimensional Hilbert spaces H(N1 )
and H(N2 ) , respectively. The Hamiltonian for the coupled kicked tops corresponds
to an N1 N2 -dimensional Hilbert space H(N1 N2 ) = H(N1 ) ⊗ H(N2 ) . Also, (Jxr , Jyr , Jyr )
are angular momentum operators for the rth top and satisfy the usual commutation
relations. The parameter kr controls the chaotic behaviour of the individual tops. The
parameter  takes care of the coupling between the two tops.
The unitary time evolution operator (Floquet operator) corresponding to the
Hamiltonian (40) is
U = (U1 ⊗ U2 )U12 ,

(43)

with


ιkr 2
ιπ
J
, r = 1, 2;
Ur = exp − Jyr −
2
2jr zr


ι
U12 = exp − √
Jz ⊗ Jz2 .
j1 j2 1

(44)
(45)

√
Here ι = −1 represents the imaginary unit. The initial state for the individual
tops is chosen as a generalized SU(2) coherent state or the directed angular
(r)
(r)
momentum state [68,
is given in |jr , mr i basis as hjr , mr |θ0 , φ0 i =
r69], which

−j
(r)
(r)
2jr
(1 + |γr |2 ) r γrjr −mr
with γr ≡ exp(ιφ0 ) tan(θ0 /2). For later use, we define
jr +mr
Nr -dimensional vectors given by
(r)

(r)

χr = [hjr , mr |θ0 , φ0 i]mr =−jr ,...,+jr .

(46)

For the coupled top, the initial state is taken as the tensor-product of the states
(1)
(1)
(2)
(2)
of the individual tops: |ψ(0)i = |θ0 , φ0 i ⊗ |θ0 , φ0 i. We can implement the time
evolution to obtain the state |ψ(ν)i starting from |ψ(0)i using the iteration scheme
|ψ(ν)i = U |ψ(ν − 1)i = (U1 ⊗ U2 )U12 |ψ(ν − 1)i, which, when written in hj1 , s1 ; j2 , s2 |
basis, is [72]



hj1 , s1 ; j2 , s2 |ψ(ν)i = exp −ι √
s1 s2
j1 j2
+j1
+j2
X
X
×
hj1 , s1 |U1 |j1 , m1 ihj2 , s2 |U2 |j2 , m2 ihj1 , m1 ; j2 , m2 |ψ(ν − 1)i.
m1 =−j1 m2 =−j2

A convenient approach for implementing this iteration scheme and eventually
calculating the reduced density matrix involves writing the states as N1 × N2 matrices:
Ψ(ν) = V ◦ (U1 Ψ(ν − 1)UT2 ).

(47)

Here ‘◦’ represents the Hadamard product and ‘T ’ the transpose. V is an N1 × N2
matrix given by




V = exp −ι √
ab
.
(48)
a=−j1 ,...,+j1
j1 j2
b=−j2 ,...,+j2

Smallest eigenvalue density for complex Wishart-Laguerre ensemble
(a)

50

(1)

(1)

(1)

(2)

(2)

(1)

(1)

(2)

(2)

θ0 =3.08,ϕ0 =2.35,
30

(c)

12

(1)

(1)

10

θ0 =1.62,ϕ0 =1.53

8

θ0 =1.71,ϕ0 =1.42,

θ0 =0.34,ϕ0 =0.47

10
0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
(i)

(2)

(1)

(2)

(2)

θ0 =1.33,ϕ0 =2.12

6
20

(2)

(1)

(d)

12

(1)

(1)

10

θ0 =2.35,ϕ0 =0.83

8

θ0 =2.53,ϕ0 =1.61,

(2)

(2)

(1)

(1)

(2)

(2)

10

θ0 =2.81,ϕ0 =1.72

8

θ0 =1.73,ϕ0 =1.21,

(2)

(2)

(1)

(1)

(2)

(2)

θ0 =1.64,ϕ0 =2.52

θ0 =1.81,ϕ0 =1.52

6

(1)

θ0 =2.55,ϕ0 =0.63,

θ0 =1.22,ϕ0 =2.13,

θ0 =2.11,ϕ0 =0.96,

θ0 =2.43,ϕ0 =0.89

40

pF ( μ)

(b)

12

θ0 =0.75,ϕ0 =1.71,

16

6

4

4

4

2

2

2

0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

μ

140

(ii)

120

(iii)

(iv)

120

120

400
100

100
100

� � (�)

300

80

80

80

60

60

60

200
40

40

40

100
20
0
0.000

0.005

0.010

0.015

0.020

0.025

0.030

0
0.000

20

20

0.005

0.010

0.015

0.020

0.025

0.030

�

0
0.000

0.005

0.010

0.015

0.020

0.025

0.030

0
0.000

0.005

0.010

0.015

0.020

0.025

0.030

Figure 7. Comparison of marginal density (top row) and the smallest eigenvalue

density (bottom row) of the coupled kicked top system with the fixed trace
ensemble results for N1 = 11, N2 = 21 and  = 1. For plots (a)-(d) as well as
(i)-(iv), the parameters (k1 , k2 ) vary as (0.5, 1), (0.5, 8), (2.5, 3), (7, 8). In
each case the solid (black) curves correspond to the analytical results, while the
dotted (blue) and dashed (red) curves correspond to different initial conditions.
The θ0 and φ0 values used for different initial conditions are mentioned in (a)(d), and hold, respectively, for (i)-(iv) also.

Also, Ur is an Nr × Nr dimensional matrix



 
kr 2
(jr ) π
Ur = exp −ι a da,b
.
2jr
2 a,b=−jr ,...,+jr

(49)

(j )

Here da,br represents the Wigner (small) d matrix elements. We use the inbuilt function
in Mathematica [47] for Wigner (big) D matrix to evaluate it. The initial N1 × N2 dimensional state matrix is given by
Ψ(0) = χ1 ⊗ χT2 .

(50)

Eventually, the reduced density matrix of dimension N1 × N1 can be constructed as
ρd = Ψ(ν)Ψ(ν)† .

(51)

The eigenvalues of this matrix are the sought after Schmidt eigenvalues, whose statistics
is of interest to us. To obtain an ensemble of states we proceed as follows. We begin
with an initial state and apply (47) iteratively. After ignoring initial 500 states to safely
avoid the transient regime [54], we start considering states separated by 20 time steps
to put off any unwanted correlations. In all, we consider 50000 states for statistical
analysis.
In figure 7 we set N1 = 11, N2 = 21,  = 1, and examine the effect of different choices
of k1 , k2 on one level density and smallest eigenvalue density for the coupled kicked tops.
For (a), (i) we have k1 = 0.5, k2 = 1 for which the classical phase spaces of the individual

Smallest eigenvalue density for complex Wishart-Laguerre ensemble
14

pF (μ )

14

(a)

12

14

(b)

12
10

10

8

8

8

6

6

6

4

4

4

2

2
0.1

0.2

0.3

0.4

0.5

0.6

25

2

0
0.0

0.1

0.2

0.3

0.4

0
0.0

0.5

25

(d)

20

20

15

10

10

10

5

5

5

0.10

0.15

0.20

0.25

0.30

0.2

0.3

0.4

0
0.00

0.05

0.10

0.15

0.20

0.25

(f)

20

15

0.05

0.1

25

(e)

15

0
0.00

(c)

12

10

0
0.0

17

0.30

0
0.00

0.05

0.10

0.15

0.20

μ
2000

350

(i)

1500

(ii)

300

(iii)

150

250
100

200

1000

150
50

100

500

f F (x)

50
0
0.000

0.001

0.002

0.003

0.004

0.005

10 000

0
0.000

0.005

0.010

(v)

(iv)

0.01

0.02

0.03

0.04

(vi)

500
400

1000

6000

0
0.00
600

1500

8000

0.015

300
4000

200

500
2000
0
0.0000

100
0.0002

0.0004

0.0006

0.0008

0
0.0000 0.0005 0.0010 0.0015 0.0020 0.0025

0
0.000

0.002

0.004

0.006

0.008

x

Figure 8. Effect of varying  on marginal density ((a)-(f)), and the smallest

eigenvalue density ((i)-(vi)). The solid lines (black) correspond to analytical
results, while the dotted (red), dot-dashed (blue) and dashed (green) curves
result from coupled top simulation for  = 0.05, 0.1 and 0.5, respectively. The
parameters k1 , k2 are fixed at 7, 8, while dimension parameters (N1 , N2 ) vary
for the figures (a)-(f) as well as (i)-(vi) as (11, 11), (11, 15), (11, 25), (21, 21),
(21, 25), (21, 35).

tops consist mostly of regular orbits [54]. In this case, we can see deviations from the
(r)
(r)
fixed trace ensemble results with strong sensitivity to initial conditions, i.e. θ0 and φ0
values. In (b), (ii) we set k1 = 0.5, k2 = 8. In this case highly chaotic phase space [54]
of the second top leads to an agreement with the results of the fixed trace ensemble,
even though the phase space of the first top is mostly regular. Moreover, there is a
weak sensitivity to initial conditions. In (c), (iii) we consider k1 = 2.5, k2 = 3, both of
which correspond to mixed type phase space [54]. Here we observe deviations, along
with some sensitivity to initial conditions. Finally, in (d), (iv) we take k1 = 7, k2 = 8,

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

18

for which phase spaces of both the tops are highly chaotic. In this case, we have very
good agreement with the random matrix results and very weak sensitivity to the initial
conditions.
In figure 8 we consider a chaotic regime (k1 = 7, k2 = 8) and examine the effect of
varying  for various combinations of n and α. We observe that for a given , increase in
n or α leads to a better agreement with the fixed trace ensemble results. Recent studies
in a similar direction have investigated the universal aspects of spectral fluctuations and
entanglement transitions in strongly chaotic subsystems [73, 74].
A quantifier to measure the fraction of close to maximally entangled states can be
R 1/n
the cumulative probability R(δ) = 1/n−δ fF (x) dx [24], that turns out to be vanishingly
small for δ << 1/n and thus, implies that the actual fraction of such states is extremely
small. For example, using (25), we obtain R(δ = 0.1/n) value to be roughly (i) 8 × 10−6
for n = 3, m = 11, (ii) 1×10−35 for n = 7, m = 19, and (iii) 5×10−91 for n = 11, m = 25.
7. Summary and conclusion
We considered complex Wishart-Laguerre ensemble, both with and without the fixed
trace condition, and provided an easily implementable recurrence scheme to obtain the
exact result for the density of the smallest eigenvalue. This method also gives access to
arbitrary moments of the smallest eigenvalue. The recursion-based approach for exact
and explicit expressions for the density is preferable to the results based on determinants
which are difficult to handle with increasing dimensionality n or rectangularity α.
We also demonstrated the equivalence of the recurrence scheme and the determinantbased results for α = 0 and 1. We validated our analytical results using Monte Carlo
simulations and also used large n and α evaluations to compare with the Tracy-Widom
density. As an application to quantum entanglement problem we explored the behaviour
of Schmidt eigenvalues of the coupled kicked top system. Among other things, we found
that in the chaotic regime, the fixed trace ensemble describes the behaviour of the
Schmidt eigenvalues very well if sufficient coupling is provided between the constituent
tops.
Acknowledgments
This work initiated from a project that was carried out at Shiv Nadar University under
the Opportunities for Undergraduate Research (OUR) scheme. The authors are grateful
to the anonymous reviewers for fruitful suggestions.
Appendix A. Recurrence scheme
We begin with (6) and apply the shift λi → λi + x. This results in
Z ∞
Z ∞
n
Y
Y
α −nx
2
f (x) = nCn,α x e
dλ2 · · ·
dλn
(λj − λk )
λ2i (λi + x)α e−λi .
0

0

2≤k<j≤n

i=2

(A.1)

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

19

To derive the recurrence relation, we will proceed parallel to the steps in chapter 4
of [25], or as in [26]. To this end, we shift the indices of the integration variables as
λi → λi−1 , and also introduce the measure dΩi = λ2i e−λi dλi . Consequently, we arrive
at the following expression:
Z ∞
Z ∞
n−1
Y
2
α −nx
dΩn−1 ∆n−1 ({λ})
(λi + x)α . (A.2)
dΩ1 · · ·
f (x) = nCn,α x e
0

0

i=1

Next, we define
α
Ii,j

Z

∞

Z

∞

dΩn−1 ∆2n−1 ({λ})u(x),

dΩ1 · · ·

=

(A.3)

0

0

where the integrand u(x) is
(λ + x)α · · · (λi + x)α (λi+1 + x)α−1 · · · (λi+j + x)α−1
| 1
{z
}|
{z
}
i terms

j terms

α−2

α−2

.
× (λi+j+1 + x)
· · · (λn + x)
{z
}
|

(A.4)

n−i−j−1 terms

We also define the operator
Z ∞
Z
α
Ii,j [v] =
dΩ1 · · ·
0

∞

dΩn−1 ∆2n−1 ({λ})u(x) v.

(A.5)

0

Using the above notation the smallest eigenvalue density can be written as
α
f (x) = nCn,α xα e−nx In−1,0
.

(A.6)

With these, Lemma 4.2 of [25] (or, equivalently, Lemma 4.1 of [26]) holds in the complex
case also:
(
α
α
Ii+1,j−1
− xIi,j
if i < k ≤ i + j,
α
Ii,j [λk ] =
(A.7)
α
α
Ii,j+1 − xIi,j
if i + j < k < n.
The above result follows by writing λk as (λk + x) − x and then using the operator
defined in (A.5). Next, if the terms (λk + x) and (λl + x) share the same exponent in
the integrals (i.e., both k and l fall within one of the closed intervals [1, i], [i + 1, i + j],
or [i + j + 1, n − 1]), then


λk λl
α
Ii,j
= 0,
(A.8)
λk − λl


λk
1 α
α
Ii,j
= Ii,j
,
(A.9)
λk − λl
2


λ2k
α
α
Ii,j
= Ii,j
[λk ].
(A.10)
λk − λl
Equation (A.8) follows because of the asymmetry in λk and λl , while (A.9) is obtained
using the identity λk /(λk −λl )+λl /(λl −λk ) = 1 and using symmetry. The result (A.10)
is obtained using the identity λ2k /(λk − λl ) = λk + λk λl /(λk − λl ) and (A.8).

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

20

The crucial difference occurs in the first equation of Lemma 4.3 [25] (or Lemma
4.2 [26]), which reads for the present case as
α
α
α
α
Ii,j
= (x + α + j + 2k + 2)Ii−1,j+1
− x[k + (α − 1)]Ii−1,j
+ (i − 1)xIi−2,j+2
α
I0,j

=

α−1
Ij,n−j−1
,

(A.11)
(A.12)

with k = n − i − j − 1. The second equation of this set, (A.12), follows readily from the
definition (A.3). This first equation of this set, (A.11), is derived using
α
α
α
Ii,j
= xIi−1,j+1
+ Ii−1,j+1
[λi ],

(A.13)

which is a consequence of (A.7). The difference in the result compared to the real case
α
occurs due to the term Ii−1,j+1
[λi ]. For the complex case, this involves observing the
following:
Z ∞
Z ∞
Y
Y
d
α−1
2 3 −λi
[(λi +x)α−1 (λl −λi )2 λ3i ]e−λi dλi .(A.14)
(λi +x)
(λl −λi ) λi e dλi =
dλi
0
0
i<l
i<l
Also, using the result
α
dIi−1,j+1
α
α
= (i − 1)αIi−2,j+2
+ (j + 1)(α − 1)Ii−1,j
(A.15)
dx
for i + j = n − 1, as given in the proof for Lemma 4.4 of [25] (or Lemma 4.3 of [26]),
yields in the present case


α
x d α
α
α
α
Ii−1,j+1 + x(i − 1) 1 +
Ii−2,j+2
. (A.16)
Ii,j = (x + α + j + 2)Ii−1,j+1 −
j + 1 dx
j+1
α−1
α
in view of (A.12). Equation (A.16)
Now, we can begin with In−1,0
, which is same as I0,n−1
α
can be used with j = n−i−1 repeatedly for i = 1 to n−1 to arrive at In−1,0
, starting from
α
α
I0,n−1 . We note that In−1,0 is the term needed to obtain the smallest eigenvalue density
expression (A.6) explicitly. This is essentially what has been employed in the recurrence
0
0
α
= 1/Cn−1,2 .
involving Si := Ii,n−i−1
/In−1,0
for gn,m (x) in (7). We also note that In−1,0
The constant cn,m of (7) is therefore nCn,α /Cn−1,2 .

Appendix B. Mathematica codes
The following code can be implemented in Mathematica [28] to obtain exact expressions
for the smallest eigenvalue density for the Wishart-Laguerre ensemble:

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

21

� = ��� � = ���
�[��_� ��_] �=

�
�����[��] �����[��]

�������

�����[� + �]
�����[� + �� - ��]

� {�� �� �� - �}�

α = � - �� �[-�] = �� �[�] = �� �[� - �] = ��
���� = �� � < α + �� � ++� 
� = � + ��
���� = �� � < �� � ++�
�[�] = ������(� + � - � + �) �[� - �] -

�
�-�

�[�[� - �]� �] + � (� - �)

�-�
�-�

�[� - �]�

�[-�] = �� �[�] = �[� - �]�
�[�_ ] = �[�� �] ⅇ-� � �α ������[�[� - �]]

For generating the smallest eigenvalue density for the fixed trace Wishart-Laguerre
ensemble, the following code can be used along with the above.
� = ���������������ⅇ� � �[�]� ��
��[�_ ] = �����������[� �] ����[[�]]

(� - � �)� �-�-� ��-�
�����[� � - �]

� {�� α + �� α � + �} ��������������[� - � �]

We can also directly implement the inverse Laplace transform function built in
Mathematica:
��[�_ ] = �����������[� �] �������������������������-� � �[� �]� �� �

The ‘Factor’ option in the above codes is for printing compact expressions on-screen.
For computation involving large n or α values, it may be removed, since factoring very
large expressions may result in a large computation time.
The moments of the smallest eigenvalue of the regular or the fixed-trace WishartLaguerre ensemble can be obtained using the following functions:
���[η_ ] �= ���

����[η_ ] �=

�[[�]]
��+η

�����[� + η ]� {�� α + �� α � + �}

�����[� �]
�����[� � + η ]

���[η ]

Appendix C. Relation with associated Laguerre polynomial
The associated Laguerre polynomials satisfy the following relations [46]:
(k)

(k+1)

(k+2)

iLi (−x) = (x + k + 1)Li−1 (−x) + xLi−2 (−x),
d (k)
(k+1)
Li (−x) = Ln−1 (−x).
dx
These two can be combined to obtain the following relation:
x d (k+1)
xk (k+2)
(k)
(k+1)
iLi (−x) = (x + k + 1)Li−1 (−x) −
Li−1 (−x) +
L
(−x).
k − 1 dx
k − 1 i−2

(C.1)
(C.2)

(C.3)

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

22

Considering k = n − i + 1 gives
(n−i+1)

iLi

(n−i+2)

(−x) = (x + n − i + 2)Li−1

(−x) −

x d (n−i+2)
L
(−x)
n − i dx i−1

x(n − i + 1) (n−i+3)
Li−2
(−x).
(C.4)
n−i
Multiplying this equation by Γ(i) and then calling Si = Γ(i + 1)Li (n − i + 1)(−x), we
get
+

x dSi−1
(n − i + 1)
+ x(i − 1)
Si−2 . (C.5)
n − i dx
n−i
This recurrence relation is the same as that given in section 2 when used for m = n + 1.
(2)
Hence, gn,n+1 (x) = Sn−1 = Γ(n)Ln−1 (−x).
Si = (n − i + 2 + x)Si−1 −

Appendix D. Some explicit results
For α = m − n = 0, the smallest eigenvalue density expressions valid for all n are quite
compact and are already provided in (13) and (28), respectively, for the regular WishartLaguerre ensemble and for the fixed trace Wishart-Laguerre ensemble. For a few other
cases we tabulate the exact results in Tables D1 and D2 using the above Mathematica
codes. This includes the α = 1 case for which closed-form results for any n are given in
(15) and (30). In the case of fixed trace ensemble there is a Θ(1 − nx) term in each of
the probability density expressions that we have not shown in the table for the sake of
compactness.

Smallest eigenvalue density for complex Wishart-Laguerre ensemble
n

23

m

f (x)

3

e−2x x (x + 3)

4


.
e−2x x2 x2 + 6x + 12 6

5



e−2x x3 x3 + 9x2 + 36x + 60 /72

6



e−2x x4 x4 + 12x3 + 72x2 + 240x + 360 /1440

4


.
e−3x x x2 + 8x + 12 2

5


.
e−3x x2 x4 + 16x3 + 96x2 + 240x + 240 48

6


.
e−3x x3 x6 + 24x5 + 252x4 + 1440x3 + 4680x2 + 8640x + 7200 2880

7


.
e−3x x4 x8 + 32x7 + 480x6 + 4320x5 + 25200x4 + 97920x3 + 253440x2 + 403200x + 302400 345600

5


.
e−4x x x3 + 15x2 + 60x + 60 6

6


.
e−4x x2 x6 + 30x5 + 360x4 + 2160x3 + 6840x2 + 10800x + 7200 720

e−4x x3 x9 + 45x8 + 900x7 + 10380x6 + 75600x5 + 360720x4 + 1130400x3 + 2268000x2
.
+2721600x + 1512000 259200

e−4x x4 x12 + 60x11 + 1680x10 + 28800x9 + 334800x8 + 2773440x7 + 16790400x6 + 74995200x5
.
+246456000x4 + 586656000x3 + 972518400x2 + 1016064000x + 508032000 217728000

2

3

4
7

8

6


.
e−5x x x4 + 24x3 + 180x2 + 480x + 360 24

7


.
e−5x x2 x8 + 48x7 + 960x6 + 10320x5 + 64800x4 + 241920x3 + 524160x2 + 604800x + 302400 17280

e−5x x3 x12 + 72x11 + 2340x10 + 45120x9 + 572400x8 + 5019840x7 + 31157280x6 + 137894400x5
.
+432734400x4 + 943488000x3 + 1371686400x2 + 1219276800x + 508032000 43545600

e−5x x4 x16 + 96x15 + 4320x14 + 120480x13 + 2323440x12 + 32780160x11 + 349493760x10 + 2870380800x9
+18353563200x8 + 91755417600x7 + 358177075200x6 + 1083937075200x5 + .
2506629888000x4

5
8

9

+4316239872000x3 + 5267275776000x2 + 4096770048000x + 1536288768000

Table D1. Results for the Wishart-Laguerre ensemble

292626432000

Smallest eigenvalue density for complex Wishart-Laguerre ensemble
n

m

fF (x)

3



2
60x 1 − x 1 − 2x

4


2 
2
420x2 1 − x
1 − 2x

5


3 
2
2520x3 1 − x
1 − 2x

6


4 
2
13860x4 1 − x
1 − 2x

4



7
660x 1 − 3x2 1 − 3x

5



7
10920x2 1 − x − x2 − 9x3 + 15x4 1 − 3x

6



7
28560x3 5 − 12x + 12x2 − 48x3 − 48x4 + 432x5 − 411x6 1 − 3x

7



7
1627920x4 1 − 4x + 8x2 − 16x3 + 320x6 − 756x7 + 489x8 1 − 3x

5



3420x 1 + 5x − 20x2 + 4x3 1 − 4x)14

2

3

6
4
7

8

6
7
5
8

9



14
106260x2 1 + 6x + x2 − 204x3 + 486x4 − 424x5 + 356x6 1 − 4x

491400x3 5 + 27x + 51x2 − 683x3 − 5286x4 + 35910x5 − 85295x6 + 116895x7

14
−79980x8 − 9196x9 1 − 4x

6796440x4 7 + 28x + 86x2 − 540x3 − 6775x4 − 18416x5 + 440876x6 − 2012008x7

14
+4901710x8 − 7145600x9 + 5855692x10 − 3288592x11 + 2386196x12 1 − 4x


23
12180x 1 + 16x − 39x2 − 140x3 + 220x4 1 − 5x


23
628320x2 1 + 22x + 142x2 − 1234x3 − 580x4 + 4676x5 + 29788x6 − 92420x7 + 75355x8 1 − 5x

23030280x3 1 + 24x + 243x2 + 280x3 − 19962x4 + 50208x5 − 31022x6 + 649056x7 − 1420095x8

23
−7867032x9 + 35763831x10 − 53675640x11 + 27627140x12 1 − 5x

97740720x4 7 + 168x + 1968x2 + 9642x3 − 75517x4 − 1457898x5 + 10143328x6 − 31939648x7
+134132583x8 − 323536148x9 − 511260568x10 + 786421818x11 + 22191959881x12

23
−105911938466x13 + 211492028376x14 − 203837200540x15 + 80216630930x16 1 − 5x

Table D2. Results for the fixed trace Wishart-Laguerre ensemble

24

Smallest eigenvalue density for complex Wishart-Laguerre ensemble

25

References
[1] Mehta M L 2004 Random Matrices (New York: Academic Press)
[2] Forrester P J 2010 Log-Gases and Random Matrices (LMS-34) (Princeton University Press,
Princeton, NJ)
[3] Anderson T W 2003 An Introduction to Multivariate Statistical Analysis (John Wiley & Sons)
[4] Muirhead R J 2005 Aspects of Multivariate Statistical Theory (Wiley Interscience)
[5] James A T 1964 Ann. Math. Statist. 35 475
[6] Gnanadesikan R 1997 Methods for Statistical Data Analysis of Multivariate Observations (John
Wiley & Sons)
[7] Plerou V, Gopikrishnan P, Rosenow B, Amaral L A N, Guhr T and Stanley H E 2002 Phys. Rev.
E 65 066126
[8] Vinayak and Pandey A 2010 Phys. Rev. E 81 036202
[9] Verbaarschot J 1994 Phys. Rev. Lett. 72 2531
[10] Verbaarschot J J M and Wettig T 2000 Ann. Rev. Nucl. Part. Sci. 50 343
[11] Tulino A M and Verdú S 2004 Random Matrix Theory and Wireless Communications (Now
Publishers Inc)
[12] Foschini G J and Gans M J 1998 Wireless Pers. Commun. 6 311
[13] Telatar I E 1999 Europ. Trans. Telecommun. 10 585
[14] Lubkin E 1978 J. Math. Phys. (N.Y.) 19 1028
[15] Lloyd S and Pagels H 1988 Ann. Phys. (NY) 188 186
[16] Page D N 1993 Phys. Rev. Lett. 71 1291
[17] Hall M J W 1998 Phys. Lett. A 242 123
[18] Życzkowski K and Sommers H-J 2001 J. Phys. A: Math. Gen. 34 7111
[19] Sommers H-J and Życzkowski K 2004 J. Phys. A: Math. Gen 37 8457
[20] Osipov V A, Sommers H-J and Życzkowski K 2010 J. Phys. A: Math. Theor. 43 055302
[21] Burel G 2002 Proc. of the WSEAS Int. Conf. on Signal, Speech and Image Processing (ICOSSIP
2002)
[22] Park C S and Lee K B 2008 IEEE Trans. Wireless Commun. 7 4432
[23] Majumdar S N, Bohigas O and Lakshminarayan A 2008 J. Stat. Phys. 131 33
[24] Majumdar S N 2011 Handbook of Random Matrix Theory, eds. Akemann G, Baik J and Di
Francesco P (Oxford Press, New York)
[25] Edelman A 1989 Eigenvalues and Condition Numbers of Random Matrices Ph.D. thesis, MIT
[26] Edelman A 1991 Lin. Alg. Appl. 159 55
[27] Khatri C G 1964 Ann. Math. Statist. 35 1807
[28] Forrester P J and Hughes T D 1994 J. Math. Phys. 35 6736
[29] Forrester P J 1993 Nucl. Phys. B 402 709
[30] Forrester P J 1994 J. Math. Phys. 35 2539
[31] Nagao T and Forrester P J 1998 Nucl. Phys. B 509 561
[32] Damgaard P H and Nishigaki S M 2001 Phys. Rev. D 63 045012
[33] Akemann G, Guhr T, Kieburg M, Wegner R and Wirtz T 2014 Phys. Rev. Lett. 113 250201
[34] Edelman A, Guionnet A and Péché S 2016 Ann. App. Prob. 26 1659
[35] Chen Y, Liu D-Z and Zhou D-S 2010 J. Phys. A: Math. Theor. 43 315303
[36] Akemann G and Vivo P 2011 J. Stat. Mech. 2011 P05020
[37] Zanella A, Chiani M and Win M Z 2009 IEEE Trans. Commun. 57 1050
[38] Nishigaki S M, Damgaard P H and Wettig T 1998 Phys. Rev. D 58 087704
[39] Tracy C A and Widom H 1993 Phys. Lett. B 305 115
[40] Tracy C A and Widom H 1994 Commun. Math. Phys. 159 151
[41] Tracy C A and Widom H 1994 Commun. Math. Phys. 161 289
[42] Borodin A and Forrester P J 2003 J. Phys. A: Math. Gen. 36 2963
[43] Forrester P J 2007 J. Phys. A: Math. Theor. 40 11093

Smallest eigenvalue density for complex Wishart-Laguerre ensemble
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]

26

Wirtz T and Guhr T 2013 Phys. Rev. Lett. 111 094101
Trail C M, Madhok V and Deutsch I H 2008 Phys. Rev. E 78 046211
Szego G 1975 Orthogonal Polynomials (American Mathematical Society, Providence)
Wolfram Research Inc. Mathematica Version 10.0 (Wolfram Research Inc.: Champaign,
Illinois)
Kumar S and Pandey A 2011 J. Phys. A: Math. Theor. 44 445301
Adachi S, Toda M and Kubotani H 2009 Ann. Phys. 324 2278
Vivo P 2010 J. Phys. A: Math. Theor. 43 405206
Marčenko V A and Pastur L A 1967 Math. USSR-Sb. 1 457
Žnidarič M 2007 J. Phys. A: Math. Theor. 40 F105
Bandyopadhyay J N and Lakshminarayan A 2002 Phys. Rev. Lett. 89 060402
Bandyopadhyay J N and Lakshminarayan A 2004 Phys. Rev. E 69 016201
Tao T and Vu V 2010 Geom. Funct. Anal. 20 260
Feldheim O N and Sodin S 2010 Geom. Funct. Anal. 20 88
Katzav E and Castillo I P 2010 Phys. Rev. E 82 040104(R)
Wirtz T, Kieburg M and Guhr T 2015 EPL 109 20005
Perret A and Schehr G 2016 Random Matrices: Theory Appl. 05 1650001
Bornemann F 2016 Ann. Appl. Probab. 26 1942
Liu D-Z and Zhou D-S 2011 Int. Math. Res. Notices 2011 725
Giraud O 2007 J. Phys. A: Math. Theor. 40 F1053
Facchi P, Marzolino U, Parisi G, Pascazio S and Scardicchio A 2008 Phys. Rev. Lett. 101 050502
Vivo P, Pato M P and Oshanin G 2016 Phys. Rev. E 93 052106
Kubotani H, Adachi S and Toda M 2013 Phys. Rev. E 87 062921
Nadal C, Majumdar S N and Vergassola M 2010 Phys. Rev. Lett. 104 110501
Nadal C, Majumdar S N and Vergassola M 2011 J. Stat. Phys. 142 403
Haake F, Kuś M and Scharf R 1987 Z. Phys. B - Condensed Matter 65 381
Haake F 2010 Quantum Signatures of Chaos 3rd ed. (Springer-Verlag, Berlin)
Chaudhury S, Smith A, Anderson B E, Ghose S and Jessen P S 2009 Nature 461 768
Fujisaki H, Miyadera T and Tanaka A 2003 Phys. Rev. E 67 066201
Miller P A and Sarkar S 1999 Phys. Rev. E 60 1542
Srivastava S C L, Tomsovic S, Lakshminarayan A, Ketzmerick R and Bäcker A 2016 Phys. Rev.
Lett. 116 054101
Lakshminarayan A, Srivastava S C L, Ketzmerick R, Bäcker A and Tomsovic S 2016 Phys. Rev.
E 94 010205(R)

