The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

Abstract

We calculate the k-point generating function of the correlated Jacobi ensemble using supersymmetric methods. We use the result for complex matrices for $k=1$ to derive a closed-form expression for the eigenvalue density. For real matrices we obtain the density in terms of a twofold integral that we evaluate numerically. For both expressions we find agreement when comparing with Monte Carlo simulations. Relations between these quantities for the Jacobi and the Cauchy–Lorentz ensemble are derived.

Appendices

Appendix 1: Supersymmetric Two-Matrix Model

Let us derive another supersymmetric integral which consists of two supermatrices entering in a symmetric way. This integral explicitly shows the symmetry of the correlated Jacobi ensemble under $n_1\leftrightarrow n_2$, $\Lambda \rightarrow \Lambda ^{-1}$, and $x\rightarrow -x$. This symmetry is not immediate in the expression (20) where we have to substitute $b=(1-x)/(1+x)$ since it is given for the correlated Cauchy ensemble. However the supersymmetric integral (20) is certainly simpler to compute than the one we present in this section since we have to deal with only one supermatrix in Eq. (20).

We start from Eq. (4). To apply the same approach as in [33, 34] we have to linearize the arguments of the characteristic polynomials in the $FF^\dagger$ and $BB^\dagger$. This can be achieved by multiplying the matrices in the determinants from the right with $(FF^\dagger + BB^\dagger )$ yielding

$$\begin{aligned} \frac{\det \left( (FF^\dagger - BB^\dagger )(FF^\dagger + BB^\dagger )^{-1} - \kappa _{a2}\mathbbm {1}_{p}\right) }{\det \left( (FF^\dagger - BB^\dagger )(FF^\dagger + BB^\dagger )^{-1}- \kappa _{b1}\mathbbm {1}_{p}\right) } = \left( \frac{1+\kappa _{a2}}{1+\kappa _{b1}}\right) ^p\frac{\det \left( FF^\dagger \widehat{\kappa }_{a2} - BB^\dagger \right) }{\det \left( FF^\dagger \widehat{\kappa }_{b1} - BB^\dagger \right) }. \end{aligned}$$

(72)

with $\widehat{\kappa }=(1-\kappa )/(1+\kappa )$. In the next step we plug Eq. (72) into Eq. (4) and express the determinants as a Gaussian integral over a rectangular supermatrix

$$\begin{aligned} A= & {} \left[ z_{ja}\,z_{ja}^*\,\zeta _{jb}\,\zeta _{jb}^*\right] ,\ \beta =1,\end{aligned}$$

(73)

$$\begin{aligned} A= & {} \left[ z_{ja}\,\zeta _{jb}\right] ,\ \beta =2, \end{aligned}$$

(74)

of dimension $p\times (\gamma k|\gamma k)$, i.e.

$$\begin{aligned} \prod _{a=1}^k\frac{\det \left( FF^\dagger \widehat{\kappa }_{a2} - BB^\dagger \right) }{\det \left( FF^\dagger \widehat{\kappa }_{a1} - BB^\dagger \right) }&= \int d[A]\exp \left( \imath {{\mathrm{tr}}}FF^\dagger A\mathbf {j}A^\dagger + \imath {{\mathrm{tr}}}BB^\dagger AA^\dagger \right) . \end{aligned}$$

(75)

Here we assume for simplicity that the imaginary parts of $\widehat{\kappa }_1=\mathrm{diag}(\widehat{\kappa }_{11},\ldots ,\widetilde{\kappa }_{k1})$ are on the complex upper half-plane. The source matrix is

$$\begin{aligned} \mathbf {j}={{\mathrm{diag}}}\left( \widehat{\kappa }_{11},\ldots ,\widetilde{\kappa }_{k1},\widehat{\kappa }_{12},\ldots ,\widetilde{\kappa }_{k2}\right) \otimes \mathbbm {1}_\gamma . \end{aligned}$$

(76)

We substitute the integral (75) into the generating function (4) and exchange the integrals over F and B with the A integral. The resulting F and B integrals are Gaussian and yield

$$\begin{aligned}&\int d[F]\int d[B]P(F|\mathbbm {1}_p)P(B|\Lambda ) \exp \left( \imath {{\mathrm{tr}}}FF^\dagger A\mathbf {j}A^\dagger + \imath {{\mathrm{tr}}}BB^\dagger AA^\dagger \right) \nonumber \\&\quad = {\det }^{-n_1/\gamma } \left( \mathbbm {1}_{p}-\imath A\mathbf {j}A^\dagger \right) {\det }^{-n_2/\gamma } \left( \mathbbm {1}_{p}-\imath \Lambda A A^\dagger \right) . \end{aligned}$$

(77)

Then the generating function becomes

$$\begin{aligned} Z_{p,\beta }^{k|k}(\kappa ) \propto {{{\mathrm{sdet}}}}^{-p}\left( \mathbbm {1}_{k}+\kappa \right) \int d[A] {\det }^{-n_1/\gamma }\left( \mathbbm {1}_{p}-\imath A\mathbf {j}A^\dagger \right) {\det }^{-n_2/\gamma }\left( \mathbbm {1}_{p}-\imath \Lambda AA^\dagger \right) . \end{aligned}$$

(78)

The normalization constant is independent of $\kappa$ and $\Lambda$. The next step is known as the duality between ordinary and superspace. Due to the invariance of the integrand in Eq. (78) under $A\rightarrow UA$ for an arbitrary $U\in \mathrm{O}(p)$ for $\beta =1$ and $U\in \mathrm{U}(p)$ for $\beta =2$, the integrand only depends on the invariants ${{\mathrm{tr}}}\left( AA^\dagger \right) ^m$ for $m\in \mathbb N$. These invariants are equal to the superinvariants ${{\mathrm{tr}}}\left( AA^\dagger \right) ^m={{\mathrm{str}}}\left( A^\dagger A\right) ^m$, see [46, 47, 52, 53]. Employing this duality in the generating function (78), we arrive at

$$\begin{aligned} Z_{p,\beta }^{k|k}(\kappa )\propto {{\mathrm{sdet}}}^{-p}\left( \mathbbm {1}_{k}+\kappa \right) \int d[A] {{{\mathrm{sdet}}}}^{-n_1/\gamma }\left( \mathbbm {1}_{{\gamma } k|\gamma k}-\imath A^\dagger A \mathbf {j}\right) {{{\mathrm{sdet}}}}^{-n_2/\gamma }\left( \mathbbm {1}_{\gamma k|\gamma k}-\imath A^\dagger \Lambda A\right) . \end{aligned}$$

(79)

The main difference of Eq. (79) to most models discussed in the literature so far is that this one includes two different products of A and $A^\dagger$. Namely, $A^\dagger A$ which arises naturally if invariant matrix models are considered and $A^\dagger \Lambda _{\text {eff}}A$ appearing due to a non-trivial correlation structure. We cannot replace both products by one supermatrix, but we can apply the generalized Hubbard–Stratonovich transformation [50, 52, 53] independently for both products. It yields the following supersymmetric two-matrix model

$$\begin{aligned} Z_{p,\beta }^{k|k}(\kappa )&\propto {{\mathrm{sdet}}}^{-p}\left( \mathbbm {1}_{k}+\kappa \right) \int d[\sigma ]d[\varrho ] I_{n_2}(\varrho )I_{n_1}(\sigma )\exp \left( -{{\mathrm{str}}}\varrho -{{\mathrm{str}}}\sigma \right) \nonumber \\&\quad \times {{\mathrm{sdet}}}^{-1/\gamma }\left( \mathbbm {1}_p\otimes \sigma - \Lambda ^{-1}\otimes \varrho \mathbf {j}\right) , \end{aligned}$$

(80)

where the function $I_{n_i}(\varrho )$, $i=1,2$, is the supersymmetric Ingham–Siegel integral (24). The $(\gamma k|\gamma k)\times (\gamma k|\gamma k)$ dimensional supermatrices $\rho$ and $\sigma$ have the same symmetries as the supermatrix in the third equality of Eq. (20).

We can completely symmetrize the integral in $\rho$ and $\sigma$ by going back to the sources $\widehat{\kappa }\rightarrow \kappa$ and the empirical matrix $\Lambda \rightarrow C_F^{-1/2}C_BC_F^{-1/2}$. Then we have the final result

$$\begin{aligned} Z_{p,\beta }^{k|k}(\kappa )&\propto \int d[\sigma ]d[\varrho ] I_{n_1}(\varrho )I_{n_2}(\sigma )\exp \left( -{{\mathrm{str}}}\varrho -{{\mathrm{str}}}\sigma \right) \nonumber \\&\quad \times {{\mathrm{sdet}}}^{-1/\gamma }\left( C_B\otimes \sigma [(1+\kappa )\otimes \mathbbm {1}_\gamma ] - C_F\otimes \varrho [(1-\kappa )\otimes \mathbbm {1}_\gamma ] \right) . \end{aligned}$$

(81)

This expression is completely invariant under the original symmetry $n_1\leftrightarrow n_2$, $\Lambda \rightarrow \Lambda ^{-1}$, and $\kappa \rightarrow -\kappa$ because the symmetry is achieved by the change $\rho \leftrightarrow \sigma$.

We again underline that the supermatrix model (80) can be in principal computed by expanding the integrand in the Grassmann variables and performing the remaining integrals. However we have now two supermatrices such that this calculation can be a highly non-trivial task. This is the reason why we use more advanced techniques which involve the relation to the correlated Cauchy–Lorentz ensemble, see Sect. 3.1. The supersymmetry result (20) can be obtained from Eq. (80) by rescaling $\sigma \rightarrow \rho \sigma$ and then integrating over $\rho$ which yields the superdeterminant ${{\mathrm{sdet}}}^{-\mu /\gamma }(\sigma +\mathbbm {1}_{\gamma k|\gamma k})$.

Appendix 2: Regularizations of the Integrals in Section 5

The numerical evaluation of the integrals (51) and (52) suffer from the non-integrable singularities of order 3 / 2 at the boundaries, in particular they are of the two forms

$$\begin{aligned} J_1=\int \limits _{b\Lambda _{j+1}^{-1}}^{b\Lambda _j^{-1}}d{r}\frac{f(r)}{\left| b\Lambda _j^{-1}-r\right| ^{3/2}},\quad J_2=\int \limits _{b\Lambda _j^{-1}}^{b\Lambda _{j-1}^{-1}}d{r}\frac{f(r)}{\left| b\Lambda _j^{-1}-r\right| ^{3/2}} \end{aligned}$$

(82)

for certain real valued functions f(r) without singularities in the interval $[b\Lambda _{j+1}^{-1},b\Lambda _j^{-1}]$. As already said the integrals are taken via Cauchy’s principal value because of the original imaginary increment $\imath \varepsilon$. Thus we can effectively regularize the integral as follows¹

$$\begin{aligned} J_1&= \lim _{\varepsilon \rightarrow 0}\mathrm{Re}\int \limits _{b\Lambda _{j+1}^{-1}}^{b\Lambda _j^{-1}+\varepsilon }d{r}\frac{f(r)}{\left( (b+\imath \varepsilon )\Lambda _j^{-1}-r\right) ^{3/2}}\nonumber \\&=\lim _{\varepsilon \rightarrow 0}\mathrm{Re}\int \limits _{b\Lambda _{j+1}^{-1}}^{b\Lambda _j^{-1}+\varepsilon }d{r}\frac{f(r)-f(b\Lambda _{j}^{-1})}{\left( (b+\imath \varepsilon )\Lambda _j^{-1}-r\right) ^{3/2}}+\lim _{\varepsilon \rightarrow 0}\mathrm{Re}\left[ \frac{2f(b\Lambda _{j}^{-1})}{\sqrt{(b+\imath \varepsilon )\Lambda _j^{-1}-r}}\right] _{r=b\Lambda _{j+1}^{-1}}^{b\Lambda _j^{-1}+\varepsilon } \nonumber \\&=\int \limits _{b\Lambda _{j+1}^{-1}}^{b\Lambda _j^{-1}}d{r}\frac{f(r)-f(b\Lambda _j^{-1})}{\left| b\Lambda _j^{-1}-r\right| ^{3/2}}-\frac{2f(b\Lambda _{j}^{-1})}{\sqrt{|b\Lambda _j^{-1}-b\Lambda _{j+1}^{-1}|}} \end{aligned}$$

(83)

and similar for the other integral (then the imaginary part is needed)

$$\begin{aligned} J_2&= \int \limits _{b\Lambda _j^{-1}}^{b\Lambda _{j-1}^{-1}}d{r}\frac{f(r)-f(b\Lambda _j^{-1})}{\left| b\Lambda _j^{-1}-r\right| ^{3/2}}-\frac{2f(b\Lambda _{j}^{-1})}{\sqrt{|b\Lambda _j^{-1}-b\Lambda _{j-1}^{-1}|}}. \end{aligned}$$

(84)

The minus sign in Eq. (84) in front of the second term results from taking the imaginary part despite it is evaluated at the upper boundary. The cut-off of the intervals can be also chosen independently of $\varepsilon$. The reason is that the other boundary term of the integration by parts vanishes due to taking the real or imaginary part, respectively.

We define the following two one-fold integrals

$$\begin{aligned} g_{a,c,l}^0(b;\Lambda )=\int _{V_l}\frac{r^{a/2}dr}{(1+r)^{c/2}\sqrt{|\det (b\Lambda ^{-1}-r\mathbbm {1}_p)|}} \end{aligned}$$

(85)

and

$$\begin{aligned}&g_{a,c,l}^1(b;\Lambda ;\Lambda _i)\\= & {} \left\{ \begin{array}{ll} \displaystyle \mathrm{sign}(p-l-i)\int _{V_l}\frac{r^{a/2}dr}{(1+r)^{c/2}|b\Lambda _i^{-1}-r|\sqrt{|\det (b\Lambda ^{-1}-r\mathbbm {1}_p)|}}, &{} \\ &{} l\ne p-i,p-i+1,\\ &{}\\ \displaystyle -\frac{2(b\Lambda _{i}^{-1})^{a/2}}{(1+b\Lambda _{i})^{c/2}\sqrt{|b\Lambda _i^{-1}-b\Lambda _{i+1}^{-1}|}\sqrt{|\det (b\Lambda _{\ne i}^{-1}-b\Lambda _{i}^{-1}\mathbbm {1}_{p-1})|}}+\int _{V_{p-i}}\frac{dr}{|b\Lambda _i-r|^{3/2}}&{} \\ \displaystyle \times \left[ \frac{r^{a/2}}{(1+r)^{c/2}\sqrt{|\det (b\Lambda _{\ne i}^{-1}-r\mathbbm {1}_{p-1})|}}-\frac{(b\Lambda _i^{-1})^{a/2}}{(1+b\Lambda _i^{-1})^{c/2}\sqrt{|\det (b\Lambda _{\ne i}^{-1}-b\Lambda _i^{-1}\mathbbm {1}_{p-1})|}}\right] ,&{}\\ &{} l=p-i,\\ &{}\\ \displaystyle \frac{2(b\Lambda _{i}^{-1})^{a/2}}{(1+b\Lambda _{i})^{c/2}\sqrt{|b\Lambda _i^{-1}-b\Lambda _{i-1}^{-1}|}\sqrt{|\det (b\Lambda _{\ne i}^{-1}-b\Lambda _{i}^{-1}\mathbbm {1}_{p-1})|}}-\int _{V_{p-i+1}}\frac{dr}{|b\Lambda _i-r|^{3/2}}&{} \\ \displaystyle \times \left[ \frac{r^{a/2}}{(1+r)^{c/2}\sqrt{|\det (b\Lambda _{\ne i}^{-1}-r\mathbbm {1}_{p-1})|}}-\frac{(b\Lambda _i^{-1})^{a/2}}{(1+b\Lambda _i^{-1})^{c/2}\sqrt{|\det (b\Lambda _{\ne i}^{-1}-b\Lambda _i^{-1}\mathbbm {1}_{p-1})|}}\right] ,&{}\\ &{} l=p-i+1. \end{array}\right. \nonumber \end{aligned}$$

(86)

Then we can combine the discussion about the splitting of the integral over $\mathbb {R}_+^2$ into disjoint sets and the regularization of the 3 / 2-singularities. Therefore we explicitly have for the imaginary parts of the integrals (50), (51), and (52)

$$\begin{aligned}&\frac{1}{\pi }\lim _{\varepsilon \rightarrow 0}\mathrm{Im}\,N_{a,c,d_1,d_2}^0(b+\imath \varepsilon ;\Lambda )\\&\quad =\frac{1}{\pi }\sum _{\begin{array}{c} 0\le l_1,l_2\le p \\ l_1+l_2\in 2\mathbb {N}_0+1 \end{array}}(-1)^{(l_1+l_2+1)/2}\mathrm{sign}(l_1-l_2) \nonumber \\&\qquad \times \,\det \left[ \begin{array}{cc} g_{a+d_1+1,c+d_1+1,l_1}^0(b;\Lambda ) &{} g_{a+d_1-1,c+d_1+1,l_1}^0(b;\Lambda ) \\ g_{a+d_2+1,c+d_2+1,l_2}^0(b;\Lambda ) &{} g_{a+d_2-1,c+d_2+1,l_2}^0(b;\Lambda ) \end{array}\right] ,\nonumber \end{aligned}$$

(87)

$$\begin{aligned}&\frac{1}{\pi }\lim _{\varepsilon \rightarrow 0}\mathrm{Im}\,N_{a,c,d}^1(\kappa ;\Lambda ;\Lambda _j)\\&\quad =\frac{1}{\pi }\sum _{\begin{array}{c} 0\le l_1,l_2\le p \\ l_1+l_2\in 2\mathbb {N}_0+1 \end{array}}(-1)^{(l_1+l_2+1)/2}\mathrm{sign}(l_1-l_2) \nonumber \\&\qquad \times \det \left[ \begin{array}{cc} g_{a+2,c,l_1}^1(b;\Lambda ;\Lambda _j) &{} g_{a,c,l_1}^1(b;\Lambda ;\Lambda _j) \\ g_{a+d+1,c+d+1,l_2}^0(b;\Lambda ) &{} g_{a+d-1,c+d+1,l_2}^0(b;\Lambda ) \end{array}\right] ,\nonumber \end{aligned}$$

(88)

$$\begin{aligned}&\frac{1}{\pi }\lim _{\varepsilon \rightarrow 0}\mathrm{Im}\,N_{a,c}^2(\kappa ;\Lambda ;\Lambda _i,\Lambda _j)\\&\quad =\frac{1}{\pi }\sum _{\begin{array}{c} 0\le l_1,l_2\le p \\ l_1+l_2\in 2\mathbb {N}_0+1 \end{array}}(-1)^{(l_1+l_2+1)/2}\mathrm{sign}(l_1-l_2)\nonumber \\&\qquad \times \,\det \left[ \begin{array}{cc} g_{a+2,c,l_1}^1(b;\Lambda ;\Lambda _i) &{} g_{a,c,l_1}^1(b;\Lambda ;\Lambda _i) \\ g_{a+2,c,l_2}^1(b;\Lambda ;\Lambda _j) &{} g_{a,c,l_2}^1(b;\Lambda ;\Lambda _j) \end{array}\right] .\nonumber \end{aligned}$$

(89)

These results can be combined with Eq. (48) to find the level density $S'_1(b)$ of the correlated Lorentz ensemble, see Eq. (59).