Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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where
a
is the absolutely continuous part of , A
is
the set of atoms of , and # is the counting measure.
The functional is set equal to þ1 on all measures
whose singular part
s
is nonatomic. A general
representation result (see Bouchitte and Buttazzo
(1992) and references therein) establishes that a
functional : M(; R
N
) ! [0, þ1], which is
sequentially lower-semicontinuous for the weak
convergence of M(; R
N
), and
local on M(; R
N
) in the sense that ( þ ) =
() þ () whenever and are mutually
singular in ,
has to be of the form
ðÞ¼
Z
ðx;
a
Þd þ
Z
1
ðx;
c
Þ
þ
Z
ðx;
#
ðxÞÞd #
where is a non-negative measure, =
a
dx þ
c
þ
#
is the decomposition of into absolutely
continuous, Cantor, and atomic parts, (x, v)isan
integrand convex in v,and
1
is its recession
function. The novelty is now represented by the
integrand (x, v) which has to be subadditive in v
and satisfying the compatibility condition
lim
t!þ1
ðx; tvÞ
t
¼ lim
t!0
þ
ðx; tvÞ
t
When has a superlinear growth the condition
above gives that the slope of (x, ) at the origin has
to be infinite. For instance, in the Mumford– Shah
case we have
ðx; vÞ¼jvj
2
; ðx; vÞ¼
1ifv 6¼ 0
0ifv ¼ 0
½10
Coming back to the case u 2 BV( ), we have the
decomposition (see [4]):
Du ¼ru L
N
þðDuÞ
c
þ½u
u
ðxÞH
N1
9S
u
where we considered, for simplicity, only the scalar
case m = 1 and denoted by [u] the jump u
þ
u
.
We have then the functional
FðuÞ¼
Z
ðx; ruÞdx þ
Z
1
ðx; ðDuÞ
c
Þ
þ
Z
S
u
ðx; ½u
u
ÞdH
N 1
For insta nce, in the homogeneous–isotropic
case, when (x, v) and (x, v) are independent
of x and depend only on jvj, the formula above
reduces to
FðuÞ¼
Z
ðjrujÞdx þ jDuj
c
ðÞ
þ
Z
S
u
ðj½ujÞdH
N1
½11
where , , satisfy the compatibility condition
¼
1
ð1Þ¼lim
t!0
þ
ðt Þ
t
½12
In the original Mumford–Shah model for computer
vision, is a rectangle of the plane, u
0
: ! [0, 1]
represents the gray level of a picture, c
1
and c
2
are
positive scale and contrast parameters, and the
variational problem under consideration is
min
Z
jruj
2
dx þ c
1
Z
ju u
0
j
2
dx
þc
2
H
N1
ðS
u
Þ: ðDuÞ
c
0
½13
The solution u then represents the reconstructed
image, whose contours are given by the jump set S
u
.
We refer to Giorgi and Ambrosio (1988) and to the
book by Morel and Solimini (1995) for further
details about this model.
Analogously, in the case of the study of fractures
of an elastic membrane, a problem similar to [13]
provides the vertical displacement u of the mem-
brane, together with its fracture set S
u
. We refer to
some recent papers (see Dal Maso and Toader
(2002) and Francroft and Marigo (1998),and
references therein) for a more detailed description
of fracture mechanics problems, even in the more
delicate vectorial setting of elasticity.
Using the functional F in [11] we have the
generalized Mumford–Shah problem,
min FðuÞþc
1
Z
ju u
0
j
2
dx: u 2 BVðÞ
where is convex, is subadditive, and the
compatibility condition [12] is fulfilled.
If we set K = S
u
and assume that it is closed, the
Mumford–Shah problem can be rewritten as
min
Z
nK
jruj
2
dx þ c
1
Z
nK
ju u
0
j
2
dx
þc
2
H
N1
ðK \ Þ : K
closed;
u 2 H
1
ð n KÞ
416 Free Interfaces and Free Discontinuities: Variational Problems
and this justifies the name ‘‘free discontinuity
problems,’’ which is often used in this setting.
The regularity properties of optimal pairs (u, K)
are far from being fully understood; some partial
results are available but the Mumford–Shah
conjecture:
in the case N = 2 for an optimal pair (u, K) the set
K is locally the finite union of C
1, 1
arcs
remains still open. We refer to Ambrosio et al.
(2000) for a list of the regularity results on the
problem above that are known thus for.
Further Reading
Alberti G (1993) Rank one properties for derivatives of functions
with bounded variation. Proc. Roy. Soc. Edinburgh A 123:
239–274.
Ambrosio L, Fusco N, and Pallara D (2000) Functions of
Bounded Variation and Free Discontinuity Problems. Oxford
Mathematical Monographs. Oxford: Clarendon.
Bouchitte G and Buttazzo G (1992) Integral representation of
nonconvex functionals defined on measures. Ann. Inst. H.
Poincare´ Anal. Non Line´aire 9: 101–117.
Buttazzo G (1989) Semicontinuity, Relaxation and Integral
Representation in the Calculus of Variations. Pitman Res.
Notes Math. Ser., vol. 207. Harlow: Longman.
Dacorogna B (1989) Direct Methods in the Calculus of Varia-
tions. Appl. Math. Sciences. vol. 78. Berlin: Springer.
Dal Maso G and Toader R (2002) A model for the quasi-static
growth of brittle fractures: existence and approximation
results. Arch. Ration. Mech. Anal. 162: 101–135.
De Giorgi E and Ambrosio L (1988) New functionals in the
calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci.
Fis. Mat. Natur. 82(2): 199–210.
Evans LC and Gariepy RF (1992) Measure Theory and Fine
Properties of Functions. Studies in Advanced Math Ann.
Harbor: CRC Press.
Federer H (1969) Geometric Measure Theory. Berlin: Springer.
Fonseca I and Mu¨ ller S (1992) Quasi-convex integrands and
lower semicontinuity in L
1
. SIAM J. Math. Anal. 23:
1081–1098.
Francfort G and Marigo J-J (1998) Revisiting brittle fracture as
an energy minimization problem. J. Mech. Phys. Solids 46:
1319–1342.
Giusti E (1984) Minimal Surfaces and Functions of Bounded
Variation. Boston: Birkha¨ user.
Massari U and Miranda M (1984) Minimal Surfaces of
Codimension One. Amsterdam: North-Holland.
Morel J-M and Solimini S (1995) Variational Methods in Image
Segmentation. Progress in Nonlinear Differential Equations
and their Applications, vol. 14. Boston: Birkha¨user.
Ziemer WP (1989) Weakly Differentiable Functions. Berlin:
Springer.
Free Probability Theory
D-V Voiculescu, University of California at Berkeley,
Berkeley, CA, USA
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Free probability is a probability theory adapted to
quantities with the highest degree of noncommutativ-
ity. A basic feature of this is that the definition of
independence is modified in such a way that the freely
independent random variables will not commute in
general. The exploration of this notion of indepen-
dence, which was initially motivated by questions
about operator algebras (Voiculescu 1985), has
produced a theory that runs parallel to an unexpect-
edly large part of classical probability theory. The
applications of the theory have also gone into
unexpected directions, once it turned out that the
large-N limit of systems of random matrices is a key
asymptotic model in the theory (Voiculescu 1991).
There are several signs like the connections to large N
for random matrices and to the combinatorics of
noncrossing partitions (Speicher 1998) (which corre-
spond to certain planar diagrams), that perhaps these
connections may go even further towards the large-N
limit of models in gauge theory.
In this article the noncommutative probability and
the random matrix angle will be emphasized and
very little will be said about the operator algebras
and the combinatorics. After discussing free inde-
pendence and models based on free products of
groups and creation and annihilation operators on
the Boltzmann full Fock space, we continue with the
semicircle law, which is the substitute for the Gauss
law in this context, and with the nonlinear free
harmonic analysis arising from addition and multi-
plication of free random variables.
We then devote two longer sections to the
asymptotic free independence of large random
matrices and to free entropy, the free probability
analog of Shannon’s information-theoretic entropy
for continuous random variables.
Freeness of Noncommutative
Random Variables
Classical probability deals with expectation values
of numerical random variables, that is, with
Free Probability Theory 417
numerical functions on a space of events and with
their integrals with respect to a probability measure
on the space of events. In noncommutative prob-
ability, the random variables, like quantum-mechan-
ical quantities, are elements of a noncommutative
algebra A over C, with unit 1 2 A, which is
endowed with a linear expectation functional
’ : A !C, so that ’(1) = 1. Frequently, A is a
-algebra of operators on some Hilbert space H and
’(T) = hT , i for some unit vector 2H. We call
(A,’) a noncommutative probability space and the
elements a 2 A, noncommutative random variables.
In this section we shall discuss the basics around the
notion of freeness (Voiculescu 1985), which plays
the role of independence in free probability.
If = (a
i
)
i2I
A is a family of noncommutative
random variables, the role of joint distribution is
played by the collection of noncommutative moments
’(a
i
1
...a
i
n
). This can also be extended by linearity to a
distribution functional
: ChX
i
ji 2 Ii!C, where
ChX
i
ji 2 Ii is the ring of polynomials in noncommu-
tative indeterminates X
i
(i 2 I)and
ðPðX
i
ji 2 IÞÞ ¼ ’ðPða
i
ji 2 IÞÞ
If A is a C
-algebra of operators on H, a = a
2 A
and ’( ) = h , i, the distribution of a can also be
identified with the probability measure
a
on R
a
ð!Þ¼hEð!; a Þ; i
where E( ; a) is the spectral measure of a. Indeed,
then
a
ðPðXÞÞ ¼
Z
PðtÞd
a
ðtÞ
A family (A
i
)
i2I
A,1 2 A
i
of subalgebras is
‘‘free’’ (which is short for freely independent) if
’ða
1
...a
n
Þ¼0
whenever a
j
2 A
i
j
,1 j n, i
j
6¼ i
jþ1
and ’(a
j
) = 0.
(Here it is only required that consecutive a
j
’s be in
different A
i
’s. Thus, we may have i
1
= i
3
, provided
i
1
6¼ i
2
.)
Afamilyofsetsofrandomvariables(!
i
)
i2I
, !
i
A
is free if the algebras A
i
generated by 1 [ { !
i
} are free
in (A,’).
Except for rather trivial situations, free random
variables in (A,’) do not commute.
Note also that, as in the case of classical
independence, if (!
i
)
i2I
are disjoint freely indepen-
dent sets of random variab les, then, if the distribu-
tions
!
i
(i 2 I) are given, the distribution
!
of
! =
S
i2I
!
i
is completely determined.
Example 1 Let the group G be the free product of
its subgroups (G
i
)
i2I
, that is, G is generated by these
subgroups and there is no nontrivial relation among
elements of different G
i
’s. Further, let be the
regular representation (g)e
h
= e
gh
of G on the
Hilbert space with orthonormal basis (e
g
)
g2G
.
Then, with respect to the expectation functional
(T) = hTe
e
, e
e
i on operators on l
2
(G), the sets
((G
i
))
i2I
are freely independent.
Example 2 If H is a complex Hilbert, let
TH=
L
k0
H
k
denote the full Boltzmann Fock
space, with vacuum vector 1 so that H
0
= C1.
If h 2H and 2TH, let l(h) = h denote the
left creat ion operator and ’(X) = hX1, 1 i the
vacuum expectation. Then, if the H
i
(i 2 I)
are pairwise orthogonal subspaces in H, the
-subalgebras of operators generated by l(H
i
) [
l
(H
i
), indexed by i 2 I, are freely independent
with respect to ’.
Free Independence with Amalgamation
over a Subalgebra
The classical notion of conditional independence
also has a free counterpart based on the notion of
free independence with amalgamation over a sub-
algebra. This subject is technically more compl icated
and we will only aim at giving an idea about what
kind of concepts are involved.
In the classical context, if (X, , ) is a probability
space with a -algebra , then the conditional
independence with respect to a -subalgeb ra of
events,
0
, amounts to replacing in the defini-
tion of independence the expectation functional
(which is the integral with respect to ) by the
conditional expectation functional L
1
(X, , )
E
!
L
1
(X,
0
, (
0
)).
In free probability, one considers an extension of
the theory, from the (A,’) framework to an (A,, B)
framework (Voiculescu 1995), where A is an algebra
with unit over C, B 3 1 is a subalgebra, and
: A !B is B–B-bilinear and j
B
= id
B
. Then the
definition of B-freeness (or free independence with
amalgamation over B) of a family of subalgebras
(A
i
)
i2I
, B A
i
A requires that
ða
1
...a
n
Þ¼0
whenever a
j
2 A
i
j
, i
j
6¼ i
jþ1
(1 j n), and (a
j
) = 0.
In the case of a unital -algebra of bounded
operators M with an expectation functional
( ) = h, i which is tracial (i.e., ([m
1
, m
2
]) = 0if
m
1
, m
2
2 M) and given a subalgebra 1 2 N M,as
in the classical theory, there is a certain canonical
construction in operator algebra theory of a ‘‘con-
ditional expectation’’ :
M !
N, where
M,
N are
418 Free Probability Theory
algebras of operators obtained as completion-
separates from M and N. With this construction, in
the trace-state setting there is complete analogy with
the classical notion of conditional independence.
Several other constructions of free probability
have been extended to the (A,, B) B-valued
context.
A group-theoretic example similar to Example 1
can be constructed from a group G which is a free
product with amalgamation over a subgroup H G
of subgroups H G
i
Gi 2 I. Then A is the
algebra constructed from the left-regular representa-
tion of G, whereas B is an algebra constructed from
the left-regular represent ation of H.
The Semicircle Law
In free probability the semicircle law appears as the
limit law in the free central limit theorem
(Voiculescu 1985). Here is a weak, rather algebraic,
version of this fact:
If (a
n
)
n2N
are freely independent in (A,’) and
satisfy the conditions that
’ða
n
Þ¼0ðn 2 NÞ
lim
N!1
N
1
X
1nN
’ a
2
n
¼ 1
sup
n2N
’ a
k
n
¼ C
k
< 1ðk 2 NÞ
then, if S
N
= N
1=2
P
1nN
a
n
, we have the conver-
gence of moments of the distribution of S
N
to the
semicircle distribution
lim
N!1
’ðS
k
N
Þ¼ð2Þ
1
Z
2
2
t
k
ð4 t
2
Þ
1=2
dt
Thus, the semicircle law, given by the density
(2)
1
(4 t
2
)
1=2
on [2, 2] is the free analog of the
(0,1) Gauss law.
Two coincidences involving the semicircle law
should be noted.
The field operators s(h) = 2
1
(l( h) þ l(h)
) on the
Boltzmann Fock space (Example 2) have semicircle
distributions with respect to the vacuum expectati on
( ) = h1, 1i. It turns out that this goes farther: if
H= H
R
R
C is the complexification of a real
Hilbert space, then the map H
R
3 h !s(h) is the
analog in free pro bability of the Gaussian pro cess
over the Hilbert space H
R
(Voiculescu 1985). It is
often called the semicircular process over H
R
. This
points to an important connection of free prob-
ability to the full Boltzmann statistics.
The other coincidence is that the semicircle law is
well known as the Wigner limit distribution of
eigenvalues of large Gaussian random matrices. As
we shall see, this is a clue to a deep connection of
free probability to the large-N limit of random
matrices (Voiculescu 1991).
Free Convolution Operations
In classical probability theory, the distribution of
the sum of two independent random variables is
computed by the convolution product of their
distributions. This has a free probability analog.
If a,b are free random variables in (A,’) with
distributions
a
,
b
: C[X] !C, then the joint
distribution
{a, b}
is completely determined by
a
,
b
andinparticular
aþb
, the distribution of a þ b,
also depends only on
a
,
b
. It follows that there
is an additive free convolution operation
þ
on
distributions so that
a
þ
b
=
aþb
whenever a, b are
free (Voiculescu 1985). The same can be done with
multiplication replacing addition, and this defines the
multiplicative free convolution operation
by
the equation
a
b
=
ab
, when a, b are free
(Voiculescu 1985). A slightly surprising feature of
is that in spite of noncommutativity of a and b,
the multiplicative operation
turns out to be
commutative, which of course is obvious for
þ
.
In the classical context, convolutions are bilinear
operations which can be computed using integrals.
The free convolutions are quite nonlinear and their
computation is via another route, which can also be
explained by a classical analogy. Classically, the
logarithm of the Fourier transform linearizes con-
volution, that is,
log Fð Þ¼log FðÞþlog FðÞ
and we may compute as the ( log F)
1
of
log F( ) þ log F(). The linearizing transform for
þ
is the R-transform (Voiculescu 1986), which is
obtained by the following procedure.
If : C[X] !C is a distribution, let G
(z) = z
1
þ
P
n1
(X
n
)z
n1
, which, in case is a compactly
supported probability measure on R, is the Laurent
series at 1 of the Cauchy transform
Z
dðtÞ
t z
From this, one obtains, by inversion at 1, the series
K
, so that G
(K
(z)) = z and one defines
R
(z) = K
(z) z
1
, which is a power series in z. Then
R
¢
¼ R
þ R
In case the distribution corresponds to a measure,
the formal inversion amounts to inverting an
analytic function.
Free Probability Theory 419
For the multiplicative operation
þ
,itismore
convenient to describe an analog of the Mellin trans-
form, that is, no logarithm will be taken. This is the
S-transform (Voiculescu 1991), obtained as follows.
If : C[X] !C is a distribution with (X) 6¼ 0,
one forms
(z) =
P
n1
(X
n
)z
n
and its inverse
so that
(
(z)) = z. Then
S
ðzÞ¼z
1
ð1 þ zÞ
ðzÞ
has the property that
S
¼ S
S
The free central limit theorem can be easily
proved using the R-transform. Another easy applica-
tion of the R-transform is to find the free analog of
the Poisson law, that is,
lim
n!1
ðð1 a=nÞ
0
þ a=n
1
Þ
¢
where a > 0. The free Poisson law is
¼
ð1 aÞ
0
þ if 0 a 1
if a > 1
where has support [(1 a
1=2
)
2
,(1þ a
1=2
)
2
] and
density (2t)
1
(4a (t (1 þ a))
2
)
1=2
. This distribu-
tion is well known in random matrix theory as the
Marchenko–Pastur distribution, again a coincidenc e
pointing to a random matrix theory connection.
Because probability measures on R are distribu-
tions of self-adjoint operators and a sum of self-
adjoint operators is again such an operator, the
additive free convolution
þ
yields an ope ration on
probability measures on R. Similarly, it can be
shown that
gives rise to operations on probability
measures on {z 2 C jjzj= 1} and on probability
measures on [0,1).
With the R-transform machinery at hand, the free
analogs of many of the classical results around
addition of independent random variables have been
developed (we recommend Voiculescu (1998c) for a
survey of these developments). This includes the
classification of infinitely divisible laws (Levy–
Khintchine type theorem), classification of stable
laws, domains of attraction, and convolution semi-
groups. Note that the free laws are rather different
from the classical ones, but the classification results
are quite parallel, that is, the indexing parameters
are almost the same. The situation is similar in the
multiplicative context. As in the classi cal case, these
results about laws yield in particular processes with
independent increments, which in the free frame-
work are free increments.
As in the classical setting, also in the free setting,
convolution semigroups are connected to differential
equations. In the additive free case, a semigroup is a
family (
t
)
t0
of probability measures on R, so that
tþs
=
t
þ
s
.IfG(t,z) is the Cauchy transform of
t
(which is an analytic function on the half-plane
Im z > 0), the equation (Voiculescu 1986)isa
semilinear complex PDE:
@G
@t
þ R
1
ðGÞ
@G
@z
¼ 0
where R
1
is the R-transform of
1
. In particular,
when
1
is the semicircle law, R
1
(z) = z >0 and
the PDE is a complex Burgers equation in the upper
half-plane.
Noncrossing Partitions
The series expansion of the R-transform
R
ðzÞ¼
X
n0
R
n
ðÞz
n
has as coefficients polynomials R
n
() in the
moments (X
k
). More precisely, assigning to (X
k
)
a degree k, R
n
() is a polynomial of degree n and
R
n
() (X
n
) = polynomial in (X
k
) with k < n.
The linearization property of the R-transform
implies that
R
n
ð
þ
Þ¼R
n
ðÞþR
n
ðÞ
For classical convolution, polynomials with simi-
lar properties satisfying
C
n
ð Þ¼C
n
ðÞþC
n
ðÞ
are called cummulants and satisfy
log ðe
zX
Þ¼
X
n1
C
m
ðÞz
n
There are combinatorial formulas involving the
lattice of all partitions of the set {1, ..., n} which give
the classical cummulants. For free cummulants, like
R
n
() and generalizations of these, there are similar
formulas provided the lattice of all partitions is
replaced by the lattice NC(n)ofnoncrossingpartitions
(Speicher 1998). A partition = (V
1
, ..., V
m
)of
{1, ..., n} is noncrossing if there are no a < b < c < d
so that {a, c} V
k
,{b, d} V
l
and k 6¼ l.
More generally, a family R
(n)
(a
1
, ..., a
n
) of free
cummulants, where a
1
, ..., a
n
are in some (A,’), is
defined recursively as follows (Speicher 1998). For
n = 1, one has R
(1)
(a) = ’(a). If = (V
1
, ..., V
m
) 2
NC(n), where V
k
= {i(1, k) < < i(n
k
, k)}, we
define
R½ða
1
; ...; a
n
Þ¼
Y
1km
R
ðjV
k
jÞ
ða
ið1;kÞ
; ...; a
iðn
k
;kÞ
Þ
420 Free Probability Theory
The recurrence relation for cummulants is then
’ða
1
...a
n
Þ¼
X
2NCðnÞ
R½ða
1
; ...; a
n
Þ
Note that the right-hand side involves only R
(k)
’s
with k n and that actually R
(n)
appears only in
and is equal to R[({1, ..., n})](a
1
, ..., a
n
) (the coars-
est partition).
A key property of R
(n)
(a
1
, ..., a
n
) is that if
{1, ..., n} = q and (a
k
)
k2
,(a
l
)
l2
are freely inde-
pendent, then R
(n)
(a
1
, ..., a
n
) = 0.
If is the distribution of a 2 (A, ’), then the
cummulants R
n
() are given by
R
n
ðÞ¼R
ðnÞ
ða; ...; aÞ
The noncrossing condition on partitions corre-
sponds to a planarity requirement for diagrams and
as such is very sug gestive of connections to planar
diagrams occurring in the constant term of large-N
expansions from random matrix theory and more
generally gauge theory.
For more details on the subject of noncrossing
partitions, we refer the reader to the memoir by
Speicher (1998).
Asymptotic Freeness of Random
Matrices
The explanation for the coincidences between certain
laws in free probability and in random matrix theory
is that freeness occurs asymptotically among random
matrices in the large-N limit (Voiculescu 1991).
Random matrices can be put in a noncommutative
probability framework (A
N
, ’
N
), where
A
N
= L
10
(, M
N
;d)(theN N complex matrix-
valued functions on the probability space (,d)
which are p-integrable for all p 2 [1,1)) and the
expectation functional is
’
N
ðXÞ¼N
1
Z
tr Xð!Þdð !Þ
The basic example is provided by an n-tuple of
Gaussian random matrices (Voiculescu 1991). Let
T
ðNÞ
j
¼ a
ðNÞ
p;q;j
1p; qN
2 N; 1 j n
where a
(N)
p, q; j
= a
(N)
q, p; j
and the a
(N)
p, q; j
1 p q N,
1 j n are (0, N
1
)-Gaussian and independent.
Then (T
(N)
j
)
1jn
as N !1 converges in noncommu-
tative distribution to the freely independent n-tuple
(l(e
j
) þ l
(e
j
))
1jn
in the Boltzmann Fock space
context of Example 2 for an orthonormal system
e
1
, ..., e
n
2H, that is, convergence of moments:
lim
N!1
’
N
T
ðNÞ
i
1
T
ðNÞ
i
k
¼hðlðe
i
1
Þþl
ðe
i
1
ÞÞðlðe
i
k
Þþl
ðe
i
k
ÞÞ1; 1i
In particular, the limit variables (l(e
j
) þ l
(e
j
))
1jn
are free.
More generally, asymptotic freeness of variables
or sets of variables in (A
N
, ’
N
) can be defined
without the existence of a limit distribution, that is,
by requi ring only that the freeness relations among
noncommutative moments hold asymptotically as
N !1.
Note that in these random matrix questions, the
joint classical distribution of an n-tuple of random
matrices (X
(N)
1
, ..., X
(N)
n
)inA
N
is a probability
measure on (M
N
)
n
which contains more informa-
tion than the collection of noncommutative
moments, which is the distribution of the noncom-
mutative variables in (A
N
, ’
N
). In particular, for one
random matrix the classical distribution gives the
joint distribution of all entries , whereas the non-
commutative distribution gives information only
about the distribution of eigenvalues.
From the Gaussian n-tuple using operator techni-
ques much more general asymptotic freeness results
have been obtained. For instance (Voiculesu 1998b):
Let (X
(N)
1
, ..., X
(N)
m
, Y
(N)
1
, ..., Y
(N)
n
)be(m þ n)-
tuples of self-adjoint N N random matrices with
classical joint distribution
N
on (M
sa
N
)
mþn
. Assume
that
N
is invariant under the action of the unitary
group U(N) which takes (X
1
, ..., X
m
, Y
1
, ..., Y
n
)
into (X
1
, ..., X
m
,UY
1
U
, ...,UY
n
U
) and assume
that there is a bound R on the operator norms
kX
(N)
j
k and kY
(N)
j
k independent of N. Then the
sets {X
(N)
1
, ..., X
(N)
m
} and {Y
(N)
1
, ..., Y
(N)
n
} are asymp-
totically free as N !1.
Note that the uniform bound on the operator
norms can be easily replaced by weaker conditions.
Once we know that certain random matrices are
asymptotically free and that the large-N limit in
noncommutative distribution exists, the results of
free probability apply. For instance, if X
(N)
and Y
(N)
are asymptotically free and have limit distributions
and , then the limit distribution of X
(N)
þ Y
(N)
and of X
(N)
Y
(N)
are the free convolutions
þ
and,
respectively,
.
Free probability techniques have also been suc-
cessful in dealing with other questions about the
asymptotic behavior of random matrices.
If T
(N)
1
, ..., T
(N)
n
is an n-tuple of i.i.d. Hermitian
Gaussian random, then the uniform operator norms
Free Probability Theory 421
of polynomials in noncommutative indeterminates
have the property that
lim
N!1
kPðT
ðNÞ
1
; ...; T
ðNÞ
n
Þk
¼kPðlðe
1
Þþlðe
1
Þ
; ...; lðe
n
Þþlðe
n
Þ
Þk
almost surely (Haagerup and Thorbjoernsen).
This result is a far-reaching generalization of the
results about largest eigenvalues of one Gaussian
random matrix. The use of operator-valued free
random variables (with respect to certain subalge-
bra) was an essential ingredient in the proof. Also, in
another direction, freeness of operator-valued free
random variables was used to obtain a free prob-
ability treatment of Gaussian random band matrices
and generalizations of these (Shlyakhtenko 1996).
Finally, quite recently, extensions of the free
probability framework have appeared which are
adapted to the study of fluctuat ions of systems of
random matrices in the large-N limit.
Free Entropy
There are free probability analogs also for information-
theoretic quantities (Voiculescu 1994, 1998a).
Let (f
1
, ..., f
n
)beann-tuple of classi cal numerical
random variables the joint distribution of which has
density p(t
1
, ..., t
n
) with respect to the n-dimensional
Lebesgue measure
n
on R
n
. The entropy quantity
associated by Shannon to (f
1
, ..., f
n
)is
Hðf
1
; ...; f
n
Þ¼
Z
R
n
p log p d
n
The free analog of H(f
1
, ..., f
n
) is the free entropy
quantity (X
1
, ..., X
n
). Here X
j
= X
j
,1 j n, are
noncommutative self-adjoint random variables in
(M,), where M is a -algebra of bounded operators
on a Hilbert space H. The expectation functional in
addition to the positivity properties, equivalent to
the requirement that it can be defined by a unit
vector ( ) = h, i, also has the property of a trace
(XY) = (YX) for all X, Y 2 M. For instance, the
noncommutative random variables arising from the
large-N limit of n-tuples of self-adjoint random
matrices live in noncommutative probability frame-
works (M,) of this kind.
There are two approaches to defining free entropy
and, since there are only partial results about the
equivalence of these approaches, the quantities
obtained are denoted by (X
1
, ..., X
n
)(Voiculescu
1994) and
(X
1
, ..., X
n
)(Voiculescu 1998a). The
quantity is often referred to as the ‘‘microstates
free entropy,’’ its definition being inspired by the
Boltzmann formula S = k log W, whereas the other
entropy, sometimes called ‘‘microstates-free free
entropy,’’ is obtained via a free probability analog
of the Fisher information (Voiculescu 1998a).
The microstates used to define are matricial and
the reason why this choice produced a quantity with
the right behavior with respect to free independence
can be found in the asymptotic freeness properties of
random matrices.
Given X
j
= X
j
2 M,1 j n and m 2 N, k 2 N,
>0 the microstates (X
1
, ..., X
n
; m, k, ) are
n-tuples (A
1
, ..., A
n
) of self-adjoint k k matrices,
such that, for noncommutative moments of order up
to m, we have
jk
1
tr
k
ðA
i
1
...A
i
p
ÞðX
i
1
...X
i
p
Þj <
where 1 p m,1 i
j
n,1 j p.
One obtains (X
1
, ..., X
n
) by taking the infimum
over >0 and m 2 N of
lim sup
k!1
k
2
log vol ð...Þþ
n
2
log k
where vol is the volume on (M
sa
k
)
n
corresponding to
the Hilbert–Schmidt norm Hilbert space structure
(Voiculescu 1994).
When n = 1, there is a simple formula for (X). If
is the probability measure on R which represents
the distribution of X = X
2 M with respe ct to the
expectation , then
ðXÞ¼
ZZ
log js tjdðsÞdðtÞþC
where the exact value of the constant C is
3/4 þ 1/2 log 2.
For n > 1 there is no simple formula for
(X
1
, ..., X
n
), but there are several properties
which provide a better understanding of this
quantity.
If X
j
are such that (X
j
) > 1, then
ðX
1
; ...; X
n
Þ¼ðX
1
ÞþþðX
n
Þ
if and only if X
1
, ..., X
n
are freely independent in
(M, ). Clearly, this property of with respect to
free independence is analogous to the property of
H(f
1
, ..., f
n
) with respect to classical independence.
Further, if F
1
, ..., F
n
are power series in n
noncommuting indeterminates, there is a change-
of-variable formula
ðF
1
ðX
1
; ...; X
n
Þ; ...; F
n
ðX
1
; ...; X
n
ÞÞ
¼ log jdet jðJðFÞÞ þ ðX
1
; ...; X
n
Þ
involving the Kadison–Fuglede positive determinant
jdet j and a certain noncommutative Jacobian
J(F), F = (F
1
, ..., F
n
) defined in M
n
M M
op
,
where M
op
is the opposite algebra of M.(For
422 Free Probability Theory
definitions and the many technical conditions under
which this formula holds, see Voiculescu (1994).)
The free entropy also satisfies semicontinui ty,
subadditivity, and a semicircular bound (analogous
to the classical Gaussian bound) pro perties.
An unexpected feature of is a degeneration of
convexity. If the trace state is a convex combina-
tion =
0
þ (1 )
00
, where
0
,
00
are trace states
and where
0
6¼
00
on the algebra generated by
X
1
, ..., X
n
,andn > 1, then
ðX
1
; ...; X
n
Þ¼1
(for a reference consult the survey Voiculescu
(2002)).
With the free entropy at hand, an important
variational problem can be formulated for the
noncommutative distribution of an n-tuple of self-
adjoint noncommutative random variables
T
1
, ..., T
n
in the tracial context. The quantity to
be maximized is
ðT
1
; ...; T
n
ÞðPðT
1
; ...; T
n
ÞÞ
where P is a given self-adjoint polynomial in
noncommutative indeterminates (see Voiculescu
(2002) for comments on this problem). If n = 1,
this is a classical problem for the logarithmic energy
ZZ
log js tjdðsÞdðtÞ
Z
PðtÞdðtÞ
where is a probability measure on R.
To explain the second approach, based on Fisher
information, we begin by recalling some facts about
Fisher information in the classical context.
If f is a numerical random variable with distribu-
tion given by the density p(t)onR, then
Fisherðf Þ¼
Z
p
0
p
2
p dt ¼
d
dt
1
L
2
ðR;p dtÞ
Here d=dt is the differential operator defined on test
functions in L
2
(R, p dt). Then
p
0
p
¼
d
dt
1
The classical connection to entropy is that the Fisher
information is a derivative of the entropy when the
variable becomes the starting point of a Brownian
motion. This can be written as
Fisherðf Þ¼
d
dt
Hðf þ t
1=2
gÞj
t¼0
where g and f are independent and g is (0, 1)
Gaussian.
The several-variables version is treated by using
partial derivatives.
The analog in free probability of the Fisher
information (Voiculescu 1998a) is obtained by
using the free difference quotient derivations,
which are the appropriate derivations in this
maximally noncommutative setting. On the poly-
nomials in n noncommutative indetermina tes, the
kth partial free difference quotient
@
k
: ChX
1
; ...; X
n
i!ChX
1
; ...; X
n
i
2
is defined on noncommutative monomials by the
formula
@
k
X
i
1
X
i
p
¼
X
fjji
j
¼kg
X
i
1
X
i
j1
X
i
jþ1
X
i
p
If X
j
= X
j
,1 j n, are noncommutative ran-
dom variables in (M, ), which do not satisfy any
nontrivial algebraic relations, to simplify matters we
can assume that M is generated by X
1
, ..., X
n
and
identify M with ChX
1
, ..., X
n
i. The trace state
gives rise to a scalar product hm
1
, m
2
i= (m
2
m
1
)on
M. Let L
2
(M, ) denote the Hilber t space obtained
from M. Then, skipping some technicalities, @
k
will
give rise to a densely defined operator of L
2
(M, )
into L
2
(M, ) L
2
(M, ). If 1 1 is in the domain of
the adjoints @
k
, the free Fisher information of the
n-tuple X
1
, ..., X
n
is defined to be
ðX
1
; ...; X
n
Þ¼
X
1kn
k@
k
ð1 1Þk
2
L
2
ðM;Þ
In case 1 1 is not in the domain of some @
k
, the
free Fisher information is given the v alue 1.
The ‘‘microstates-free free entropy’’
is then
defined by
ðX
1
; ...; X
n
Þ¼
n
2
log 2e
þ
Z
1
0
n
1 þ t
ðX
1
þt
1=2
S
1
; ...; X
n
þt
1=2
S
n
Þ
dt
where S
1
, ..., S
n
are (0, 1)-semicircular and freely
independent and also freely independent of
{X
1
, ..., X
n
}.
For n = 1 it is known that
(X) = (X) and the
free Fisher information is
ðXÞ¼
2
2
3
Z
p
3
ðtÞdt
if p(t) is the density with respect to the Lebesgue
measure of the distribution of X. The computation
of @
1 1 is possible in the one-variable case and up
to a factor the result is (Hp)(X), where Hp is the
Hilbert transform of p.
Free Probability Theory 423
Several of the classical inequalities for the Fisher
information have free probability analogs
(Voiculescu 1998a) (Cramer–Rao inequality, Stam
inequality, information-log-Sobolev inequality, and
others).
For n > 1only
, the easier of the inequal-
ities among and
, has been established (Biane
et al. 2003). This result was obtained based on an
important connection of and
to large deviations.
The deviations studied are for the noncommutative
distributions of n-tuples of matrices in the case of an
n-tuple of Gaussian random matrices. In this context
is related to the quantity to be estimated and
is
related to the rate function.
For more details on free entropy, the reader is
referred to the survey articles by Voiculescu (1998c,
2002).
Concluding Comments
For more details, additional results, an d
bibliography, w e r efer th e reader to the e xposi-
tions in Voiculescu (1998c), Voiculescu et al.
(1992) and Speicher (1998). To get even more
detail, the reader may con sult, beside s the original
papers of the p resent autho r, those of P Biane,
R Speicher, D Shlyakhtenko, K J Dykema, A N ica,
U Haagerup, H Bercovici, L Ge, F Radulescu,
A Guionnet, T Cabanal–Duvillard, M Anshelevich,
to name a few of the main contributors.
Also, via random matrices, there are connections
to physics models (especially large-N 2D Yang–Mills
QCD) in work of I M Singer, M Douglas, D Gross–
R Gopakumar, P Zinn–Justin. In a loose sense, one
may view the noncrossing partitions combinatorics
as related to the work on planar diagrams and the
large-N limit of t’Hooft and Brezin–Itzykson–Parisi–
Zuber in the 1970s.
See also: Large Deviations in Equilibrium Statistical
Mechanics; Large-N and Topological Strings; Random
Matrix Theory in Physics.
Further Reading
Biane P, Capitaine M, and Guionnet A (2003) Large deviation
bounds for matrix Brownian motion. Inventiones Mathematicae
152: 433–459.
Haagerup U and Thorbjoernsen S (2005) A new application of
random matrices: Ext(C
r
(F
2
)) is not a group. Annals of
Mathematics 162(2).
Shlyakhtenko D (1996) Random Gaussian band matrices and
freeness with amalgamation. International Mathematics
Research Notices 20: 1013–1025.
Speicher R (1998) Combinatorial theory of the free product with
amalgamation and operator-valued free probability. Memoirs
of the American Mathematical Society 627.
Voiculescu D (1985) Symmetries of some reduced free product
C
-algebras. In: Araki H, Moore CC, Stratila S-V, and
Voiculescu DV (eds.) Operator Algebras and Their Connec-
tions with Topology and Ergodic Theory, Lecture Notes in
Mathematics,, vol. 1132, pp. 556–588. Springer.
Voiculescu D (1986) Addition of certain non-commuting random
variables. Journal of Functional Analysis 66: 323–346.
Voiculescu D (1987) Multiplication of certain non-commuting
random variables. Journal of Operator Theory 18: 223–235.
Voiculescu D (1991) Limit laws for random matrices and free
products. Inventiones Mathematicae 104: 201–220.
Voiculescu D (1994) The analogues of entropy and of Fisher’s
information measure in free probability theory II. Inventiones
Mathematicae 118: 411–440.
Voiculescu D (1995) Operations on certain noncommuting
operator-valued random variables. Asterisque 232: 243–275.
Voiculescu D (1998a) The analogues of entropy and of Fisher’s
information measure in free probability theory V: Noncommuta-
tive Hilbert transforms. Inventiones Mathematicae 132: 182–227.
Voiculescu D (1998b) A strengthened asymptotic freeness result
for random matrices with applications to free entropy.
International Mathematics Research Notices 1: 41–63.
Voiculescu D (1998c) Lectures on free probability theory. In:
Lectures on Probability Theory and Statistics, Ecole d’Ete de
Probabilites de Saint-Flour XXVIII–1998, Lecture Notes in
Mathematics,, vol. 1738, pp. 280–349. Berlin: Springer.
Voiculescu D (2002) Free entropy. Bulletin of London Mathema-
tical Society 34: 257–278.
Voiculescu D, Dykema KJ, and Nica A (1992) Free Random
Variables, CRM Monograph Series, vol. 1. Providence:
American Mathematical Society.
Frobenius Manifolds see WDVV Equations and Frobenius Manifolds
424 Free Probability Theory
Functional Equations and Integrable Systems
H W Braden, University of Edinburgh, Edinburgh, UK
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Functional equations have a long and interesting
history in connection with math ematical physics and
touch upon many branches of mathematics. The y
have arisen in the context of both classical and
quantum completely integrable systems in several
different ways and we shall survey some of these.
In the great majority of cases functional equations
appear in the integrable system setting as the result
of an ansatz: a particular form of a solution is either
guessed or postulated, the consi stency of which
yields a functional equation. What the ansatz is for
can vary significantly. As outlined below, amongst
others, one may postulate algebraic structures in the
form of the existence of a Lax pair or of conserved
quantities; in the quantum setting, one may postu-
late properties of a ground-state wave function or
the ring of commuting differential operators.
Appearing in this way, functional equations are
really just another of the (significant) tools-of-the-
trade for constructing and disc overing new integr-
able systems. However, as one surveys both the
functional equations and the functions they describe
one sees certain comm on features. The functions are
most frequently associated with an elliptic curve, a
genus-1 abelian variety. One can seek to associate
these to another fundamental ingredient of modern
integrable systems, the Baker–Akhiezer function.
Indeed, very few of the ansa¨ tze made directly
suggest that the systems being constructed will be
completely integrable. This very desirable property
usually is a bonus of the construction and hints of
more fundamental connections. Another fundamen-
tal connection we shall mention is that with
topology. The phase space of a completely integr-
able system is rather special, admitting (generically)
a foliation by tori. The functional equations we
encounter often also characterize the Hirzebruch
genera associated with the index theorems of known
elliptic operators. These are typically evaluated by
Atiyah–Bott fixed-point theorems for circle actions
on the manifold. A general understanding of the
various interconnections has yet to be achieved.
To bring to focus our discussion we shall concen-
trate on functional equations arising from studying
systems with an arbitrary number of particles
(n below). In principle, there could be many different
interactions between the particles and symmetry will
be used to limit these. The use of symmetry is a key
ingredient, often implicit, in the various ansa¨tze we
shall describe. For simplicity, we shall most often focus
on the situation where the particles are identical. In
algebraic terms, we focus on the symmetric group S
n
and root systems of type a
n
; generalizations frequently
exist for other root systems and Weyl groups and we
shall simply note this at the outset.
Lax Pairs
The modern approach to integrable systems is to
utilize a Lax pair, that is, a pair of matrices L, M such
that the zero curvature condition
_
L = [L, M]is
equivalent to the equations of motion. By construc-
tion, Lax pairs produce the conserved quantities tr L
k
.
To establish integrability, one must further show both
that there are enough functionally independent con-
served quantities and that these are in involution.
(R-matrices are the additional ingredient of the
modern approach to establishing involutivity.) Lax
pairs can fail on both counts, and so the construction
of a Lax pair is but the first step in establishing a
system to be completely integrable. The great merit of
the modern approach is that it provides a unified
framework for treating the many disparate completely
integrable systems known. Unfortunately the construc-
tion of a Lax pair is often far from straightforward and
typically hides the ‘‘clever tricks’’ frequently employed
in establishing integrability. In the present context, we
shall outline how functional equations have been used
to construct Lax pairs. The paradigm for this approach
is the Calogero–Moser system.
Beginning with the ansatz (for n n matrices)
L
jk
¼ p
j
jk
þ gð1
jk
ÞAðq
j
q
k
Þ
M
jk
¼ g
jk
X
l 6¼j
Bðq
j
q
l
Þð1
jk
ÞCðq
j
q
k
Þ
2
4
3
5
one finds
_
L = [L, M] yields the equation s of motion
for the Hamiltonian system (n 3)
H ¼
1
2
X
j
p
2
j
þ g
2
X
j<k
Uðq
j
q
k
Þ
UðxÞ¼AðxÞAðxÞþconst:
½1
provided C(x) = A
0
(x), and that A(x) and B(x)
satisfy the functional equation
Aðx þ yÞ½BðxÞBðyÞ ¼ AðxÞA
0
ðyÞAðyÞA
0
ðxÞ½2
This is a particular example of a more general
functional equation whose solution will be described
Functional Equations and Integrable Systems 425