8b. Asymptotic moments

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Moving ahead now, we would first like to study the distribution of the arbitrary block-modified Wishart matrices [math]\widetilde{W}=(id\otimes\varphi)W[/math]. We will use as before the moment method. However, things will be more tricky in the present setting, and we will need:

Definition

The generalized colored moments of a random matrix

[[math]] W\in M_N(L^\infty(X)) [[/math]]

with respect to a colored integer [math]e=e_1\ldots e_p[/math], and a permutation [math]\sigma\in S_p[/math], are the numbers

[[math]] M^\sigma_e(W)=\frac{1}{N^{|\sigma|}}\,E\left(\sum_{i_1,\ldots,i_p}W^{e_1}_{i_1i_{\sigma(1)}}\ldots W^{e_p}_{i_pi_{\sigma(p)}}\right) [[/math]]

where [math]|\sigma|[/math] is the number of cycles of [math]\sigma[/math].

This is something quite technical, in the spirit of the free probability and free cumulant work in , that we will need in what follows. In order to understand how these generalized moments work, consider the standard cycle in [math]S_p[/math], namely:

[[math]] \gamma=(1\to2\to\ldots\to p\to 1) [[/math]]

If we use this cycle [math]\gamma\in S_p[/math] as our permutation [math]\sigma\in S_p[/math] in the above definition, the corresponding generalized moment of a random matrix [math]W[/math] is then the usual moment:

[[math]] \begin{eqnarray*} M^\gamma_e(W) &=&\frac{1}{N}\,E\left(\sum_{i_1,\ldots,i_p}W^{e_1}_{i_1i_2}\ldots W^{e_p}_{i_pi_1}\right)\\ &=&(E\circ tr)(W^{e_1}\ldots W^{e_p}) \end{eqnarray*} [[/math]]

In general, we can decompose the computation of [math]M^\sigma_e(W)[/math] over the cycles of [math]\sigma[/math], and we obtain in this way a certain product of moments of [math]W[/math]. See .

As a second illustration now, in relation with the usual square matrices, and more specifically with the square matrices [math]\Lambda\in M_n(\mathbb C)\otimes M_n(\mathbb C)[/math] as in Proposition 8.3, we have the following formula, that we will use many times in what follows:

Proposition

Given a usual square matrix, of composed size,

[[math]] \Lambda\in M_n(\mathbb C)\otimes M_n(\mathbb C) [[/math]]

we have the following generalized moment formula,

[[math]] (M^\sigma_e\otimes M^\tau_e)(\Lambda)=\frac{1}{n^{|\sigma|+|\tau|}}\sum_{i_1,\ldots, i_p}\sum_{j_1,\ldots,j_p}\Lambda_{i_1j_1,i_{\sigma(1)}j_{\tau(1)}}^{e_1}\ldots\ldots\Lambda_{i_pj_p,i_{\sigma(p)}j_{\tau(p)}}^{e_p} [[/math]]

valid for any two permutations [math]\sigma,\tau\in S_p[/math], and any colored integer [math]e=e_1\ldots e_p[/math].

Show Proof

This is something obvious, applying the construction in Definition 8.8 with [math]N=n^2[/math], [math]X=\{.\}[/math], [math]W=\Lambda[/math], and then making a tensor product of the corresponding moments [math]M^\sigma_e[/math], [math]M^\tau_e[/math], regarded as linear functionals on [math]M_n(\mathbb C)\otimes M_n(\mathbb C)[/math].

■

Consider now the embedding [math]NC(p)\subset S_p[/math] obtained by “cycling inside each block”. That is, each block [math]b=\{b_1,\ldots,b_k\}[/math] with [math]b_1 \lt \ldots \lt b_k[/math] of a given noncrossing partition [math]\sigma\in NC(p)[/math] produces by definition the cycle [math](b_1\ldots b_k)[/math] of the corresponding permutation [math]\sigma\in S_p[/math]. Observe that the one-block partition [math]\gamma\in NC(p)[/math] corresponds in this way to the standard cycle [math]\gamma\in S_p[/math]. Also, the number of blocks [math]|\sigma|[/math] of a partition [math]\sigma\in NC(p)[/math] corresponds to the number of cycles [math]|\sigma|[/math] of the corresponding permutation [math]\sigma\in S_p[/math].

With these conventions, we have the following result, from , , generalizing our various Wishart matrix moment computations, that we did so far in this book:

Theorem

The asymptotic moments of a block-modified Wishart matrix

[[math]] \widetilde{W}=(id\otimes\varphi)W [[/math]]

with parameters [math]d,m,n\in\mathbb N[/math] as before, are given by the formula

[[math]] \lim_{d\to\infty}M_e\left(\frac{\widetilde{W}}{d}\right)=\sum_{\sigma\in NC(p)}(mn)^{|\sigma|}(M^\sigma_e\otimes M^\gamma_e)(\Lambda) [[/math]]

where [math]\Lambda\in M_n(\mathbb C)\otimes M_n(\mathbb C)[/math] is the square matrix associated to [math]\varphi:M_n(\mathbb C)\to M_n(\mathbb C)[/math].

Show Proof

We use the formula for the matrix entries of [math]\widetilde{W}[/math], directly in terms of the matrix [math]\Lambda[/math] associated to the map [math]\varphi[/math], from Proposition 8.4, namely:

[[math]] \widetilde{W}_{ia,jb}=\sum_{cd}\Lambda_{ca,db}W_{ic,jd} [[/math]]

By conjugating this formula, we obtain the following formula for the entries of the adjoint matrix [math]\widetilde{W}^*[/math], that we will need as well, in what follows:

[[math]] \widetilde{W}_{ia,jb}^* =\sum_{cd}\bar{\Lambda}_{db,ca}\bar{W}_{jd,ic} =\sum_{cd}\Lambda^*_{ca,db}W_{ic,jd} [[/math]]

Thus, we have the following global formula, valid for any exponent [math]e\in\{1,*\}[/math]:

[[math]] \widetilde{W}_{ia,jb}^e=\sum_{cd}\Lambda^e_{ca,db}W_{ic,jd} [[/math]]

In order to compute the moments of [math]\widetilde{W}[/math], observe first that we have:

[[math]] \begin{eqnarray*} tr(\widetilde{W}^{e_1}\ldots\widetilde{W}^{e_p}) &=&\frac{1}{dn}\sum_{i_ra_r}\prod_s\widetilde{W}_{i_sa_s,i_{s+1}a_{s+1}}^{e_s}\\ &=&\frac{1}{dn}\sum_{i_ra_rc_rd_r}\prod_s\Lambda_{c_sa_s,d_sa_{s+1}}^{e_s}W_{i_sc_s,i_{s+1}d_s}\\ &=&\frac{1}{dn}\sum_{i_ra_rc_rd_rj_rb_r}\prod_s\Lambda_{c_sa_s,d_sa_{s+1}}^{e_s}Y_{i_sc_s,j_sb_s}\bar{Y}_{i_{s+1}d_s,j_sb_s} \end{eqnarray*} [[/math]]

The average of the general term can be computed by the Wick rule, which gives:

[[math]] E\left(\prod_sY_{i_sc_s,j_sb_s}\bar{Y}_{i_{s+1}d_s,j_sb_s}\right) =\#\left\{\sigma\in S_p\Big|i_{\sigma(s)}=i_{s+1},c_{\sigma(s)}=d_s,j_{\sigma(s)}=j_s,b_{\sigma(s)}=b_s\right\} [[/math]]

Let us look now at the above sum. The [math]i,j,b[/math] indices range over sets having respectively [math]d,d,m[/math] elements, and they have to be constant under the action of [math]\sigma\gamma^{-1},\sigma,\sigma[/math]. Thus when summing over these [math]i,j,b[/math] indices we simply obtain a factor as follows:

[[math]] f=d^{|\sigma\gamma^{-1}|}d^{|\sigma|}m^{|\sigma|} [[/math]]

Thus, we obtain the following moment formula:

[[math]] (E\circ tr)(\widetilde{W}^{e_1}\ldots\widetilde{W}^{e_p}) =\frac{1}{dn}\sum_{\sigma\in S_p}d^{|\sigma\gamma^{-1}|}(dm)^{|\sigma|}\sum_{a_rc_r}\prod_s\Lambda_{c_sa_s,c_{\sigma(s)}a_{s+1}}^{e_s} [[/math]]

On the other hand, we know from Proposition 8.9 that the generalized moments of the matrix [math]\Lambda\in M_n(\mathbb C)\otimes M_n(\mathbb C)[/math] are given by the following formula:

[[math]] (M^\sigma_e\otimes M^\tau_e)(\Lambda)=\frac{1}{n^{|\sigma|+|\tau|}}\sum_{i_1\ldots i_p}\sum_{j_1\ldots j_p}\Lambda_{i_1j_1,i_{\sigma(1)}j_{\tau(1)}}^{e_1}\ldots\ldots\Lambda_{i_pj_p,i_{\sigma(p)}j_{\tau(p)}}^{e_p} [[/math]]

By combining the above two formulae, we obtain the following moment formula:

[[math]] (E\circ tr)(\widetilde{W}^{e_1}\ldots\widetilde{W}^{e_p}) =\sum_{\sigma\in S_p}d^{|\sigma|+|\sigma\gamma^{-1}|-1}(mn)^{|\sigma|}(M^\sigma_e\otimes M^\gamma_e)(\Lambda) [[/math]]

We use now the standard fact, that we know well from before, that for [math]\sigma\in S_p[/math] we have an inequality as follows, with equality precisely when [math]\sigma\in NC(p)[/math]:

[[math]] |\sigma|+|\sigma\gamma^{-1}|\leq p+1 [[/math]]

Thus with [math]d\to\infty[/math] the sum restricts over the partitions [math]\sigma\in NC(p)[/math], and we get:

[[math]] \lim_{d\to\infty}M_e\big(\widetilde{W}\big)=d^p\sum_{\sigma\in NC(p)}(mn)^{|\sigma|}(M^\sigma_e\otimes M^\gamma_e)(\Lambda) [[/math]]

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

■

With the above result in hand, we are left with the question of recovering the asymptotic law of [math]\widetilde{W}=(id\otimes\varphi)W[/math], out of the asymptotic moments found there. The question here only involves the matrix [math]\Lambda\in M_n(\mathbb C)\otimes M_n(\mathbb C)[/math], and to be more precise, given such a matrix, we would like to find the real or complex probability measure, or abstract distribution, having as colored moments the following numbers:

[[math]] M_e=\sum_{\sigma\in NC_p}(mn)^{|\sigma|}(M^\sigma_e\otimes M^\gamma_e)(\Lambda) [[/math]]

Although this is basically a linear algebra problem, the underlying linear algebra is of quite difficult type, and this question cannot really be solved, in general. We will see however that this question can be solved for our basic examples, coming from Theorem 8.7, and more generally, for a certain joint generalization of all these examples.

General references

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