8b. Free probability

[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

As a solution to the difficulties met in the previous section, let us turn to probability. This is surely not geometry, in a standard sense, but at a more advanced level, geometry that is. For instance if you have a quantum manifold [math]X[/math], and you want to talk about its Laplacian, or its Dirac operator, you will certainly need to know a bit about [math]L^2(X)[/math]. And isn't advanced measure theory the same as probability theory, hope we agree on this.

Let us start our discussion with something that we know since chapter 5:

Definition

Let [math]A[/math] be a [math]C^*[/math]-algebra, given with a trace [math]tr:A\to\mathbb C[/math].

• The elements [math]a\in A[/math] are called random variables.
• The moments of such a variable are the numbers [math]M_k(a)=tr(a^k)[/math].
• The law of such a variable is the functional [math]\mu:P\to tr(P(a))[/math].

Here the exponent [math]k=\circ\bullet\bullet\circ\ldots[/math] is as before a colored integer, with the powers [math]a^k[/math] being defined by multiplicativity and the usual formulae, namely:

[[math]] a^\emptyset=1\quad,\quad a^\circ=a\quad,\quad a^\bullet=a^* [[/math]]

As for the polynomial [math]P[/math], this is a noncommuting [math]*[/math]-polynomial in one variable:

[[math]] P\in\mathbb C \lt X,X^* \gt [[/math]]

Generally speaking, the above definition is something quite abstract, but there is no other way of doing things, at least at this level of generality. However, in the special case where our variable [math]a\in A[/math] is self-adjoint, or more generally normal, we have:

Proposition

The law of a normal variable [math]a\in A[/math] can be identified with the corresponding spectral measure [math]\mu\in\mathcal P(\mathbb C)[/math], according to the following formula,

[[math]] tr(f(a))=\int_{\sigma(a)}f(x)d\mu(x) [[/math]]

valid for any [math]f\in L^\infty(\sigma(a))[/math], coming from the measurable functional calculus. In the self-adjoint case the spectral measure is real, [math]\mu\in\mathcal P(\mathbb R)[/math].

Show Proof

This is something that we again know well, either from chapter 5, or simply from chapter 3, coming from the spectral theorem for normal operators.

■

Let us discuss now independence, and its noncommutative versions. As a starting point, we have the following update of the classical notion of independence:

Definition

We call two subalgebras [math]B,C\subset A[/math] independent when the following condition is satisfied, for any [math]x\in B[/math] and [math]y\in C[/math]:

[[math]] tr(xy)=tr(x)tr(y) [[/math]]

Equivalently, the following condition must be satisfied, for any [math]x\in B[/math] and [math]y\in C[/math]:

[[math]] tr(x)=tr(y)=0\implies tr(xy)=0 [[/math]]

Also, [math]b,c\in A[/math] are called independent when [math]B= \lt b \gt [/math] and [math]C= \lt c \gt [/math] are independent.

It is possible to develop some theory here, but this leads to the usual CLT. As a much more interesting notion now, we have Voiculescu's freeness :

Definition

Given a pair [math](A,tr)[/math], we call two subalgebras [math]B,C\subset A[/math] free when the following condition is satisfied, for any [math]x_i\in B[/math] and [math]y_i\in C[/math]:

[[math]] tr(x_i)=tr(y_i)=0\implies tr(x_1y_1x_2y_2\ldots)=0 [[/math]]

Also, [math]b,c\in A[/math] are called free when [math]B= \lt b \gt [/math] and [math]C= \lt c \gt [/math] are free.

As a first observation, there is a certain lack of symmetry between Definition 8.12 and Definition 8.13, because the latter does not include an explicit formula for quantities of type [math]tr(x_1y_1x_2y_2\ldots)[/math]. But this can be done, the precise result being as follows:

Proposition

If [math]B,C\subset A[/math] are free, the restriction of [math]tr[/math] to [math] \lt B,C \gt [/math] can be computed in terms of the restrictions of [math]tr[/math] to [math]B,C[/math]. To be more precise, we have

[[math]] tr(x_1y_1x_2y_2\ldots)=P\Big(\{tr(x_{i_1}x_{i_2}\ldots)\}_i,\{tr(y_{j_1}y_{j_2}\ldots)\}_j\Big) [[/math]]

where [math]P[/math] is certain polynomial, depending on the length of [math]x_1y_1x_2y_2\ldots\,[/math], having as variables the traces of products [math]x_{i_1}x_{i_2}\ldots[/math] and [math]y_{j_1}y_{j_2}\ldots\,[/math], with [math]i_1 \lt i_2 \lt \ldots[/math] and [math]j_1 \lt j_2 \lt \ldots[/math]

Show Proof

With [math]x'=x-tr(x)[/math], we can start our computation as follows:

[[math]] \begin{eqnarray*} tr(x_1y_1x_2y_2\ldots) &=&tr\big[(x_1'+tr(x_1))(y_1'+tr(y_1))(x_2'+tr(x_2))\ldots\big]\\ &=&tr(x_1'y_1'x_2'y_2'\ldots)+{\rm other\ terms}\\ &=&{\rm other\ terms} \end{eqnarray*} [[/math]]

Thus, we are led to a kind of recurrence, and this gives the result.

■

Let us discuss now some examples of independence and freeness. We first have the following result, from , which is something elementary:

Proposition

Given two algebras [math](A,tr)[/math] and [math](B,tr)[/math], the following hold:

• [math]A,B[/math] are independent inside their tensor product [math]A\otimes B[/math], endowed with its canonical tensor product trace, given on basic tensors by [math]tr(a\otimes b)=tr(a)tr(b)[/math].
• [math]A,B[/math] are free inside their free product [math]A*B[/math], endowed with its canonical free product trace, given by the formulae in Proposition 8.14.

Show Proof

Both the assertions are indeed clear from definitions, with just some standard discussion needed for (2), in connection with the free product trace. See .

■

More concretely now, we have the following result, also from Voiculescu :

Proposition

We have the following results, valid for group algebras:

• [math]L(\Gamma),L(\Lambda)[/math] are independent inside [math]L(\Gamma\times\Lambda)[/math].
• [math]L(\Gamma),L(\Lambda)[/math] are free inside [math]L(\Gamma*\Lambda)[/math].

Show Proof

In order to prove these results, we can use the general results in Proposition 8.15, along with the following two isomorphisms, which are both standard:

[[math]] L(\Gamma\times\Lambda)=L(\Lambda)\otimes L(\Gamma)\quad,\quad L(\Gamma*\Lambda)=L(\Lambda)*L(\Gamma) [[/math]]

Alternatively, we can check the independence and freeness formulae on group elements, which is something trivial, and then conclude by linearity. See .

■

We have already seen limiting theorems in classical probability, in chapter 6. In order to deal now with freeness, let us develop some tools. First, we have:

Proposition

We have a well-defined operation [math]\boxplus[/math], given by

[[math]] \mu_a\boxplus\mu_b=\mu_{a+b} [[/math]]

with [math]a,b[/math] being free, called free convolution.

Show Proof

We need to check here that if [math]a,b[/math] are free, then the distribution [math]\mu_{a+b}[/math] depends only on the distributions [math]\mu_a,\mu_b[/math]. But for this purpose, we can use the formula in Proposition 8.14. Indeed, by plugging in arbitrary powers of [math]a,b[/math] as variables [math]x_i,y_j[/math], we obtain a family of formulae of the following type, with [math]Q[/math] being certain polyomials:

[[math]] tr(a^{k_1}b^{l_1}a^{k_2}b^{l_2}\ldots)=P\Big(\{tr(a^k)\}_k,\{tr(b^l)\}_l\Big) [[/math]]

Thus the moments of [math]a+b[/math] depend only on the moments of [math]a,b[/math], and the same argument shows that the same holds for [math]*[/math]-moments, and this gives the result.

■

In order to advance now, we would need an analogue of the Fourier transform, or rather of the log of the Fourier transform. Quite remarkably, such a transform exists indeed, the precise result here, due to Voiculescu , being as follows:

Theorem

Given a probability measure [math]\mu[/math], define its [math]R[/math]-transform as follows:

[[math]] G_\mu(\xi)=\int_\mathbb R\frac{d\mu(t)}{\xi-t}\implies G_\mu\left(\ R_\mu(\xi)+\frac{1}{\xi}\right)=\xi [[/math]]

The free convolution operation is then linearized by the [math]R[/math]-transform.

Show Proof

This is something quite tricky, the idea being as follows:

(1) In order to model the free convolution, the best is to use creation operators on free Fock spaces, corresponding to the semigroup von Neumann algebras [math]L(\mathbb N^{*k})[/math]. Indeed, we have some freeness here, a bit in the same way as in the free group algebras [math]L(F_k)[/math].

(2) The point now, motivating this choice, is that the variables of type [math]S^*+f(S)[/math], with [math]S\in L(\mathbb N)[/math] being the shift, and with [math]f\in\mathbb C[X][/math] being an arbitrary polynomial, are easily seen to model in moments all the possible distributions [math]\mu:\mathbb C[X]\to\mathbb C[/math].

(3) Now let [math]f,g\in\mathbb C[X][/math] and consider the variables [math]S^*+f(S)[/math] and [math]T^*+g(T)[/math], where [math]S,T\in L(\mathbb N*\mathbb N)[/math] are the shifts corresponding to the generators of [math]\mathbb N*\mathbb N[/math]. These variables are free, and by using a [math]45^\circ[/math] argument, their sum has the same law as [math]S^*+(f+g)(S)[/math].

(4) Thus the operation [math]\mu\to f[/math] linearizes the free convolution. We are therefore left with a computation inside [math]L(\mathbb N)[/math], which is elementary, and whose conclusion is that [math]R_\mu=f[/math] can be recaptured from [math]\mu[/math] via the Cauchy transform [math]G_\mu[/math], as in the statement.

■

With the above linearization technology in hand, we can now establish the following remarkable free analogue of the CLT, also due to Voiculescu :

Theorem (Free CLT)

Given self-adjoint variables [math]x_1,x_2,x_3,\ldots,[/math] which are f.i.d., centered, with variance [math]t \gt 0[/math], we have, with [math]n\to\infty[/math], in moments,

[[math]] \frac{1}{\sqrt{n}}\sum_{i=1}^nx_i\sim\gamma_t [[/math]]

\ where [math]\gamma_t=\frac{1}{2\pi t}\sqrt{4t-x^2}dx[/math] is the Wigner semicircle law of parameter [math]t[/math].

Show Proof

We follow the same idea as in the proof of the CLT:

(1) At [math]t=1[/math], the [math]R[/math]-transform of the variable in the statement can be computed by using the linearization property from Theorem 8.18, and is given by:

[[math]] R(\xi) =nR_x\left(\frac{\xi}{\sqrt{n}}\right) \simeq\xi [[/math]]

(2) On the other hand, some standard computations show that the Cauchy transform of the Wigner law [math]\gamma_1[/math] satisfies the following equation:

[[math]] G_{\gamma_1}\left(\xi+\frac{1}{\xi}\right)=\xi [[/math]]

Thus, by using Theorem 8.18, we have the following formula:

[[math]] R_{\gamma_1}(\xi)=\xi [[/math]]

(3) We conclude that the laws in the statement have the same [math]R[/math]-transforms, and so they are equal. The passage to the general case, [math]t \gt 0[/math], is routine, by dilation.

■

In the complex case now, we have a similar result, also from , as follows:

Theorem (Free CCLT)

Given random variables [math]x_1,x_2,x_3,\ldots[/math] which are f.i.d., centered, with variance [math]t \gt 0[/math], we have, with [math]n\to\infty[/math], in moments,

[[math]] \frac{1}{\sqrt{n}}\sum_{i=1}^nx_i\sim\Gamma_t [[/math]]

where [math]\Gamma_t=law\big((a+ib)/\sqrt{2}\big)[/math], with [math]a,b[/math] being free, each following the Wigner semicircle law [math]\gamma_t[/math], is the Voiculescu circular law of parameter [math]t[/math].

Show Proof

This follows indeed from the free CLT, established before, simply by taking real and imaginary parts of all the variables involved.

■

Now that we are done with the basic results in continuous case, let us discuss as well the discrete case. We can establish a free version of the PLT, as follows:

Theorem (Free PLT)

The following limit converges, for any [math]t \gt 0[/math],

[[math]] \lim_{n\to\infty}\left(\left(1-\frac{t}{n}\right)\delta_0+\frac{t}{n}\delta_1\right)^{\boxplus n} [[/math]]

and we obtain the Marchenko-Pastur law of parameter [math]t[/math],

[[math]] \pi_t=\max(1-t,0)\delta_0+\frac{\sqrt{4t-(x-1-t)^2}}{2\pi x}\,dx [[/math]]

also called free Poisson law of parameter [math]t[/math].

Show Proof

Let [math]\mu[/math] be the measure in the statement, appearing under the convolution sign. The Cauchy transform of this measure is elementary to compute, given by:

[[math]] G_{\mu}(\xi)=\left(1-\frac{t}{n}\right)\frac{1}{\xi}+\frac{t}{n}\cdot\frac{1}{\xi-1} [[/math]]

By using Theorem 8.18, we want to compute the following [math]R[/math]-transform:

[[math]] R =R_{\mu^{\boxplus n}}(y) =nR_\mu(y) [[/math]]

We know that the equation for this function [math]R[/math] is as follows:

[[math]] \left(1-\frac{t}{n}\right)\frac{1}{y^{-1}+R/n}+\frac{t}{n}\cdot\frac{1}{y^{-1}+R/n-1}=y [[/math]]

With [math]n\to\infty[/math] we obtain from this the following formula:

[[math]] R=\frac{t}{1-y} [[/math]]

But this being the [math]R[/math]-transform of [math]\pi_t[/math], via some calculus, we are done.

■

As a first application now of all this, following Voiculescu , we have:

Theorem

Given a sequence of complex Gaussian matrices [math]Z_N\in M_N(L^\infty(X))[/math], having independent [math]G_t[/math] variables as entries, with [math]t \gt 0[/math], we have

[[math]] \frac{Z_N}{\sqrt{N}}\sim\Gamma_t [[/math]]

in the [math]N\to\infty[/math] limit, with the limiting measure being Voiculescu's circular law.

Show Proof

We know from chapter 6 that the asymptotic moments are:

[[math]] M_k\left(\frac{Z_N}{\sqrt{N}}\right)\simeq t^{|k|/2}|\mathcal{NC}_2(k)| [[/math]]

On the other hand, the free Fock space analysis done in the proof of Theorem 8.18 shows that we have, with the notations there, the following formulae:

[[math]] S+S^*\sim\gamma_1\quad,\quad S+T^*\sim\Gamma_1 [[/math]]

By doing some combinatorics, this shows that an abstract noncommutative variable [math]a\in A[/math] is circular, following the law [math]\Gamma_t[/math], precisely when its moments are:

[[math]] M_k(a)=t^{|k|/2}|\mathcal{NC}_2(k)| [[/math]]

Thus, we are led to the conclusion in the statement. See .

■

Next in line, comes the main result of Voiculescu in , as follows:

Theorem

Given a family of sequences of Wigner matrices,

[[math]] Z^i_N\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

with pairwise independent entries, each following the complex normal law [math]G_t[/math], with [math]t \gt 0[/math], up to the constraint [math]Z_N^i=(Z_N^i)^*[/math], the rescaled sequences of matrices

[[math]] \frac{Z^i_N}{\sqrt{N}}\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

become with [math]N\to\infty[/math] semicircular, each following the Wigner law [math]\gamma_t[/math], and free.

Show Proof

We can assume that we are dealing with 2 sequences of matrices, [math]Z_N,Z_N'[/math]. In order to prove the asymptotic freeness, consider the following matrix:

[[math]] Y_N=\frac{1}{\sqrt{2}}(Z_N+iZ_N') [[/math]]

This is then a complex Gaussian matrix, so by using Theorem 8.22, we have:

[[math]] \frac{Y_N}{\sqrt{N}}\sim\Gamma_t [[/math]]

We are therefore in the situation where [math](Z_N+iZ_N')/\sqrt{N}[/math], which has asymptotically semicircular real and imaginary parts, converges to the distribution of a free combination of such variables. Thus [math]Z_N,Z_N'[/math] become asymptotically free, as desired.

■

Getting now to the complex case, we have a similar result here, as follows:

Theorem

Given a family of sequences of complex Gaussian matrices,

[[math]] Z^i_N\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

with pairwise independent entries, each following the law [math]G_t[/math], with [math]t \gt 0[/math], the matrices

[[math]] \frac{Z^i_N}{\sqrt{N}}\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

become with [math]N\to\infty[/math] circular, each following the Voiculescu law [math]\Gamma_t[/math], and free.

Show Proof

This follows indeed from Theorem 8.23, which applies to the real and imaginary parts of our complex Gaussian matrices, and gives the result.

■

Finally, we have as well a similar result for the Wishart matrices, as follows:

Theorem

Given a family of sequences of complex Wishart matrices,

[[math]] Z^i_N=Y^i_N(Y^i_N)^*\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

with each [math]Y^i_N[/math] being a [math]N\times M[/math] matrix, with entries following the normal law [math]G_1[/math], and with all these entries being pairwise independent, the rescaled sequences of matrices

[[math]] \frac{Z^i_N}{N}\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

become with [math]M=tN\to\infty[/math] Marchenko-Pastur, each following the law [math]\pi_t[/math], and free.

Show Proof

Here the first assertion is the Marchenko-Pastur theorem, from chapter 6, and the second assertion follows from Theorem 8.23, or from Theorem 8.24.

■

Let us develop now some further limiting theorems, classical and free. We have the following definition, extending the Poisson limit theory developed before:

Definition

Associated to any compactly supported positive measure [math]\rho[/math] on [math]\mathbb C[/math] are the probability measures

[[math]] p_\rho=\lim_{n\to\infty}\left(\left(1-\frac{c}{n}\right)\delta_0+\frac{1}{n}\rho\right)^{*n}\quad,\quad \pi_\rho=\lim_{n\to\infty}\left(\left(1-\frac{c}{n}\right)\delta_0+\frac{1}{n}\rho\right)^{\boxplus n} [[/math]]

where [math]c=mass(\rho)[/math], called compound Poisson and compound free Poisson laws.

In what follows we will be interested in the case where [math]\rho[/math] is discrete, as is for instance the case for [math]\rho=t\delta_1[/math] with [math]t \gt 0[/math], which produces the Poisson and free Poisson laws. The following result allows one to detect compound Poisson/free Poisson laws:

Proposition

For [math]\rho=\sum_{i=1}^sc_i\delta_{z_i}[/math] with [math]c_i \gt 0[/math] and [math]z_i\in\mathbb C[/math], we have

[[math]] F_{p_\rho}(y)=\exp\left(\sum_{i=1}^sc_i(e^{iyz_i}-1)\right)\quad,\quad R_{\pi_\rho}(y)=\sum_{i=1}^s\frac{c_iz_i}{1-yz_i} [[/math]]

where [math]F,R[/math] denote respectively the Fourier transform, and Voiculescu's [math]R[/math]-transform.

Show Proof

Let [math]\mu_n[/math] be the measure appearing in Definition 8.26. We have:

[[math]] \begin{eqnarray*} F_{\mu_n}(y)=\left(1-\frac{c}{n}\right)+\frac{1}{n}\sum_{i=1}^sc_ie^{iyz_i} &\implies&F_{\mu_n^{*n}}(y)=\left(\left(1-\frac{c}{n}\right)+\frac{1}{n}\sum_{i=1}^sc_ie^{iyz_i}\right)^n\\ &\implies&F_{p_\rho}(y)=\exp\left(\sum_{i=1}^sc_i(e^{iyz_i}-1)\right) \end{eqnarray*} [[/math]]

In the free case we can use a similar method, and we obtain the above formula.

■

We have the following result, providing an alternative to Definition 8.26, which will be our formulation here of the Compond Poisson Limit Theorem, classical and free:

Theorem (CPLT)

For [math]\rho=\sum_{i=1}^sc_i\delta_{z_i}[/math] with [math]c_i \gt 0[/math] and [math]z_i\in\mathbb C[/math], we have

[[math]] p_\rho/\pi_\rho={\rm law}\left(\sum_{i=1}^sz_i\alpha_i\right) [[/math]]

where the variables [math]\alpha_i[/math] are Poisson/free Poisson[math](c_i)[/math], independent/free.

Show Proof

This follows indeed from the fact that the the Fourier/[math]R[/math]-transform of the variable in the statement is given by the formulae in Proposition 8.27.

■

Following , , we will be interested here in the main examples of classical and free compound Poisson laws, which are constructed as follows:

Definition

The Bessel and free Bessel laws are the compound Poisson laws

[[math]] b^s_t=p_{t\varepsilon_s}\quad,\quad \beta^s_t=\pi_{t\varepsilon_s} [[/math]]

where [math]\varepsilon_s[/math] is the uniform measure on the [math]s[/math]-th roots unity. In particular:

• At [math]s=1[/math] we obtain the usual Poisson and free Poisson laws, [math]p_t,\pi_t[/math].
• At [math]s=2[/math] we obtain the “real” Bessel and free Bessel laws, denoted [math]b_t,\beta_t[/math].
• At [math]s=\infty[/math] we obtain the “complex” Bessel and free Bessel laws, denoted [math]B_t,\mathfrak B_t[/math].

There is a lot of theory regarding these laws, and we refer here to , , where these laws were introduced. We will be back to these laws, in a moment.

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].