14c. Hyperspherical laws

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Changing topics now, we know from Theorem 14.13, dealing with the Meixner/free Meixner correspondence, that doing probability in the free geometry setting can lead us to unexplored territory, beyond what the Bercovici-Pata bijection says. As a continuation of that material, we will discuss here the classical and free hyperspherical laws. In the classical case, we will need the following result, that we know well from chapter 1:

Theorem

The even moments of the hyperspherical variables are

[[math]] \int_{S^{N-1}_\mathbb R}z_i^kdx=\frac{(N-1)!!k!!}{(N+k-1)!!} [[/math]]

and the variables [math]y_i=\sqrt{N}z_i[/math] become normal and independent with [math]N\to\infty[/math].

Show Proof

The moment formula in the statement is something that we know from chapter 1. Now observe that with [math]N\to\infty[/math] we have the following estimate:

[[math]] \int_{S^{N-1}_\mathbb R}z_i^kdz \simeq N^{-k/2}\times k!! =N^{-k/2}M_k(g_1) [[/math]]

Thus we have, as claimed, [math]\sqrt{N}z_i\sim g_1[/math]. Finally, the asymptotic independence assertion follows as well from the formulae in chapter 1, via standard probability theory.

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In the case of the free real sphere now, the computations are substantially more complicated than those in the classical case. Let us start with the following result:

Theorem

For the free sphere [math]S^{N-1}_{\mathbb R,+}[/math], the rescaled coordinates

[[math]] y_i=\sqrt{N}z_i [[/math]]

become semicircular and free, in the [math]N\to\infty[/math] limit.

Show Proof

The Weingarten formula for the free sphere, together with the standard fact that the Gram matrix is asymptotically diagonal, gives the following estimate:

[[math]] \int_{S^{N-1}_{\mathbb R,+}}z_{i_1}\ldots z_{i_k}\,dz\simeq N^{-k/2}\sum_{\sigma\in NC_2(k)}\delta_\sigma(i_1,\ldots,i_k) [[/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}_{\mathbb R,+}}z_i^k\,dz\simeq N^{-k/2}|NC_2(k)| [[/math]]

Thus the rescaled coordinates [math]y_i=\sqrt{N}z_i[/math] become semicircular in the [math]N\to\infty[/math] limit, as claimed. As for the asymptotic freeness result, this follows as well from the above general joint moment estimate, via standard free probability theory. See , , .

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Summarizing, we have good results for the free sphere, with [math]N\to\infty[/math]. The problem now, which is non-trivial, is that of computing the moments of the coordinates of the free sphere at fixed values of [math]N\in\mathbb N[/math]. The answer here, from , which is based on advanced quantum group techniques, that we will briefly explain here, is as follows:

Theorem

The moments of the free hyperspherical law are given by

[[math]] \int_{S^{N-1}_{\mathbb R,+}}z_1^{2l}=\frac{1}{(N+1)^l}\cdot\frac{q+1}{q-1}\cdot\frac{1}{l+1}\sum_{r=-l-1}^{l+1}(-1)^r\begin{pmatrix}2l+2\cr l+r+1\end{pmatrix}\frac{r}{1+q^r} [[/math]]

where [math]q\in [-1,0)[/math] is such that [math]q+q^{-1}=-N[/math].

Show Proof

The idea is that [math]z_1\in C(S^{N-1}_{\mathbb R,+})[/math] has the same law as [math]v_{11}\in C(O_N^+)[/math], which has the same law as a certain variable [math]w\in C(SU^q_2)[/math], which can modelled by an explicit operator on [math]l^2(\mathbb N)[/math], whose law can be computed by using advanced calculus.

(1) Let us first explain the relation between [math]O_N^+[/math] and [math]SU^q_2[/math]. To any matrix [math]F\in GL_N(\mathbb R)[/math] satisfying [math]F^2=1[/math] we associate the following universal algebra:

[[math]] C(O_F^+)=C^*\left((v_{ij})_{i,j=1,\ldots,N}\Big|v=F\bar{v}F={\rm unitary}\right) [[/math]]

Observe that we have [math]O_{I_N}^+=O_N^+[/math]. In general, the above algebra satisfies Woronowicz' generalized axioms in , which do not include the antipode axiom [math]S^2=id[/math].

(2) At [math]N=2[/math] now, up to a trivial equivalence relation on the matrices [math]F[/math], and on the quantum groups [math]O_F^+[/math], we can assume that [math]F[/math] is as follows, with [math]q\in [-1,0)[/math]:

[[math]] F=\begin{pmatrix}0&\sqrt{-q}\\ 1/\sqrt{-q}&0\end{pmatrix} [[/math]]

Our claim is that for this matrix we have [math]O_F^+=SU^q_2[/math]. Indeed, the relations [math]v=F\bar{v}F[/math] tell us that [math]v[/math] must be of the following form:

[[math]] v=\begin{pmatrix}\alpha&-q\gamma^*\\ \gamma&\alpha^*\end{pmatrix} [[/math]]

Thus [math]C(O_F^+)[/math] is the universal algebra generated by two elements [math]\alpha,\gamma[/math], with the relations making the above matrix [math]v[/math] a unitary. But these unitarity conditions are:

[[math]] \alpha\gamma=q\gamma\alpha\quad,\quad \alpha\gamma^*=q\gamma^*\alpha\quad,\quad \gamma\gamma^*=\gamma^*\gamma [[/math]]

[[math]] \alpha^*\alpha+\gamma^*\gamma=1\quad,\quad \alpha\alpha^*+q^2\gamma\gamma^*=1 [[/math]]

We recognize here the relations in defining the algebra [math]C(SU^q_2)[/math], and it follows that we have an isomorphism of Hopf algebras, as follows:

[[math]] C(O_F^+)\simeq C(SU^q_2) [[/math]]

(3) Now back to the general case, where [math]F\in GL_N(\mathbb R)[/math] satisifes [math]F^2=1[/math], let us try to understand the integration over [math]O_F^+[/math]. Given [math]\pi\in NC_2(2k)[/math] and [math]i=(i_1,\ldots,i_{2k})[/math], we set:

[[math]] \delta_\pi^F(i)=\prod_{s\in\pi}F_{i_{s_l}i_{s_r}} [[/math]]

Here the product is over all the strings [math]s=\{s_l\curvearrowright s_r\}[/math] of [math]\pi[/math]. Our claim is that the following family of vectors, with [math]\pi\in NC_2(2k)[/math], spans the space of fixed vectors of [math]v^{\otimes 2k}[/math]:

[[math]] \xi_\pi=\sum_i\delta_\pi^F(i)e_{i_1}\otimes\ldots\otimes e_{i_{2k}} [[/math]]

Indeed, having [math]\xi_\cap[/math] fixed by [math]v^{\otimes 2}[/math] is equivalent to assuming that [math]v=F\bar{v}F[/math] is unitary. By using now these vectors, as in , we obtain the following Weingarten formula:

[[math]] \int_{O_F^+}v_{i_1j_1}\ldots v_{i_{2k}j_{2k}}=\sum_{\pi\sigma}\delta_\pi^F(i)\delta_\sigma^F(j)W_{kN}(\pi,\sigma) [[/math]]

(4) With these preliminaries in hand, we can now start the computation that we are interested in. Let [math]N\in\mathbb N[/math], and consider the number [math]q\in [-1,0)[/math] satisfying:

[[math]] q+q^{-1}=-N [[/math]]

Our claim is that we have the following formula:

[[math]] \int_{O_N^+}\varphi(\sqrt{N+2}\,v_{ij})=\int_{SU^q_2}\varphi(\alpha+\alpha^*+\gamma-q\gamma^*) [[/math]]

Indeed, according to the above, the moments of the variable on the left are given by:

[[math]] \int_{O_N^+}v_{ij}^{2k}=\sum_{\pi\sigma}W_{kN}(\pi,\sigma) [[/math]]

On the other hand, the moments of the variable on the right, which in terms of the fundamental corepresentation [math]u=(u_{ij})[/math] is given by [math]w=\sum_{ij}u_{ij}[/math], are as follows:

[[math]] \int_{SU^q_2}w^{2k}=\sum_{ij}\sum_{\pi\sigma}\delta_\pi^F(i)\delta_\sigma^F(j)W_{kN}(\pi,\sigma) [[/math]]

We deduce that [math]w/\sqrt{N+2}[/math] has the same moments as [math]v_{ij}[/math], which proves our claim.

(5) In order to do the computation over [math]SU^q_2[/math], we can use a well-known matrix model, due to Woronowicz , where the standard generators [math]\alpha,\gamma[/math] are mapped as follows:

[[math]] \pi_u(\alpha)e_k=\sqrt{1-q^{2k}}e_{k-1}\quad,\quad \pi_u(\gamma)e_k=uq^ke_k [[/math]]

Here [math]u\in\mathbb T[/math] is a parameter, and [math](e_k)[/math] is the standard basis of [math]l^2(\mathbb N)[/math]. The point with this representation is that it allows the computation of the Haar functional. Indeed, if [math]D[/math] is the diagonal operator given by [math]D(e_k)=q^{2k}e_k[/math], then we have the following formula:

[[math]] \int _{SU^q_2}x=(1-q^2)\int_{\mathbb T}tr(D\pi_u(x))\frac{du}{2\pi iu} [[/math]]

With the above explicit model in hand, we conclude that the law of the variable that we are interested in is subject to the following formula:

[[math]] \int_{SU^q_2}\varphi(\alpha+\alpha^*+\gamma-q\gamma^*)=(1-q^2)\int_{\mathbb T}tr(D\varphi(M))\frac{du}{2\pi iu} [[/math]]

To be more precise, this formula holds indeed, with [math]M[/math] being as follows:

[[math]] M(e_k)=e_{k+1}+q^k(u-qu^{-1})e_k+(1-q^{2k})e_{k-1} [[/math]]

(6) The point now is that the integral on the right in the above can be computed, by using advanced calculus methods, and this gives the result. We refer here to .

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The computation of the joint free hyperspherical laws remains an open problem. Open as well is the question of finding a more conceptual proof for the above formula.

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].