Page 108 - Special Topic Session (STS) - Volume 3

STS518 Steen T.
2.2 Connection to random matrices. As mentioned above one of the most
important aspects of free independence is its significance for large random
matrices. To briefly indicate this, recall that for a Hermitian × −matrix ℋ,
the empirical spectral distribution µ is the probability measure on ℝ given by:

1
= ∑ =1 , where ≤ ··· ≤ are the n real eigenvalues of ℋ (counted

1

with multiplicity), and δc denotes the Dirac measure at a real number c. Now,
let µ and ν be probability measures on ℝ, and for each n in ℕ assume that An
and Bn are two Hermitian random matrices such that the entries of An are
(jointly) independent of those of Bn, and such that µAn(ω) → µ and µBn(ω) → ν
weakly as n → ∞ for almost all ω. Then under various additional (but rather
general) conditions on the distribution of the entries of An and Bn it holds that
() + () ⟶ ⊞ weakly as → ∞ for almost all ω (see e.g. [22] or

[1]). This (meta-) result illustrates the phenomenon that as dimension increases

the assumed (classical) independence between the random matrices An and Bn
is transformed into free independence. Thus free probability provides a
concrete model for the asymptotics of large (classically independent) random
matrices, and the analytic function tools of free probability (described in the
following) can be used to determine these asymptotics; a fact that has been
exploited recently e.g. in the theory of wireless communication.
As we shall demonstrate, the theory of free additive convolution is in many
respects completely parallel to the classical theory. For example we have the
following analog of the classical CLT:

2.3 Free Central Limit Theorem ([20], [8], [23], [24]). Let µ be a probability
measure on R with zero mean and finite variance σ . Then
2

In fact, D 1√nσ 2 (μ ⊞n ) is Lebesgue absolutely continuous for large n, and the
densities converge uniformly to that of the semi-circle distribution.
In Theorem 2.3 we use the notation Dcµ for the scaling (or dilation) of a
measure µ by the constant c. Theorem 2.3 illustrates the phenomenon that the
role of the Gaussian distribution in classical probability is played by the
semicircle distribution in free probability. In a similar fashion the role of the
Poisson distribution in classical probability theory is in many respects played
by the Marchenko-Pastur distribution in free probability.

2.4 Free Poisson Limit Theorem ([20]). For any positive number λ, it holds
that

where ν is the Marchenko-Pastur Law:

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