Free integration

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5a. Weingarten formula

We have seen so far that [math]\mathbb R^N,\mathbb C^N[/math] have no free analogues, in an analytic sense, but that the “basic geometry” of [math]\mathbb R^N,\mathbb C^N[/math], taken in a somewhat abstract sense, does have free analogues, that we can informally call “basic geometry” of [math]\mathbb R^N_+,\mathbb C^N_+[/math]. Thus, we have 4 main geometries, classical/free, real/complex, forming a diagram as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ \mathbb R^N_+\ar[r]&\mathbb C^N_+\\ \mathbb R^N\ar[u]\ar[r]&\mathbb C^N\ar[u] } [[/math]]

In this second part of the present book, we develop the geometry of [math]\mathbb R^N_+,\mathbb C^N_+[/math]. To be more precise, each of these free geometries consists so far of 4 objects, namely a sphere [math]S[/math], a torus [math]T[/math], a unitary group [math]U[/math], and a reflection group [math]K[/math]. We must on one hand study [math]S,T,U,K[/math], from a geometric perspective, and on the other hand construct other “free manifolds”, as for instance suitable homogeneous spaces, and study them too.

Observe that all this is not exactly related to the axiomatization work from chapters 1-4. We will of course heavily use the various things that we learned there, but basically, what we want to do here is something new. We want to develop free geometry, real and complex, and our goals will be very explicit. As a basic question here, we have: \begin{question} What is a free manifold? \end{question} Unfortunately this is a difficult question, whose solution is not known yet. As an illustration, even in the quantum group case, after 30 long years of work on free quantum groups, it is still not known what a free quantum group exactly is. A quite reasonable definition seems to be [math]S_N^+\subset G\subset U_N^+[/math], but then comes the conjecture that such a quantum group must be easy, with this conjecture being guaranteed to be non-trivial.

In short, modesty. We have as starting point [math]S,T,U,K[/math], and this is not that bad, and we will slowly enlarge our menagery of free manifolds, not to the point of solving Question 5.1, but at least to the point of understanding what this question says.

For going ahead with more modesty, let us take [math]N=2[/math], in the real case. We know what the free circle [math]\bigcirc[/math] and the free square [math]\square[/math] are, and we also know what the symmetries of these free [math]\bigcirc[/math] and free [math]\square[/math] are, and the question is, shall we be awarded a PhD in noncommutative geometry for that. Ironically, probably yes, such basic things being not necessarily known by everyone. But leaving now aside academia and politics, the truth is that we are somewhere at the level of the ancient Greeks. Or even below, because the Greeks knew for instance what a conic is, along with many other things.

Which brings us into a second question, what kind of manifolds shall we look at, and what kind of geometry do we want to develop. A look at what we have, [math]S,T,U,K[/math], does not help much, because these are really very basic manifolds, having all geometric properties that you can ever dream of, and therefore belonging to all geometric theories that you can ever imagine. And so, we need some kind of plan here.

Looking at the story of classical geometry, you would say why not doing some algebraic geometry, reaching first to the level of the ancient Greeks, and then going up into more complicated things, towards analogues of what was doing the Italian school. But this is in fact completely unreasonable, because somewhere between ancient Greeks and more modern Italians we had Newton, who axiomatized classical mechanics by using geometry. And are we here for axiomatizing quantum mechanics by using free geometry, most likely leading to a Nobel Prize in physics, or shall we aim for something more modest.

Well, looks like we are completely lost. Again, we know what the free analogues of [math]\bigcirc[/math] and [math]\square[/math] are, and the question is, with this tremendous piece of knowledge, what's next. Fortunately there are already people who have thought about such things, in slightly different noncommutative geometry contexts, and we have, as a key piece of advice: \begin{fact}[Connes principle] Manifolds should be Riemannian. \end{fact} Here we are talking of course about noncommutative manifolds, because in what concerns the classical manifolds, this principle goes back to Riemann himself. Or perhaps to Weyl, who is usually credited for pointing out the beauty and importance of the Riemannian manifolds, among all sorts of other manifolds, available at that time.

This being said, how can a free manifold be Riemannian, because we already know, as explained on several occasions in chapters 1-4, that such free manifolds are not smooth. And checking the mathematics and physics literature here, in look for new ideas, does not help much, because Riemannian geometry is always associated, in mathematics and physics, with all kinds of complicated differential geometry computations.

As a last-ditch attempt, let us ask instead an engineer. And the engineer actually answers something which is very interesting for us, namely: \begin{fact}[Engineer's take] Riemannian means that you can integrate over it. \end{fact} This is of course, technically speaking, not exactly correct. But hey, we are deep into the mud, and open to any piece of valuable advice. And valuable advice this is, because we know how to integrate over [math]S,T,U,K[/math], and so we should look for similar manifolds, having a sort of Haar measure, that we can compute via a Weingarten formula.

Be said in passing, if you're not familiar with engineers and engineering, what our engineer friend says in Fact 5.3 is in fact something quite subtle, with “you can integrate” rather meaning “your computer can integrate”. Which is something which really fits with what we are doing, because the Weingarten formula can be implemented on a computer, and so is ready for “quantum engineering”, whatever that might mean.

All this looks good, but as a last piece of philosophy now, aren't we going into some kind of extreme, with respect to smoothness, by following engineer's advice. Indeed, if we adopt this viewpoint, what shall we then think of the Riemannian manifolds which are smooth, but lack an efficient integration formula over them, such as a Weingarten formula? This is not en easy question, to put it this way, and in the lack of any academic willing to discuss such things, we will have to ask the cat. And cat says: \begin{fact}[Cat's take] Some manifolds are more Riemannian than other. \end{fact} Which sounds very wise, there is now agreement between everyone involved so far. Be said in passing, what cat says agrees as well with Nash , suggesting that the noncommutative Riemannian manifolds having coordinates are “more Riemannian” that those not having coordinates. And also with von Neumann , teaching us the noncommutative spaces having trace functionals [math]tr:L^\infty(X)\to\mathbb C[/math] are perhaps “more Riemannian” than those lacking this property. And also with Jones , and with Voiculescu , whose theories need trace functionals [math]tr:L^\infty(X)\to\mathbb C[/math], and with the “quality” of such a trace functional being directly related to the quality of your investigations.

In short, problem solved. As a last thing, however, more on smoothness. Classical geometry teaches us that smoothness comes in several flavors, [math]C^1,C^2,\ldots,C^\infty[/math], with the mathematical reasons behind this being usually complicated, such as solutions of PDE, or singularities of algebraic manifolds, and so on, and with the physical reasons behind this being even more complicated, basically coming from statistical mechanics. So if there is one thing to be said, “some manifolds are more smooth than other”. Obviously.

In what regards now our free manifolds, let us not forget that these have, as mentioned on several occasions in chapters 1-4, a Laplacian [math]\Delta[/math]. However, no one knows how to construct this Laplacian in general, nor how to use it in relation to integration, nor how to use it in order to have some PDE running, on these manifolds. But one day, all this will be done, and our free manifolds will be entitled to be called “half-smooth”.

So, this will be our philosophy, for the next 100 pages to follow. As usual when regarding controversies, we can only recommend more reading on this, geometry at large, story of geometry, and Riemannian manifolds and related topics. Good references here are Shafarevich , do Carmo and Arnold for geometry, then von Neumann or Blackadar for operator algebras, and then Connes and Connes-Marcolli for a mix of operator algebras and geometry, complemented perhaps with Gracia-Bond\'ia-Várilly-Figueroa and Landi . And don't forget about Nash .

Back to work now, our first task will be that of explaining how to integrate over [math]S,T,U,K[/math]. In order to integrate over [math]U,K[/math], we can use the Weingarten formula , , whose general quantum group formulation, from , is as follows:

Theorem

Assuming that a closed subgroup [math]G\subset U_N^+[/math] is easy, coming from a category of partitions [math]D\subset P[/math], we have the Weingarten formula

[[math]] \int_Gu_{i_1j_1}^{e_1}\ldots u_{i_kj_k}^{e_k}=\sum_{\pi,\sigma\in D(k)}\delta_\pi(i)\delta_\sigma(j)W_{kN}(\pi,\sigma) [[/math]]

where [math]\delta[/math] are Kronecker type symbols, and where the Weingarten matrix

[[math]] W_{kN}=G_{kN}^{-1} [[/math]]

is the inverse of the Gram matrix [math]G_{kN}(\pi,\sigma)=N^{|\pi\vee\sigma|}[/math]. This formula applies to all classical and free unitary and reflection groups [math]U,K[/math], which are all easy.

Show Proof

We know the Weingarten formula from chapter 3, the idea being that the integrals in the statement form the projection on the following space:

[[math]] Fix(u^{\otimes k})=span\left(\xi_\pi\Big|\pi\in D(k)\right) [[/math]]

As for the easiness property of our various classical and free unitary and reflection groups [math]U,K[/math], this is something that we know too from chapter 3.

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Regarding now the integration over the tori [math]T[/math], this is something very simple, because we can use here the following fact, coming from the definition of group algebras:

Theorem

Given a finitely generated discrete group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], the integrals over the corresponding torus [math]T=\widehat{\Gamma}[/math] are given by

[[math]] \int_Tg_{i_1}^{e_1}\ldots g_{i_k}^{e_k}=\delta_{g_{i_1}^{e_1}\ldots g_{i_k}^{e_k},1} [[/math]]

for any indices [math]i_r\in\{1,\ldots,N\}[/math] and any exponents [math]e_r\in\''ptyset,*\''[/math], with the Kronecker symbol on the right being a usual one, computed inside the group [math]\Gamma[/math].

Show Proof

This is something clear, coming from the fact that the Haar integration over the torus [math]T=\widehat{\Gamma}[/math] is given by the following formula:

[[math]] \int_Tg=\delta_{g1} [[/math]]

Indeed, this formula defines a functional on the algebra [math]C(T)=C^*(\Gamma)[/math], which is obviously left and right invariant, and so is the Haar functional.

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Finally, regarding [math]S[/math], here the integrals appear as particular cases of the integrals over [math]U[/math], as explained in chapter 3, and we have a Weingarten formula, as follows:

Theorem

The integration over a sphere [math]S[/math], which is such that [math]U=G^+(S)[/math] is easy, coming from a category of pairings [math]D[/math], is given by the Weingarten formula

[[math]] \int_Sx_{i_1}^{e_1}\ldots x_{i_k}^{e_k}=\sum_\pi\sum_{\sigma\leq\ker i}W_{kN}(\pi,\sigma) [[/math]]

with [math]\pi,\sigma\in D(k)[/math], where [math]W_{kN}=G_{kN}^{-1}[/math] is the inverse of [math]G_{kN}(\pi,\sigma)=N^{|\pi\vee\sigma|}[/math]. This formula applies to all classical and free spheres [math]S[/math], whose [math]U=G^+(S)[/math] are easy.

Show Proof

This is something that we know too from chapter 3, coming from the definition of the integration functional over [math]S[/math], as being the following composition:

[[math]] \int_S:C(S)\to C(U)\to\mathbb C [[/math]]

Indeed, with this description of the integration functional in mind, we can compute this functional via the Weingarten formula for [math]U[/math], from Theorem 5.5, as follows:

[[math]] \begin{eqnarray*} \int_Sx_{i_1}^{e_1}\ldots x_{i_k}^{e_k} &=&\int_Uu_{1i_1}^{e_1}\ldots u_{1i_k}^{e_k}\\ &=&\sum_{\pi,\sigma\in D(k)}\delta_\pi(1)\delta_\sigma(i)W_{kN}(\pi,\sigma)\\ &=&\sum_\pi\sum_{\sigma\leq\ker i}W_{kN}(\pi,\sigma) \end{eqnarray*} [[/math]]

Thus, we are led to the formula in the statement.

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Summarizing, we know how to integrate over [math]S,T,U,K[/math], and with the remark, in relation with Fact 5.3, that our integration methods can be implemented on a computer, as engineers love them. Indeed, after implementing [math]D[/math], and then [math]G_{kN}(\pi,\sigma)=N^{|\pi\vee\sigma|}[/math], the big problem, which is that of inverting, [math]W_{kN}=G_{kN}^{-1}[/math], can be solved by the computer, and then you obtain all the integrals that you want just by summing. For some numerics here, you can check for instance the various papers citing Collins-\'Sniady .

5b. Free probability

We are not over with integration, because we have now to apply our various results above, to some suitable variables, and see what we get. The range of applications here is potentially infinite, and in the lack of a good high energy physics problem to be solved, and let us put that on our to-do list, we will do some pure mathematics.

The point indeed is that Voiculescu came in the 80s with a beautiful theory of free probability, explained in his book with Dykema and Nica , and we would like to know if our liberation considerations fit with this. More specifically, the correspondence between classical and free probability was axiomatized by Bercovici-Pata in , and we would like to know if our constructions [math]X\to X^+[/math] fit with this correspondence.

And there is a long way to go here. First we must explain free probability, following Voiculescu-Dykema-Nica , so let start with the following standard definition:

Definition

Let [math]A[/math] be a [math]C^*[/math]-algebra, given with a positive trace [math]tr[/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_a:P\to tr(P(a))[/math].

Here [math]k=\circ\bullet\bullet\circ\ldots[/math] is as usual a colored integer, and the powers [math]a^k[/math] are defined by the usual formulae, namely [math]a^\emptyset=1,a^\circ=a,a^\bullet=a^*[/math] and multiplicativity. As for the polynomial [math]P[/math], this is by definition a noncommuting [math]*[/math]-polynomial in one variable:

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

Observe that the law is uniquely determined by the moments, because:

[[math]] P(X)=\sum_k\lambda_kX^k\implies\mu_a(P)=\sum_k\lambda_kM_k(a) [[/math]]

In the self-adjoint case, [math]a=a^*[/math] the law is a usual probability measure, supported by the spectrum of [math]a[/math]. This follows indeed from the Gelfand theorem, and the Riesz theorem. More generally, the same happens in the normal case, [math]aa^*=a^*a[/math], with the spectrum being now complex. However, in the non-normal case, [math]aa^*\neq a^*a[/math], such a probability measure describing the law [math]\mu_a[/math] does not exist, due to the following computation:

[[math]] \begin{eqnarray*} aa^*-a^*a\neq0 &\implies&(aa^*-a^*a)^2 \gt 0\\ &\implies&aa^*aa^*-aa^*a^*a-a^*aaa^*+a^*aa^*a \gt 0\\ &\implies&tr(aa^*aa^*-aa^*a^*a-a^*aaa^*+a^*aa^*a) \gt 0\\ &\implies&tr(aa^*aa^*+a^*aa^*a) \gt tr(aa^*a^*a+a^*aaa^*)\\ &\implies&tr(aa^*aa^*) \gt tr(aaa^*a^*) \end{eqnarray*} [[/math]]

Indeed, assuming that [math]a[/math] has a probability measure as law, the above quantities would both appear by integrating [math]|z|^2[/math] with respect to this measure, which is contradictory.

Talking now probability, in a general sense, if there is one thing to be known here, this is the Central Limit Theorem (CLT). So, let us start with this:

Theorem (CLT)

Given real random variables [math]x_1,x_2,x_3,\ldots,[/math] which are i.i.d., centered, and 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 g_t [[/math]]

where [math]g_t[/math] is the Gaussian law of parameter [math]t[/math], having as density:

[[math]] g_t=\frac{1}{\sqrt{2\pi t}}e^{-x^2/2t}dx [[/math]]

Show Proof

This is something standard, the proof being in three steps, as follows:

(1) Linearization of the convolution. It well-known that the log of the Fourier transform [math]F_x(\xi)=\mathbb E(e^{i \xi x})[/math] does the job, in the sense that if [math]x,y[/math] are independent, then:

[[math]] F_{x+y}=F_xF_y [[/math]]

(2) Study of the limit. We have the following formula for a general Fourier transform [math]F_x(\xi)=\mathbb E(e^{i \xi x})[/math], in terms of moments:

[[math]] F_x(\xi)=\sum_{k=0}^\infty\frac{i^kM_k(x)}{k!}\,\xi^k [[/math]]

It follows that the Fourier transform of the variable in the statement is:

[[math]] \begin{eqnarray*} F(\xi) &=&\left[F_x\left(\frac{\xi}{\sqrt{n}}\right)\right]^n\\ &=&\left[1-\frac{t\xi^2}{2n}+O(n^{-2})\right]^n\\ &\simeq&e^{-t\xi^2/2} \end{eqnarray*} [[/math]]

(3) Gaussian laws. The Fourier transform of the Gaussian law is given by:

[[math]] \begin{eqnarray*} F_{g_t}(x) &=&\frac{1}{\sqrt{2\pi t}}\int_\mathbb Re^{-y^2/2t+ixy}dy\\ &=&\frac{1}{\sqrt{2\pi t}}\int_\mathbb Re^{-(y/\sqrt{2t}-\sqrt{t/2}ix)^2-tx^2/2}dy\\ &=&\frac{1}{\sqrt{2\pi t}}\int_\mathbb Re^{-z^2-tx^2/2}\sqrt{2t}dz\\ &=&e^{-tx^2/2} \end{eqnarray*} [[/math]]

Thus the variables on the left and on the right in the statement have the same Fourier transform, and so these variables follow the same law, as claimed.

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Following Voiculescu , in order to extend the CLT to the free setting, our starting point will be the following definition:

Definition

Given a pair [math](A,tr)[/math], two subalgebras [math]B,C\subset A[/math] are called free when the following condition is satisfied, for any [math]b_i\in B[/math] and [math]c_i\in C[/math]:

[[math]] tr(b_i)=tr(c_i)=0\implies tr(b_1c_1b_2c_2\ldots)=0 [[/math]]

Also, two noncommutative random variables [math]b,c\in A[/math] are called free when the [math]C^*[/math]-algebras [math]B= \lt b \gt [/math], [math]C= \lt c \gt [/math] that they generate inside [math]A[/math] are free, in this sense.

As a first observation, there is a similarity here with the classical notion of independence. Indeed, modulo some standard identifications, two subalgebras [math]B,C\subset L^\infty(X)[/math] are independent when the following condition is satisfied, for any [math]b\in B[/math] and [math]c\in C[/math]:

[[math]] tr(bc)=tr(b)tr(c) [[/math]]

But this is equivalent to the following condition, which is similar to freeness:

[[math]] tr(b)=tr(c)=0\implies tr(bc)=0 [[/math]]

In short, freeness appears by definition as a kind of “free analogue” of independence. As a first result now regarding this notion, clarifying the basics, we have:

Proposition

Assuming that [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,

[[math]] tr(b_1c_1b_2c_2\ldots)=P\Big(\{tr(b_{i_1}b_{i_2}\ldots)\}_i,\{tr(c_{j_1}c_{j_2}\ldots)\}_j\Big) [[/math]]

where [math]P[/math] is certain polynomial in several variables, depending on the length of the word [math]b_1c_1b_2c_2\ldots[/math], and having as variables the traces of products of type

[[math]] b_{i_1}b_{i_2}\ldots\quad,\quad c_{j_1}c_{j_2}\ldots [[/math]]

with the indices being chosen increasing, [math]i_1 \lt i_2 \lt \ldots[/math] and [math]j_1 \lt j_2 \lt \ldots[/math]

Show Proof

We can start indeed our computation as follows:

[[math]] \begin{eqnarray*} tr(b_1c_1b_2c_2\ldots) &=&tr\big[(b_1'+tr(b_1))(c_1'+tr(c_1))(b_2'+tr(b_2))(c_2'+tr(c_2))\ldots\ldots\big]\\ &=&tr(b_1'c_1'b_2'c_2'\ldots)+{\rm other\ terms}\\ &=&{\rm other\ terms} \end{eqnarray*} [[/math]]

Observe that we have used here the freeness condition, in the following form:

[[math]] tr(b_i')=tr(c_i')=0\implies tr(b_1'c_1'b_2'c_2'\ldots)=0 [[/math]]

Thus, we are led into some sort of recurrence, as desired. For more on all this, including examples, we refer to the book of Voiculescu-Dykema-Nica .

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As a second result regarding the notion of freeness, which provides us with a useful class of examples, which can be used for various modelling purposes, we have:

Proposition

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

• [math]B,C[/math] are independent inside their tensor product [math]B\otimes C[/math], endowed with its canonical tensor product trace, given on basic tensors by [math]tr(b\otimes c)=tr(b)tr(c)[/math].
• [math]B,C[/math] are free inside their free product [math]B*C[/math], endowed with its canonical free product trace, given by the formulae in Proposition 5.11.

Show Proof

Both the assertions are clear from definitions, as follows:

(1) This is clear, because we have by construction of the trace:

[[math]] \begin{eqnarray*} tr(bc) &=&tr[(b\otimes1)(1\otimes c)]\\ &=&tr(b\otimes c)\\ &=&tr(b)tr(c) \end{eqnarray*} [[/math]]

(2) This is clear again, the only point being that of showing that the notion of freeness, or the recurrence formulae in Proposition 5.11, can be used in order to construct a canonical free product trace, on the free product of the two algebras involved:

[[math]] tr:B*C\to\mathbb C [[/math]]

But this can be done for instance by using a GNS construction. Indeed, by taking the free product of the GNS constructions for [math](B,tr)[/math] and [math](C,tr)[/math], we obtain a representation as follows, with the [math]*[/math] on the right being a free product of pointed Hilbert spaces:

[[math]] B*C\to B(l^2(B)*l^2(C)) [[/math]]

Now by composing with the linear form [math]T\to \lt T\xi,\xi \gt [/math], where [math]\xi=1_B=1_C[/math] is the common distinguished vector of [math]l^2(B)[/math] and [math]l^2(C)[/math], we obtain a linear form, as follows:

[[math]] tr:B*C\to\mathbb C [[/math]]

It is routine then to check that [math]tr[/math] is indeed a trace, and this is the “canonical free product trace” from the statement. Then, an elementary computation shows that [math]B,C[/math] are indeed free inside [math]B*C[/math], with respect to this trace, as desired.

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Finally, again following , we have the following more explicit modelling result:

Theorem

We have the following results, valid for group algebras:

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

Show Proof

We can use here the general results in Proposition 5.12, along with the following two isomorphisms, which are both standard:

[[math]] C^*(\Gamma\times\Lambda)=C^*(\Lambda)\otimes C^*(\Gamma) [[/math]]

[[math]] C^*(\Gamma*\Lambda)=C^*(\Lambda)*C^*(\Gamma) [[/math]]

Alternatively, we can prove this directly, starting from definitions, by using the fact that each group algebra is spanned by the corresponding group elements.

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There are many things that can be said about the analogy between independence and freeness. We have in particular the following result, due to Voiculescu :

Theorem

Given a real probability measure [math]\mu[/math], consider its Cauchy transform

[[math]] G_\mu(\xi)=\int_\mathbb R\frac{d\mu(t)}{\xi-t} [[/math]]

and define its [math]R[/math]-transform as being the solution of the following equation:

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

The operation [math]\mu\to R_\mu[/math] linearizes then the free convolution operation.

Show Proof

In order to prove this, we need a good model for the free convolution. The best here is to use the semigroup algebra of the free semigroup on two generators:

[[math]] A=C^*(\mathbb N*\mathbb N) [[/math]]

Indeed, we have some freeness in the semigroup setting, a bit in the same way as for the group algebras [math]C^*(\Gamma*\Lambda)[/math], from Theorem 5.13 (2), and in addition to this fact, and to what happens in the group algebra case, the following two key things happen:

(1) The variables of type [math]S^*+f(S)[/math], with [math]S\in C^*(\mathbb N)[/math] being the shift, and with [math]f\in\mathbb C[X][/math] being a polynomial, model in moments all the distributions [math]\mu:\mathbb C[X]\to\mathbb C[/math]. This is indeed something elementary, which can be checked via a direct algebraic computation.

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

With these results in hand, we can see that the operation [math]\mu\to f[/math] linearizes the free convolution. We are therefore left with a computation inside [math]C^*(\mathbb N)[/math], 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 stated.

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We can now state and prove a free analogue of the CLT, from , as follows:

Theorem (FCLT)

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[/math] is the Wigner semicircle law of parameter [math]t[/math], having density:

[[math]] \gamma_t=\frac{1}{2\pi t}\sqrt{4t^2-x^2}dx [[/math]]

Show Proof

At [math]t=1[/math], the [math]R[/math]-transform of the variable in the statement can be computed by using the linearization property with respect to the free convolution, and is:

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

On the other hand, some elementary computations show that the Cauchy transform of the Wigner law [math]\gamma_1[/math] from the statement satisfies the following equation:

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

Thus we have [math]R_{\gamma_1}(\xi)=\xi[/math], which by the way follows as well from [math]S^*+S\sim\gamma_1[/math], and this gives the result. The passage to the general case, [math]t \gt 0[/math], is routine.

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In the complex case now, we recall that the complex Gaussian law of parameter [math]t \gt 0[/math] is defined as follows, with [math]a,b[/math] being independent, each following the law [math]g_t[/math]:

[[math]] G_t=law\left(\frac{1}{\sqrt{2}}(a+ib)\right) [[/math]]

With this convention, we have the following result:

Theorem (CCLT)

Given complex random variables [math]x_1,x_2,x_3,\ldots,[/math] which are i.i.d., centered, and 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 G_t [[/math]]

where [math]G_t[/math] is the complex Gaussian law of parameter [math]t[/math].

Show Proof

This follows indeed from the real CLT, established above, without new computations needed, just by taking real and imaginary parts.

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In the free case, the Voiculescu circular law of parameter [math]t \gt 0[/math] is defined as follows, with [math]\alpha,\beta[/math] being independent, each following the law [math]\gamma_t[/math]:

[[math]] \Gamma_t=law\left(\frac{1}{\sqrt{2}}(\alpha+i\beta)\right) [[/math]]

With this convention, we have the following result, again from Voiculescu :

Theorem (FCCLT)

Given noncommutative random variables [math]x_1,x_2,x_3,\ldots,[/math] which are f.i.d., centered, and 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[/math] is the Voiculescu circular law of parameter [math]t[/math].

Show Proof

This follows indeed from the FCLT, by taking real and imaginary parts.

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With these ingredients in hand, let us go back now to our quantum groups. According to the Peter-Weyl theory from chapter 2, if there are some variables that we should look at, these are the characters. And here, for the unitary quantum groups, we have:

Theorem

With [math]N\to\infty[/math], the main characters

[[math]] \chi=\sum_{i=1}^Nu_{ii} [[/math]]

for the basic unitary quantum groups are as follows:

• [math]O_N[/math]: real Gaussian, following [math]g_1[/math].
• [math]O_N^+[/math]: semicircular, following [math]\gamma_1[/math].
• [math]U_N[/math]: complex Gaussian, following [math]G_1[/math].
• [math]U_N^+[/math]: circular, following [math]\Gamma_1[/math].

Show Proof

We use the moment method, and combinatorics. For a closed subgroup [math]G_N\subset U_N^+[/math], we have, according to the Peter-Weyl type results of Woronowicz in :

[[math]] \int_{G_N}\chi^k=\dim(Fix(u^{\otimes k})) [[/math]]

In the easy case now, where [math]G=(G_N)[/math] comes from a certain category of partitions [math]D[/math], the fixed point space on the right is spanned by the vectors [math]T_\pi[/math] with [math]\pi\in D(k)[/math]. Now since by Lindstöm these vectors are linearly independent with [math]N\to\infty[/math], we have:

[[math]] \lim_{N\to\infty}\int_{G_N}\chi^k=|D(k)| [[/math]]

Thus, we are led into some combinatorics, and the continuation is as follows:

(1) For [math]O_N[/math] we have [math]D=P_2[/math], so we obtain as even asymptotic moments the numbers [math]|P_2(2k)|=k!![/math], which are well-known to be the moments of the Gaussian law.

(2) For [math]O_N^+[/math] we have [math]D=NC_2[/math], so we obtain as even asymptotic moments the Catalan numbers [math]|NC_2(2k)|=C_k[/math], which are the moments of the Wigner semicircle law.

(3) For [math]U_N[/math] we have [math]D=\mathcal P_2[/math], and we can conclude as in the real case, involving [math]O_N[/math], by using this time moments with respect to colored integers, as in Definition 5.8.

(4) For [math]U_N^+[/math] we have [math]D=\mathcal{NC}_2[/math], and again we can conclude as in the real case, involving [math]O_N^+[/math], by using moments with respect to colored integers, as in Definition 5.8.

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The above result is of course just the tip of the iceberg, and there are countless things that can be done, as a continuation of this. In what follows we will orient the discussion towards something rather theoretical, namely the Bercovici-Pata bijection .

5c. Truncated characters

We have seen so far that for [math]O_N,O_N^+,U_N,U_N^+[/math], the asymptotic laws of the main characters are the laws [math]g_1,\gamma_1,G_1,\Gamma_1[/math] coming from the various classical and free CLT. This is certainly nice, but there is still one conceptual problem, coming from:

Proposition

The above convergences [math]law(\chi_u)\to g_1,\gamma_1,G_1,\Gamma_1[/math] are as follows:

• They are non-stationary in the classical case.
• They are stationary in the free case, starting from [math]N=2[/math].

Show Proof

This is something quite subtle, which can be proved as follows:

(1) Here we can use an amenability argument, based on the Kesten criterion. Indeed, [math]O_N,U_N[/math] being coamenable, the upper bound of the support of the law of [math]Re(\chi_u)[/math] is precisely [math]N[/math], and we obtain from this that the law of [math]\chi_u[/math] itself depends on [math]N\in\mathbb N[/math].

(2) Here the result follows from the well-known fact that the linear maps [math]T_\pi[/math] associated to the noncrossing pairings are linearly independent, at any [math]N\geq2[/math], which fact, which is non-trivial, follows itself either from the general theory developed by Jones in , in relation with the Temperley-Lieb algebra, or from Di Francesco .

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Fortunately, the solution to the convergence question raised by Proposition 5.19 is quite simple. The idea will be that of improving our [math]g_1,\gamma_1,G_1,\Gamma_1[/math] results with certain [math]g_t,\gamma_t,G_t,\Gamma_t[/math] results, which will require [math]N\to\infty[/math] in both the classical and free cases, in order to hold at any [math]t[/math]. Following , the definition that we will need is as follows:

Definition

Given a Woronowicz algebra [math](A,u)[/math], the variable

[[math]] \chi_t=\sum_{i=1}^{[tN]}u_{ii} [[/math]]

is called truncation of the main character, with parameter [math]t\in(0,1][/math].

Our purpose in what follows will be that of proving that for [math]O_N,O_N^+,U_N,U_N^+[/math], the asymptotic laws of the truncated characters [math]\chi_t[/math] with [math]t\in(0,1][/math] are the laws [math]g_t,\gamma_t,G_t,\Gamma_t[/math]. This is something quite technical, but natural, motivated by the findings in Proposition 5.19, and also by a number of more advanced considerations, to become clear later on. So, let us do this. In order to study the truncated characters, we can use:

Theorem

The moments of the truncated characters are given by

[[math]] \int_G(u_{11}+\ldots +u_{ss})^k=Tr(W_{kN}G_{ks}) [[/math]]

and with [math]N\to\infty[/math] this quantity equals [math](s/N)^k|D(k)|[/math].

Show Proof

The first assertion follows from the following computation:

[[math]] \begin{eqnarray*} \int_G(u_{11}+\ldots +u_{ss})^k &=&\sum_{i_1=1}^{s}\ldots\sum_{i_k=1}^s\int u_{i_1i_1}\ldots u_{i_ki_k}\\ &=&\sum_{\pi,\sigma\in D(k)}W_{kN}(\pi,\sigma)\sum_{i_1=1}^{s}\ldots\sum_{i_k=1}^s\delta_\pi(i)\delta_\sigma(i)\\ &=&\sum_{\pi,\sigma\in D(k)}W_{kN}(\pi,\sigma)G_{ks}(\sigma,\pi)\\ &=&Tr(W_{kN}G_{ks}) \end{eqnarray*} [[/math]]

We have [math]G_{kN}(\pi,\sigma)=N^k[/math] for [math]\pi=\sigma[/math], and [math]G_{kN}(\pi,\sigma)\leq N^{k-1}[/math] for [math]\pi\neq\sigma[/math]. Thus with [math]N\to\infty[/math] we have [math]G_{kN}\sim N^k1[/math], which gives:

[[math]] \begin{eqnarray*} \int_G(u_{11}+\ldots +u_{ss})^k &=&Tr(G_{kN}^{-1}G_{ks})\\ &\sim&Tr((N^k1)^{-1} G_{ks})\\ &=&N^{-k}Tr(G_{ks})\\ &=&N^{-k}s^k|D(k)| \end{eqnarray*} [[/math]]

Thus, we have obtained the formula in the statement.

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In order to process the above moment formula, we will need some more probability theory, both classical and free. Given a random variable [math]a[/math], we write:

[[math]] \log F_a(\xi)=\sum_nk_n(a)\xi^n\quad,\quad R_a(\xi)=\sum_n\kappa_n(a)\xi^n [[/math]]

We call the above coefficients [math]k_n(a),\kappa_n(a)[/math] the cumulants, respectively free cumulants of [math]a[/math]. With this notion in hand, we can define then more general quantities [math]k_\pi(a),\kappa_\pi(a)[/math], depending on arbitrary partitions [math]\pi\in P(k)[/math], which coincide with the above ones for the 1-block partitions, and then by multiplicativity over the blocks, and we have:

Theorem

We have the classical and free moment-cumulant formulae

[[math]] M_k(a)=\sum_{\pi\in P(k)}k_\pi(a)\quad,\quad M_k(a)=\sum_{\pi\in NC(k)}\kappa_\pi(a) [[/math]]

where [math]k_\pi(a),\kappa_\pi(a)[/math] are the generalized cumulants and free cumulants of [math]a[/math].

Show Proof

This is something very standard, due to Rota in the classical case, and to Speicher in the free case, obtained either by using the formulae of [math]F_a,R_a[/math], or by doing some direct combinatorics, based on the Möbius inversion formula. See .

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We can now improve our results about characters, as follows:

Theorem

With [math]N\to\infty[/math], the laws of truncated characters are as follows:

• For [math]O_N[/math] we obtain the Gaussian law [math]g_t[/math].
• For [math]O_N^+[/math] we obtain the Wigner semicircle law [math]\gamma_t[/math].
• For [math]U_N[/math] we obtain the complex Gaussian law [math]G_t[/math].
• For [math]U_N^+[/math] we obtain the Voiculescu circular law [math]\Gamma_t[/math].

Show Proof

With [math]s=[tN][/math] and [math]N\to\infty[/math], the formula in Theorem 5.21 gives:

[[math]] \lim_{N\to\infty}\int_{G_N}\chi_t^k=\sum_{\pi\in D(k)}t^{|\pi|} [[/math]]

By using now the formulae in Theorem 5.22, this gives the results.

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As an interesting consequence, related to , let us formulate as well:

Theorem

The asymptotic laws of truncated characters for the operations

[[math]] O_N\to O_N^+\quad,\quad U_N\to U_N^+ [[/math]]

are in Bercovici-Pata bijection, in the sense that the classical cumulants in the classical case equal the free cumulants in the free case.

Show Proof

This follows indeed from Theorem 5.23, and from the standard combinatorial interpretation of the Bercovici-Pata bijection , in terms of cumulants.

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Let us discuss now the integration over the spheres. Following , we have:

Theorem

With [math]N\to\infty[/math], the rescaled coordinates of the various spheres

[[math]] \sqrt{N}x_i\in C(S^{N-1}_{\times}) [[/math]]

are as follows, with respect to the uniform integration:

• [math]S^{N-1}_\mathbb R[/math]: real Gaussian.
• [math]S^{N-1}_{\mathbb R,+}[/math]: semicircular.
• [math]S^{N-1}_\mathbb C[/math]: complex Gaussian.
• [math]S^{N-1}_{\mathbb C,+}[/math]: circular.

Show Proof

This follows from Theorem 5.23, but we can use as well the Weingarten formula for the spheres, from Theorem 5.7. Indeed, we have the following estimate:

[[math]] \int_{S^{N-1}_\times}x_{i_1}\ldots x_{i_k}\,dx\simeq N^{-k/2}\sum_{\sigma\in P_2^\times(k)}\delta_\sigma(i) [[/math]]

With this formula in hand, we can compute the asymptotic moments of each coordinate [math]x_i[/math]. Indeed, by setting [math]i_1=\ldots=i_k=i[/math], all Kronecker symbols are 1, and we obtain:

[[math]] \int_{S^{N-1}_\times}x_i^k\,dx\simeq N^{-k/2}|P_2^\times(k)| [[/math]]

But this gives the results, via the same combinatorics as before. See .

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5d. Poisson laws

In order to discuss now the quantum reflection groups, we will need some more theory, namely Poisson limit theorems. In the classical case, we have the following result:

Theorem (PLT)

We have the following convergence, in moments,

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

the limiting measure being

[[math]] p_t=\frac{1}{e^t}\sum_{k=0}^\infty\frac{t^k\delta_k}{k!} [[/math]]

which is the Poisson law of parameter [math]t \gt 0[/math].

Show Proof

We recall that the Fourier transform is given by [math]F_f(x)=\mathbb E(e^{ixf})[/math]. We therefore obtain the following formula:

[[math]] \begin{eqnarray*} F_{p_t}(x) &=&e^{-t}\sum_k\frac{t^k}{k!}F_{\delta_k}(x)\\ &=&e^{-t}\sum_k\frac{t^k}{k!}\,e^{ikx}\\ &=&e^{-t}\sum_k\frac{(e^{ix}t)^k}{k!}\\ &=&\exp(-t)\exp(e^{ix}t)\\ &=&\exp\left((e^{ix}-1)t\right) \end{eqnarray*} [[/math]]

Let us denote by [math]\mu_n[/math] the measure under the convolution sign, namely:

[[math]] \mu_n=\left(1-\frac{t}{n}\right)\delta_0+\frac{t}{n}\delta_1 [[/math]]

We have the following computation:

[[math]] \begin{eqnarray*} F_{\delta_r}(x)=e^{irx} &\implies&F_{\mu_n}(x)=\left(1-\frac{t}{n}\right)+\frac{t}{n}e^{ix}\\ &\implies&F_{\mu_n^{*n}}(x)=\left(\left(1-\frac{t}{n}\right)+\frac{t}{n}e^{ix}\right)^n\\ &\implies&F_{\mu_n^{*n}}(x)=\left(1+\frac{(e^{ix}-1)t}{n}\right)^n\\ &\implies&F(x)=\exp\left((e^{ix}-1)t\right) \end{eqnarray*} [[/math]]

Thus, we obtain the Fourier transform of [math]p_t[/math], as desired.

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In the free case, the result is as follows, with [math]\boxplus[/math] being the free convolution operation:

Theorem (FPLT)

We have the following convergence, in moments,

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

the limiting measure being the Marchenko-Pastur law of parameter [math]t \gt 0[/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 \gt 0[/math].

Show Proof

Consider the measure in the statement, under the convolution sign:

[[math]] \mu=\left(1-\frac{t}{n}\right)\delta_0+\frac{t}{n}\delta_1 [[/math]]

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]]

In order to prove the result, we want to compute the following [math]R[/math]-transform:

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

But 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]]

By multiplying by [math]n/y[/math], this equation can be written as:

[[math]] \frac{t+yR}{1+yR/n}=\frac{t}{1+yR/n-y} [[/math]]

With [math]n\to\infty[/math] we obtain [math]t+yR=t/(1-y)[/math], so [math]R=t/(1-y)=R_{\pi_t}[/math], as desired.

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In order to get beyond this, let us introduce the following notions:

Definition

Associated to any compactly supported positive measure [math]\rho[/math] on [math]\mathbb C[/math], not necessarily of mass [math]1[/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} [[/math]]

[[math]] \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:

Theorem

For a discrete measure, [math]\rho=\sum_{i=1}^sc_i\delta_{z_i}[/math] with [math]c_i \gt 0[/math] and [math]z_i\in\mathbb R[/math],

[[math]] F_{p_\rho}(y)=\exp\left(\sum_{i=1}^sc_i(e^{iyz_i}-1)\right) [[/math]]

[[math]] 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 in Definition 5.28, under the convolution signs:

[[math]] \mu_n=\left(1-\frac{c}{n}\right)\delta_0+\frac{1}{n}\rho [[/math]]

In the classical case, we have the following computation:

[[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 now, we use a similar method. The Cauchy transform of [math]\mu_n[/math] is:

[[math]] G_{\mu_n}(\xi)=\left(1-\frac{c}{n}\right)\frac{1}{\xi}+\frac{1}{n}\sum_{i=1}^s\frac{c_i}{\xi-z_i} [[/math]]

Consider now the [math]R[/math]-transform of the measure [math]\mu_n^{\boxplus n}[/math], which is given by:

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

The above formula of [math]G_{\mu_n}[/math] shows that the equation for [math]R=R_{\mu_n^{\boxplus n}}[/math] is as follows:

[[math]] \begin{eqnarray*} &&\left(1-\frac{c}{n}\right)\frac{1}{y^{-1}+R/n}+\frac{1}{n}\sum_{i=1}^s\frac{c_i}{y^{-1}+R/n-z_i}=y\\ &\implies&\left(1-\frac{c}{n}\right)\frac{1}{1+yR/n}+\frac{1}{n}\sum_{i=1}^s\frac{c_i}{1+yR/n-yz_i}=1 \end{eqnarray*} [[/math]]

Now multiplying by [math]n[/math], rearranging the terms, and letting [math]n\to\infty[/math], we get:

[[math]] \begin{eqnarray*} &&\frac{c+yR}{1+yR/n}=\sum_{i=1}^s\frac{c_i}{1+yR/n-yz_i}\\ &\implies&c+yR_{\pi_\rho}(y)=\sum_{i=1}^s\frac{c_i}{1-yz_i}\\ &\implies&R_{\pi_\rho}(y)=\sum_{i=1}^s\frac{c_iz_i}{1-yz_i} \end{eqnarray*} [[/math]]

This finishes the proof in the free case, and we are done.

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We also have the following technical result, providing a useful alternative to Definition 5.28, in order to detect the classical and free compound Poisson laws:

Theorem

For a discrete measure, written as [math]\rho=\sum_{i=1}^sc_i\delta_{z_i}[/math] with [math]c_i \gt 0[/math] and [math]z_i\in\mathbb R[/math], we have the classical/free formulae

[[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

Let [math]\alpha[/math] be the sum of Poisson/free Poisson variables in the statement:

[[math]] \alpha=\sum_{i=1}^sz_i\alpha_i [[/math]]

By using some well-known Fourier transform formulae, we have:

[[math]] \begin{eqnarray*} F_{\alpha_i}(y)=\exp(c_i(e^{iy}-1)) &\implies&F_{z_i\alpha_i}(y)=\exp(c_i(e^{iyz_i}-1))\\ &\implies&F_\alpha(y)=\exp\left(\sum_{i=1}^sc_i(e^{iyz_i}-1)\right) \end{eqnarray*} [[/math]]

Also, by using some well-known [math]R[/math]-transform formulae, we have:

[[math]] \begin{eqnarray*} R_{\alpha_i}(y)=\frac{c_i}{1-y} &\implies&R_{z_i\alpha_i}(y)=\frac{c_iz_i}{1-yz_i}\\ &\implies&R_\alpha(y)=\sum_{i=1}^s\frac{c_iz_i}{1-yz_i} \end{eqnarray*} [[/math]]

Thus we have indeed the same formulae as those which are needed.

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We refer to , for the general theory here, to , for representation theory aspects, and to , for random matrix aspects. In what follows we will only need the main examples of classical and free compound Poisson laws, which are the classical and free Bessel laws, 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, involving classical and quantum reflection groups, subfactors and planar algebras, and free probability and random matrices. We refer here to , where these laws were introduced. Let us just record here:

Theorem

The moments of the various central limiting measures, namely

[[math]] \xymatrix@R=20pt@C=22pt{ &\mathfrak B_t\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_t\ar@{-}[dd]\\ \beta_t\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_t\ar@{-}[dd]\ar@{-}[ur]\\ &B_t\ar@{-}[rr]\ar@{-}[uu]&&G_t\ar@{-}[uu]\\ b_t\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_t\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

are always given by the same formula, involving partitions, namely

[[math]] M_k=\sum_{\pi\in D(k)}t^{|\pi|} [[/math]]

with the sets of partitions [math]D(k)[/math] in question being respectively

[[math]] \xymatrix@R=20pt@C=5pt{ &\mathcal{NC}_{even}\ar[dl]\ar[dd]&&\ \ \ \mathcal{NC}_2\ \ \ \ar[ll]\ar[dd]\ar[dl]\\ NC_{even}\ar[dd]&&NC_2\ar[ll]\ar[dd]\\ &\mathcal P_{even}\ar[dl]&&\mathcal P_2\ar[ll]\ar[dl]\\ P_{even}&&P_2\ar[ll] } [[/math]]

and with [math]|.|[/math] being the number of blocks.

Show Proof

This follows indeed from our various moment results. See .

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Getting back now to our quantum reflection groups, we have:

Theorem

With [math]N\to\infty[/math], the laws of truncated characters are as follows:

• For [math]H_N[/math] we obtain the Bessel law [math]b_t[/math].
• For [math]H_N^+[/math] we obtain the free Bessel law [math]\beta_t[/math].
• For [math]K_N[/math] we obtain the complex Bessel law [math]B_t[/math].
• For [math]K_N^+[/math] we obtain the complex free Bessel law [math]\mathfrak B_t[/math].

Also, we have the Bercovici-Pata bijection for truncated characters.

Show Proof

At [math]t=1[/math] this follows by counting the partitions, a bit as in the continuous case, in the proof of Theorem 5.18. At [math]t\in(0,1)[/math] this is routine, by using the Weingarten formula, as in the continuous case, in the proof of Theorem 5.23. See .

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The results that we have so far, for the quantum unitary and refelection groups, are quite interesting, from a theoretical probability perspective, because we have:

Theorem

The laws of the truncated characters for the basic quantum groups,

[[math]] \xymatrix@R=18pt@C=18pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

and the various classical and free central limiting measures, namely

[[math]] \xymatrix@R=20pt@C=22pt{ &\mathfrak B_t\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_t\ar@{-}[dd]\\ \beta_t\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_t\ar@{-}[dd]\ar@{-}[ur]\\ &B_t\ar@{-}[rr]\ar@{-}[uu]&&G_t\ar@{-}[uu]\\ b_t\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_t\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

in the [math]N\to\infty[/math] limit.

Show Proof

This follows indeed by putting together the various results discussed above, concerning general free probability theory, and our computations here.

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Regarding now the tori, the situation here is more complicated, no longer involving the Bercovici-Pata bijection. Let us recall indeed that our tori and their duals are:

[[math]] \xymatrix@R=50pt@C=50pt{ T_N^+\ar[r]&\mathbb T_N^+\\ T_N\ar[r]\ar[u]&\mathbb T_N\ar[u] }\qquad \item[a]ymatrix@R=25pt@C=50pt{\\ :\\} \qquad \item[a]ymatrix@R=50pt@C=50pt{ \mathbb Z_2^{*N}\ar[d]&F_N\ar[l]\ar[d]\\ \mathbb Z_2^N&\mathbb Z^N\ar[l] } [[/math]]

We are interested in the computation of the laws of the associated truncated characters, which are the following variables, with [math]g_1,\ldots,g_N[/math] being the group generators:

[[math]] \chi_t=g_1+g_2+\ldots+g_{[tN]} [[/math]]

By dilation we can assume [math]t=1[/math]. For the complex tori, [math]\mathbb T_N\subset\mathbb T_N^+[/math], we are led into the computation of the Kesten measures for [math]F_N\to\mathbb Z^N[/math], and so into the Meixner/free Meixner correspondence. As for the real tori, [math]T_N\subset T_N^+[/math], here we are led into the computation of the Kesten measures for [math]\mathbb Z_2^{*N}\to\mathbb Z_2^N[/math], and so into a real version of this correspondence. These are both quite technical questions, that we will not get into, here.

General references

Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].