The Circular Law (1)

This page gives some basic results and prove the spectral law of an Gaussian i.i.d random matrix.^[1]

All random variables are defined on a unique common probability space \((\Omega,\mathcal{A},\mathbb{P})\). A typical element of \(\Omega\) is denoted \(\omega\). We write \(a.s.\), \(a.a.\), and \(a.e.\) for almost surely, Lebesgue almost all, and Lebesgue alomost everwhere respectively.

Two kinds of spectra

Label the eigenvalues of \(A \in \mathcal{M}_{n}(\mathbb{C})\) as \(\lambda_1(A),\cdots ,\lambda_{n}(A)\) so that $1(A) {n}(A) $. The singular values of \(A\) are defined by \[ s_k(A):=\lambda_{k}(\sqrt{AA^{*}}). \]

We have \[ s_1(A)\geqslant \cdots \geqslant s_{n}(A)\geqslant 0. \]

The matrices \(A,A^{\mathsf{T}},A^{*}\) have the same singular values. The Hermitian matrix \[ H_{A}:= \begin{pmatrix} 0 & A \\ A^{*} & 0 \\ \end{pmatrix} \]

has eigenvalues \(s_1(A),-s_1(A),\cdots ,s_n(A),-s_n(A)\). Notice the mapping \(A \mapsto H_{A}\) is linear in \(A\), in contrast with the mapping \(A \mapsto \sqrt{AA^{*}}\).

Define the operator norm or spectral norm of \(A\) as \[ \left\| A \right\|_{2\rightarrow 2}:=\max _{\left\| x \right\|_{2}=1}\left\| Ax \right\|_{2}=s_1(A) \]

Notice that if \(A\) is non-singular then \(s_i(A ^{-1})=s_{n-i}(A) ^{-1}\) for \(1\leqslant i\leqslant n\) and \(s_n(A)=s_1(A ^{-1}) ^{-1}=\left\| A^{-1} \right\|_{2\rightarrow 2} ^{-1}\).

Courant-Fischer variational formulas Denoting \(\mathcal{G}_{n,i}\) the Grassmannian of all \(i\)-dimensional subspaces, we have \[ \begin{aligned} s_i(A)&=\min _{E\in \mathcal{G}_{n,i-1}} \max _{\left\| x \right\|_{2}=1,x\perp E} \left\| Ax \right\|_{2}\\ &=\max_{E \in \mathcal{G}_{n,n-i}} \min _{\left\| x \right\|_{2}=1,x\perp E}\left\| Ax \right\|_{2} \\ &=\min _{E\in \mathcal{G}_{n,n-i+1}} \max_{\left\| x \right\|_{2}=1,x\in E}\left\| Ax \right\|_{2} \\ &=\max _{E\in \mathcal{G}_{n,i}} \min _{\left\| x \right\|_{2}=1,x \in E} \left\| Ax \right\|_{2} \end{aligned} \]

Corollary Let \(A \in \mathcal{m,n}\) be given, and let \(A_r\) denote a submatrix of \(A\) obtained by deleting a total of \(r\) rows and/or columns from \(A\). Then \[ s_k(A)\geqslant s_k(A_r)\geqslant s_{k+r}(A), k=1,\cdots ,\min\{m,n\} \tag{1.1} \]

where for \(X \in \mathcal{M}_{p,q}\) we set \(s_j(X)\equiv 0\) if \(j>\min\{p,q\}\).

proof It suffices to consider the case \(r=1\), i.e., \(s_k(A)\geqslant s_{k}(A_1)\geqslant s_{k+1}(A)\). If \(A_1\) is formed from \(A\) by deleting column \(s\), denote by \(e_s\) the standard unit basis vector with a \(1\) in position \(s\). If \(x\in \mathbb{C}^{n}\), denote by \(\xi \in \mathbb{C}^{n-1}\) the vector obtained by deleting entry \(s\) from \(x\). \[ \begin{aligned} s_k(A)&=\min_{E\in \mathcal{G}_{n,k-1}} \max_{x\perp E,\left\| x \right\|_{2}=1}\left\| Ax \right\|_{2} \\ &\geqslant \min_{E\in \mathcal{G}_{n,k-1}} \max_{x\perp E,x\perp e_s,\left\| x \right\|_{2}=1}\left\| Ax \right\|_{2}\\ &= \min_{E\in \mathcal{G}_{n-1,k-1}} \max_{\xi\perp E,\left\| x \right\|_{2}=1}\left\| A_1\xi \right\|_{2}=s_k(A_1) \end{aligned} \]

\[ \begin{aligned} s_{k+1}(A)&=\max _{\mathcal{G}_{n,n-k-1}}\min _{\left\| x \right\|_{2}=1,x\perp E} \left\| Ax \right\|_{2} \\ &\leqslant \max _{\mathcal{G}_{n,n-k-1}}\min _{x\perp E,x\perp e_s,\left\| x \right\|_{2}=1} \left\| Ax \right\|_{2} \\ &= \max _{\mathcal{G}_{n-1,n-k-1}}\min _{\xi\perp E,\left\| \xi \right\|_{2}=1} \left\| A_1\xi \right\|_{2}=s_k(A_1) \end{aligned} \]

If a row of \(A\) is deleted, apply the same argument to \(A^{*}\), which has the same singular values as \(A\).

Lemma Let \(C \in \mathcal{M}_{m,n}\), \(V_{k} \in \mathcal{M}_{m,k}\) and \(W_{k} \in \mathcal{M}_{n,k}\) be given, where \(k\leqslant \min\{m,n\}\) and \(V_k,W_k\) have orthonormal columns. Then - \(s_i(V_k^{*}CW_{k})\leqslant s_i(C)\), \(i=1,\cdots,k\) - \(\lvert \det V_k^{*}CW_{k} \rvert \leqslant s_1(C)\cdots s_k(C)\).

proof There are unitary matrices \(V\in \mathcal{M}_{m}\) and \(W \in \mathcal{M}_{n}\) such that \(V=[V_k \ *]\) and \(W=[W_k \ *]\). Since \(V_k^{*}CW_{k}\) is the upper left \(k\)-by-\(k\) submatrix of \(V^{*}CW\), (1.1) and unitary invariance of singular values ensure that \(s_i(V_k^{*}CW_{k})\leqslant s_i(V^{*}CW)=s_i(C),i=1,\cdots ,k\), and hence \(\lvert \det V_k^{*}CW_{k} \rvert =s_1(V_k^{*}CW_{k})\cdots s_k(V_{k}^{*}CW_{k})\leqslant s_1(C)\cdots s_k(C)\).

Weyl inequalities For every \(A \in \mathcal{M}_{n}(\mathbb{C})\) and \(1\leqslant k\leqslant n\), \[ \prod_{i=1}^{k} \lvert \lambda_{i}(A) \rvert \leqslant \prod_{i=1}^{k} s_i(A). \tag{1.2} \]

proof By the Schur triangularization theorem, there is a unitary \(U \in \mathcal{M}_{n}(\mathbb{C})\) such that \(U^{*}AU=\Delta\) is upper triangular and \(diag \Delta=(\lambda_1,\cdots ,\lambda_n)\). Let \(U_k \in \mathcal{M}_{n,k}\) denote the first \(k\) columns of \(U\). It's easy to know that \(U_k^{*}AU_{k}=\Delta_{k}\) is the upper left \(k\)-by-\(k\) principal submatrix of \(\Delta\), and \(diag \Delta_k=(\lambda_1,\cdots ,\lambda_n)\). Apply the lemma with \(C=A\) and \(V_k=W_k=U_k\) to conclude that \[ \lvert \lambda_1(A)\cdots \lambda_k(A) \rvert =\lvert \det \Delta_k \rvert =\lvert \det U_k^{*}AU_{k} \rvert \leqslant s_1(A)\cdots s_k(A) \]

It's easy to know that (1.2) holds quality when \(k=n\).

The reversed form ${i=n-k+1}^{n} s_i(A){i=n-k+1}^{n} _i(A) $ for every \(1\leqslant k\leqslant n\) can be deduced easily.

Theorem Let \(A\in \mathcal{M}_{m,p}\) and \(B\in \mathcal{M}_{p,n}\) be given, let \(q\equiv \min\{n,p,m\}\). Then \[ \prod_{i=1}^{k} s_i(AB)\leqslant \prod_{i=1}^{k} s_i(A)s_i(B), \ k=1,\cdots ,q \tag{1.3} \]

If \(n=p=m\), then equality holds in (1.3) for \(k=n\).

proof Let \(AB=V\Sigma W^{*}\), and let \(V_k \in \mathcal{M}_{m,k}\) and \(W_k\in \mathcal{M}_{n,k}\) denote the first \(k\) columns of \(V\) and \(W\), respectively. Then \(V_k^{*}(AB)W_k=diag(s_1(AB),\cdots ,s_k(AB))\). Since \(p\geqslant k\), use the polar decomposition to write the product \(BW\in \mathcal{M}_{p,k}\) as \(BW_{k}=X_{k}Q\), where \(X_k\in \mathcal{M}_{p,k}\) has orthonormal columns, \(Q\in \mathcal{M}_{k}\) is positive semidefinite, \(\det Q^{2}=\det W_k^{*}(B^{*}B)W_k\leqslant s_1(B^{*}B)\cdots s_k(B^{*}B)=s_1(B)^{2}\cdots s_k(B)^{2}\) by lemma. Use lemma again to compute \[ s_1(AB)\cdots s_k(AB)= \lvert \det V_k^{*}(AB)W_k \rvert = \lvert \det V_k^{*}AX_{k} \det Q \rvert \leqslant (s_1(A)\cdots s_k(A))(s_1(B)\cdots s_k(B)) \]

If \(n=p=m\), then \(s_1(AB)\cdots s_n(AB)=\lvert \det AB \rvert =\lvert \det A \rvert \lvert \det B \rvert =s_1(A)\cdots s_k(A)s_1(B)\cdots s_k(B)\).

(Strong) majorization inequalities If \(A\) is nonsingular, then

\[ \sum_{i=1}^{k} \log \lvert \lambda_i(A) \rvert \leqslant \sum_{i=1}^{k} \log s_i(A), k=1,\cdots ,n, \]

with equality for \(k=n\).

proof It's just a corollary of the previous theorem.

Using some classic inequality trick, we can introduce Schur-convex or isotone functions and prove weak majorization inequality.

Definition Let \(x=[x_i],y=[y_i]\in \mathbb{R}^{n}\) be given vectors, \(x_{[1]}\geqslant \cdots \geqslant x_{[n]}\) and \(y_{[1]}\geqslant \cdots \geqslant y_{[n]}\). We say that \(y\) weakly majorizes \(x\) if \[ \sum_{i=1}^{k} x_{[i]}\leqslant \sum_{i=1}^{k} y_{[i]}, \quad k=1,2,\cdots,n \tag{1.4} \]

If (1.4) holds equality when \(k=n\), then we say \(y\) majorizes \(x\), denoted as \(x \prec y\).

Schur Suppose \(A=(a_{ij})\in \mathbb{C}^{n\times n}\) is Hermitian with eigenvalues \(\{\lambda_i\}_{1\leqslant i\leqslant n}\), then \(\{a_{ii}\}\prec \{\lambda_{i}\}\).

proof \(\sum_{i=1}^{r} \lambda_{[i]}=\sup _{E\in \mathcal{G}_{n,r}}\operatorname{tr}(A|_{E})\geqslant \sum_{i=1}^{r} a_{[ii]}\).

Horn Suppose \(\{d_i\}\prec \{\mu_i\}\), then there exists a symmetric matrix with primary diagonal entries \(\{d_i\}\), eigenvalues \(\{\mu_i\}\).

proof We construct orthogonal matrix \(Q\), such that \(Q^{\mathsf{T}}diag(\mu_1,\cdots ,\mu_n)Q\) has primary diagonal entries \(\{d_i\}\). Use induction to prove it, refer to https://zhuanlan.zhihu.com/p/76422480.

Hardy-Littlewood-Polya Suppose \(x,y\in \mathbb{R}^{n}\), then \(x\prec y\) if and only if there is a doubly substochastic \(A\in \mathcal{M}_{n}(\mathbb{R})\) such that \(x=Ay\).

proof If \(x\prec y\), then the Horn theorem gives an orthogonal matrix \(Q=(q_{ij})\) such that \(x_i=\sum_{j=1}^{n} q_{ij}^{2}y_j,\forall 1\leqslant i\leqslant n\). \(A=(q_{ij}^{2})\) satisfies the condition. For the rest, refer to https://zhuanlan.zhihu.com/p/76422480 as well.

Theorem \(x\) and \(y\) are two given vectors with nonnegative entries. Then \(y\) weakly majorizes \(x\) if and only if there is a doubly substochastic \(Q\in \mathcal{M}_{n}(\mathbb{R})\) such that \(x=Qy\).

proof If \(Q\in \mathcal{M}_{n}(\mathbb{R})\) is doubly substochastic and \(Qy=x\) with \(x,y\geqslant 0\), let \(S\in \mathcal{M}_{n}(\mathbb{R})\) be doubly stochastic and such that \(0\leqslant Q\leqslant S\). If \((Qy)_{i_1}=(Qy)_{[1]},\cdots ,(Qy)_{i_k}=(Qy)_{[k]}\), then \[ \sum_{i=1}^{k} (Qy)_{[i]}=\sum_{j=1}^{k} (Qy)_{i_j}\leqslant \sum_{j=1}^{k} (Sy)_{i_j}\leqslant \sum_{i=1}^{k} (Sy)_{[i]}\leqslant \sum_{i=1}^{k} y_{[i]} \]

for \(k=1,2,\cdots,n\). Thus, \(y\) weakly majorizes \(Qy=x\).

Conversely, suppose \(y\) weakly majorizes \(x\) and \(x,y\geqslant 0\). If \(x=0\), let \(Q\equiv 0\). If \(x\neq 0\), let \(\varepsilon\) denote the smallest positive entry of \(x\), set \(\delta\equiv (y_1-x_1)+\cdots +(y_n-x_n)\geqslant 0\), and let \(m\) be any positive integer such that \(\varepsilon\geqslant \delta/m\). Let \[ \xi\equiv [x_1,\cdots ,x_n,\delta/m,\cdots ,\delta/m]^{\mathsf{T}}\in \mathbb{R}^{n+m} \]

and let \[ \eta=[y_1,\cdots ,y_n,0,\cdots ,0]^{\mathsf{T}}\in \mathbb{R}^{n+m} \]

Then \[ \sum_{i=1}^{k} \xi_{[i]}\leqslant \sum_{i=1}^{k} \eta_{[i]}, \quad k=1,2,\cdots,m+n \]

with equality for \(k=m+n\). Thus, there is strong majorization relationship between \(\xi\) and \(\eta\). By Hardy-Littlewood-Polya theorem, there is a doubly stochastic \(S\in \mathcal{M}_{m+n}(\mathbb{R})\) such that \(\xi=S\eta\). If we let \(Q\) denote the upper left \(n\)-by-\(n\) principal submatrix of \(S\), \(Q\) is doubly substochastic and \(x=Qy\).

Lemma Let \(x_1,\cdots ,x_n\), \(y_1,\cdots ,y_n\) be \(2n\) given real numbers such that \(x_1\geqslant x_2\geqslant \cdots \geqslant x_n\),\(y_1\geqslant y_2\geqslant \cdots \geqslant y_n\), and \[ \sum_{i=1}^{k} x_i\leqslant \sum_{i=1}^{k} y_i, \quad k=1,2,\cdots,n \tag{1.5} \]

If \(f(t)\) is a given real-valued increasing convex function on the interval \([\min\{x_n,y_n\},y_1]\), then \(f(x_1)\geqslant \cdots \geqslant f(x_n)\), \(f(y_1)\geqslant \cdots f(y_n)\), and \[ \sum_{i=1}^{k} f(x_i)\leqslant \sum_{i=1}^{k} f(y_i), \quad k=1,2,\cdots,n \]

If equality holds for \(k=n\) in (1.4) and if \(f(\cdot)\) is convex (but not necessarily increasing) on the interval \([y_n,y_1]\), then \[ \sum_{i=1}^{n} f(x_i)\leqslant \sum_{i=1}^{n} f(y_i) \tag{1.6} \]

proof The asserted inequality is trivial for \(k=1\). Let \(k\) be a given integer with \(2\leqslant k\leqslant n\). Since \(x\) is weakly majorized by \(y\), there is a doubly stochastic \(S\in \mathcal{M}_{k}(\mathbb{R})\) such that \(x\leqslant Sy\).

Notice that the entries of \(Sy\) lie in the interval \([y_n,y_1]\) and hence are in the domain of \(f(\cdot)\). Using the monotonicity and convexity of \(f(\cdot)\), we have

\[ \sum_{i=1}^{k} f(x_i)\leqslant \sum_{i=1}^{k} f\left( \sum_{j=1}^{k} s_{ij}y_j \right) \leqslant \sum_{i=1}^{k} \sum_{j=1}^{k} s_{ij}f(y_j)=\sum_{j=1}^{k} \left( \sum_{i=1}^{k} s_{ij} \right) f(y_j)=\sum_{j=1}^{k} f(y_j) \]

If equality holds for \(k=n\), there is a doubly stochastic \(S\in \mathcal{M}_{n}(\mathbb{R})\) such that \(x=Sy\) and hence convexity of \(f(\cdot)\) gives \[ \sum_{i=1}^{n} f(x_i)=\sum_{i=1}^{n} f\left( \sum_{j=1}^{n} s_{ij}y_j \right) \leqslant \sum_{i=1}^{n} \sum_{j=1}^{n} s_{ij}f(y_j)=\sum_{j=1}^{n} f(y_j) \]

Theorem Let \(A\in \mathcal{M}_{n}(\mathbb{C})\), then \[ \sum_{i=1}^{k} \lvert \lambda_{i}(A) \rvert^{p}\leqslant \sum_{i=1}^{k} s_i(A)^{p} \ \text{for} \ k=1,\cdots ,n, \forall p>0 \]

\[ \lvert \operatorname{tr} \ A \rvert \leqslant \sum_{i=1}^{n} s_i(A) \]

We have a more general version of this theorem, please refer to [HJ, Theorem 3.3.13].^[2]

We can deduce from Weyl's inequalities that \[ \sum_{i=1}^{n} \lvert \lambda_i(A) \rvert ^{2}\leqslant \sum_{i=1}^{n} s_i(A)^{2}=\operatorname{tr}(AA^{*})=\sum_{i,j=1}^{n} \lvert A_{i,j} \rvert ^{2}. \tag{1.7} \]

Since \(s_1(\cdot)=\left\| \cdot \right\|_{2\rightarrow 2}\) we have for any \(A,B\in \mathcal{M}_{n}(\mathbb{C})\) that \[ s_1(AB)\leqslant s_1(A)s_1(B), \quad s_1(A+B)\leqslant s_1(A)+s_1(B). \tag{1.8} \]

We define tha empirical eigenvalues and singular values measures by \[ \mu_{A}:=\frac{1}{n}\sum_{k=1}^{n} \delta_{\lambda_k(A)}, \quad \nu_{A}:=\frac{1}{n}\sum_{k=1}^{n} \delta_{s_k(A)}. \]

\(\mu_{A}\) ad \(\nu_{A}\) are supported respectively in \(\mathbb{C}\) and \(\mathbb{R}_{+}\). From (1.2) we get \[ \begin{aligned} \int_{}^{} \log \lvert \lambda \rvert \mathrm{d}\mu_{A}(\lambda)&= \frac{1}{n}\sum_{i=1}^{n} \log \lvert \lambda_i(A) \rvert \\ &=\frac{1}{n}\sum_{i=1}^{n} \log (s_i(A)) \\ &=\int_{}^{} \log (s) \mathrm{d}\nu_{A}(s). \end{aligned} \]

The map \(A\mapsto (s_1(A),\cdots s_n(A))\) is 1-Lipschitz: for any \(A,B\in \mathcal{M}_{n}(\mathbb{C})\),

\[ \max_{1\leqslant i\leqslant n}\lvert s_i(A)-s_i(B) \rvert \leqslant s_1(A-B). \]

The Hilbert-Schimidt norm/ Schur norm/ Frobenius norm, is

\[ \left\| A \right\|_{2}^{2}=\operatorname{tr}(AA^{*})=\sum_{i=1}^{n} s_i(A)^{2}=n \int_{}^{} s^{2} \mathrm{d}\nu_{A}(s). \]

In the sequel, we say that a sequence of (possibly signed) measures \((\eta_{n})_{n\geqslant 1}\) on \(\mathbb{C}\) (respectively on \(\mathbb{R}\)) tends weakly to a (possibly signed) measure \(\eta\), and we denote \[ \eta_{n} \rightsquigarrow \eta, \]

when for all continuous and bounded function \(f\colon \mathbb{C}\rightarrow \mathbb{R}\) (respectively \(f\colon \mathbb{R}\rightarrow \mathbb{R}\)), \[ \lim_{n \to \infty}\int_{}^{} f \mathrm{d}\eta_n=\int_{}^{} f \mathrm{d}\eta. \]

Spectral Law

Since \(\mathbb{C}\equiv \mathbb{R}^{2}\), we now consider the case where \(X_{11}\sim \mathcal{N}(0,\frac{1}{2}I_2)\). We denote \(G\) instead of \(X\) in order to distinguish the Gaussian case from the general case. We say \(G\) belongs to the Complex Ginibre Ensemble.

The Lebesgue density of the \(n\times n\) random matrix \(G=(G_{ij})\) in \(\mathcal{M}_{n}(\mathbb{C})\equiv \mathbb{C}^{n\times n}\) is \[ A\in \mathcal{M}_{n}(\mathbb{C})\mapsto \pi^{-n^{2}}\mathrm{e}^{-\sum_{i,j=1}^{n} \lvert A_{ij} \rvert ^{2}} . \tag{2.1} \]

Notice that it's a Boltzmann distribution with energy \[ A\mapsto \sum_{i,j=1}^{n} \lvert A_{ij} \rvert ^{2}=\operatorname{tr}(AA^{*})=\left\| A \right\|_{2}^{2}=\sum_{i=1}^{n} s_i^{2}(A). \]

So this law is unitary invariant, in the sense that if \(U\) and \(V\) are \(n\times n\) unitary matrices then \(UGV\) and \(G\) are equally distributed.

We set \[ \Delta_n:=\{(z_1,\cdots ,z_n)\in \mathbb{C}^{n}\colon \lvert z_1 \rvert \geqslant \cdots \geqslant \lvert z_n \rvert \}. \]

Recall the Lebesgue density of the \(n\times n\) random matrix \(G=(G_{ij})\) in \(\mathcal{M}_{n}(\mathbb{C})\equiv \mathbb{C}^{n\times n}\) is \[ G\in \mathcal{M}_{n}(\mathbb{C})\mapsto \pi^{-n^{2}}\mathrm{e}^{-\sum_{i,j=1}^{n} \lvert G_{ij} \rvert ^{2}} . \]

Theorem \((\lambda_1(G),\cdots ,\lambda_n(G))\) has density \(n!\varphi_{n} \mathbf{1}_{\Delta_n}\) where \[ \varphi_{n}(z_1,\cdots ,z_n)=\frac{\pi^{-n}}{1!2!\cdots n!}\exp \left( -\sum_{k=1}^{n} \lvert z_k \rvert ^{2} \right) \prod_{1\leqslant i<j\leqslant n}^{} \lvert z_i-z_j \rvert ^{2}. \]

In particular, for every symmetric Borel function \(F\colon \mathbb{C}^{n}\rightarrow \mathbb{R}\), \[ \mathbb{E}[F(\lambda_1(G),\cdots ,\lambda_n(G))]=\int_{\mathbb{C}^{n}}^{} F(z_1,\cdots ,z_n)\varphi_{n}(z_1,\cdots ,z_n) \mathrm{d}z_1\cdots \mathrm{d}z_n \]

There are two ways to get \(\varphi_n\), using diagonalization or Schur decomposition respectively.^[3] We apply the first one since it is more understandable.

We may suppose that \(G\) has distinct eigenvalues since matrices with multiple eigenvalues have measure \(0\). Let \(X\) be the \(N\times N\) matrix whose columns are the eigenvectors of \(G\) so that \(G=XEX^{-1}\). By differentiation, \[ X^{-1}\mathrm{d}GX=\mathrm{d}E+\mathrm{d}BE-E\mathrm{d}B \tag{2.2} \]

with \[ \mathrm{d}B=X^{-1}\mathrm{d}X. \]

\[ (X^{-1}\mathrm{d}GX)_{jj}^{r}=\mathrm{d}z_j^{r}=\mathrm{d}x_j, \quad (X^{-1}\mathrm{d}GX)_{jj}^{i}=\mathrm{d}z_j^{i}=\mathrm{d}y_j, \]

\[ (X^{-1}\mathrm{d}GX)_{jk}^{r}=(x_k-x_j)\mathrm{d}B_{jk}^{r}-(y_k-y_j)\mathrm{d}B_{jk}^{i}, \quad j\neq k \]

\[ (X^{-1}\mathrm{d}GX)_{jk}^{i}=(y_k-y_j)\mathrm{d}B_{jk}^{r}+(x_k-x_j)\mathrm{d}B_{jk}^{i},\quad j\neq k \]

where \(x_j,y_j\) are the real and imaginary parts of \(z_j\), the diagonal elements of \(E\). Denote \(\mathrm{d}\hat{G}=X^{-1}\mathrm{d}GX\).

The Jacobian \[ \frac{\partial (\mathrm{d}\hat{G}_{jj}^{r},\mathrm{d}\hat{G}_{jj}^{i})}{\partial (x_j,y_j)}=1 \]

\[ \frac{\partial (\mathrm{d}G_{jk}^{r},\mathrm{d}G_{jk}^{i})}{\partial (\mathrm{d}B_{jk}^{r},\mathrm{d}B_{jk}^{i})}=(x_k-x_j)^{2}+(y_k-y_j)^{2}=\lvert z_k-z_j \rvert ^{2} \]

The volume element is therefore given by \[ \mu(\mathrm{d}G)=\mu(\mathrm{d}\hat{G})=\prod_{j\neq k}^{} \lvert z_k-z_j \rvert ^{2}\mathrm{d}B_{jk}^{r}\mathrm{d}B_{jk}^{i}\prod_{i}^{} \mathrm{d}x_i\mathrm{d}y_i. \tag{2.3} \]

We now evaluate the integral \[ J=\int_{}^{} \exp [-\operatorname{tr}(G^{*}G)]\prod_{j\neq k}^{} \mathrm{d}B_{jk}^{r}\mathrm{d}B_{jk}^{i}. \tag{2.4} \]

Evaluation of (2.4)

Similar to QR decomposition, any complex nonsingular \(N\times N\) matrix \(X\) can be expressed in one and only one way as \[ X=UYV \tag{2.5} \]

where \(U\) is a unitary matrix, \(Y\) is a triangular matrix with all diagonal elements equal to unity, \(y_{ij}=0,i>j\), \(y_{ii}=1\), and \(V\) is a diagonal matrix with real positive diagonal elements.

Using the fact that \(G=XEX^{-1}\), \(U^{*}U=I\), and \(EV=VE\), we can write \[ \operatorname{tr}(G^{*}G)=\operatorname{tr}[E^{*}Y^{*}YE(Y^{*}Y)^{-1}], \tag{2.6} \]

\[ \mathrm{d}B=X^{-1}\mathrm{d}X=V^{-1}Y^{-1}(U^{-1}\mathrm{d}U)YV+V^{-1}Y^{-1}\mathrm{d}YV+V^{-1}\mathrm{d}V. \tag{2.7} \]

From (3.1.6) and the structure of \(Y\) and \(V\) we see that \[ \prod_{j\neq k}^{} \mathrm{d}B_{jk}^{r}\mathrm{d}B_{jk}^{i}=\prod_{j<k}^{} (Y^{-1}\mathrm{d}Y)_{jk}^{r}(Y^{-1}\mathrm{d}Y)_{jk}^{i}a, \tag{2.8} \]

where \(a\) depends only on \(U\) and \(V\).

\[ \int_{}^{} \exp [-\operatorname{tr}(E^{*}HEH^{-1})]\prod_{j<k}^{} (Y^{-1}\mathrm{d}Y)_{jk}^{r}(Y^{-1}\mathrm{d}Y)_{jk}^{i}, \tag{2.9} \]

where \[ H=Y^{*}Y. \tag{2.10} \]

The matrix \(H\) is Hermitian. Any of its upper left diagonal block of size \(m\) is obtained from the upper left diagonal block of \(Y\) of the same size: \(H_n=Y_n^{*}Y_n\). Therefore, for every \(n\), \(\det H_n=1\).

We make a change of variables. First, because \(\det Y=1\), \[ \prod_{j<k}^{} (Y^{-1}\mathrm{d}Y)_{jk}^{r}(Y^{-1}\mathrm{d}Y)_{jk}^{i}=\prod_{j<k}^{} \mathrm{d}Y_{jk}^{r}\mathrm{d}Y_{jk}^{i}. \tag{2.11} \]

Next, we take \(H_{jk}^{r}\), \(H_{jk}^{i}\) for \(j<k\) as independent variables from \[ H_{jk}=Y_{jk}+\sum_{l>j}^{} Y_{jl}^{*}Y_{lk}, \quad j<k, \]

The Jacobian of the transformation from \(Y\) to \(H\) is \(1\). So \[ J=C \int_{}^{} \exp [-\operatorname{tr}(E^{*}HEH^{-1})]\prod_{j<k}^{} \mathrm{d}H_{jk}^{r} \mathrm{d}H_{jk}^{i}, \tag{2.12} \]

where \(C\) is a constant.

The integration over \(H\) is done in \(N\) steps. At every step we integrate over the variables of the last column and thus decrease by one the size of the matrix, whose structure remains the same.

Let \(H'=Y_n^{*}Y_n\), \(E'=[z_i\delta_{jk}]_{j,k=1,2,\cdots ,n}\), be the relevant matrices of order \(m\) and \(H\), \(E\) be those obtained from \(H'\), \(E'\) by removing the last row and last column. Let the Greek indices run from \(1\) to \(n\), and the Latin indices from \(1\) to \(n-1\). Let \(\Delta_{\alpha\beta}'\) be the cofactor of \(H_{\alpha\beta}'\) in \(H'\) and \(\Delta_{jk}\), the cofactor of \(H_{jk}\) in \(H\). Let \(g_i=H_{in}'\). Because \(\det H'=\det H=1\), we have \[ \Delta_{\alpha\beta}'=(H'^{-1})_{\beta\alpha}, \quad \Delta_{jk}=(H^{-1})_{kj}. \tag{2.13} \]

Expanding \(\det H'\), \(\Delta_{in}'\),\(\Delta_{jk}'\) by the last row and last column, we have \[ 1=H_{nn}'-\sum_{j,k}^{} g_{j}^{*}g_{k}\Delta_{kj}, \tag{2.14} \]

\[ \Delta_{in}'=-\sum_{l}^{} \Delta_{il}g_{l}^{*}, \tag{2.15} \]

\[ \Delta_{ij}'=H_{nn}'\Delta_{ij}-\sum_{l,k}^{} g_k^{*}g_l\Delta_{ij}^{lk}, \tag{2.16} \]

where \(\Delta_{ij}^{lk}\) is the cofactor obtained from \(H\) by removing the \(i\)th and \(l\)th rows and the \(j\)th and \(k\)th columns. Sylvester's theorem (also known as Desnanot–Jacobi identity) expresses \(\Delta_{ij}^{lk}\) in terms of \(\Delta_{rs}\) \[ \Delta_{ij}^{lk}=\Delta_{ij}\Delta_{lk}-\Delta_{ik}\Delta_{lj}. \tag{2.17} \]

In writing (2.17), we have replaced \(\det H\) by unity on the LHS. Let \[ \phi_{n}=\operatorname{tr}(E'^{*}H'E'H'^{-1})=\sum_{\alpha,\beta}^{} z_{\alpha}^{*}z_{\beta}H_{\alpha\beta}'\Delta_{\alpha\beta}'. \tag{2.18} \]

Separating the last row and last column and making use of (2.13) to (2.17), we get \[ \phi_{n}=\lvert z_n \rvert ^{2}+\phi_{n-1}+\langle g^{*} | H^{-1}(E^{*}-z_n^{*}I)H(E-z_nI)H^{-1}|g \rangle , \tag{2.19} \]

where \[ \langle g^{*}|B|g \rangle =\sum_{i,j}^{} g_{i}^{*}B_{ij}g_j. \tag{2.20} \]

Substituting (2.19) for \(n=N\) in (2.12), we get \[ \begin{aligned} J&=C\mathrm{e}^{-\lvert z_{N} \rvert ^{2}} \int_{}^{} \mathrm{e}^{-\phi_{N-1}} \prod_{1\leqslant i<j\leqslant N-1}^{} \mathrm{d}H_{ij}^{r}\mathrm{d}H_{ij}^{i} \\ &\times \int_{}^{} \exp [-\langle g^{*}| H^{-1}(E^{*}-z_{N}^{*}I)H(E-z_{N}I)H^{-1}|g \rangle ]\prod_{1\leqslant i\leqslant N-1}^{} \mathrm{d}g_i^{r}\mathrm{d}g_i^{i}. \end{aligned} \]

\(B\in \mathcal{M}_{N}(\mathbb{C})\), \(B\) is Hermitian. Then \[ \int_{}^{} \exp [-\langle g^{*}| B |g \rangle ] \prod_{j=1}^{N} \mathrm{d}g_{j}^{r}\mathrm{d}g_{j}^{i}=\pi^{N}(\det B)^{-1} \]

The last integral is immediate and gives \[ \pi^{N-1}\{\det [H^{-1}(E^{*}-z_{N}^{*}I)H(E-z_{N}I)H^{-1}]\}^{-1}=\pi^{N-1}\prod_{i=1}^{N-1} \lvert z_i-z_{N} \rvert ^{-2}. \tag{2.21} \]

The process can be repeated \(N\) times and we finally get \[ J=C\exp \left( -\sum_{i=1}^{N} \lvert z_i \rvert ^{2} \right) \prod_{1\leqslant i<j\leqslant N}\lvert z_i-z_j \rvert ^{-2}, \tag{2.22} \]

where \(C\) is a new constant.

The joint probability density for \(G\)

We have proved the joint probability density for the eigenvalues of \(G\) belonging to the Complex Ginibre Ensemble is given by \[ P_c(z_1,\cdots ,z_{N})=\mu(\mathrm{d}G)=C\exp \left( -\sum_{i=1}^{N} \lvert z_i \rvert ^{2} \right) \prod_{1\leqslant i<j\leqslant N}\lvert z_i-z_j \rvert ^{2}, \tag{2.23} \]

where \(C\) is the normalization constant. We now compute \(C\).

The probability that all the eigenvalues \(z_i\) will lie outside a circle of radius \(\alpha\) centered at \(z=0\) is \[ E_{N_c}(\alpha)=\int_{\lvert z_i \rvert \geqslant \alpha}^{} P_c(z_1,\cdots ,z_{N})\prod_{i}^{} \mathrm{d}x_i\mathrm{d}y_i. \]

By writing (Here we don't require $z_1 z_2 z_{N} $) \[ \begin{aligned} \prod_{i<j}^{} \lvert z_i-z_j \rvert ^{2}&=\prod_{i<j}^{} (z_i-z_j)(z_i^{*}-z_j^{*}) \\ &=\det \begin{bmatrix} 1 & \cdots & 1 \\ z_1 & \cdots & z_{N} \\ \vdots & & \vdots \\ z_1^{N-1} & \cdots & z_{N}^{N-1} \\ \end{bmatrix} \det \begin{bmatrix} 1 & \cdots & 1 \\ z_1^{*} & \cdots & z_{N}^{*} \\ \vdots & & \vdots \\ z_1^{*N-1} & \cdots & z_{N}^{*N-1} \\ \end{bmatrix} \\ &=\det \begin{bmatrix} N & \sum_{i}^{} z_i & \cdots & \sum_{i}^{} z_i^{N-1} \\ \sum_{i}^{} z_i^{*} & \sum_{i}^{} z_i^{*}z_i & \cdots & \sum_{i}^{} z_i^{*}z_i^{N-1} \\ \vdots & & & \vdots \\ \sum_{i}^{} z_i^{*N-1} & \sum_{i}^{} z_i^{*N-1}z_i & \cdots & \sum_{i}^{} z_i^{*N-1}z_i^{N-1} \end{bmatrix} \end{aligned} \]

The integrand is symmetric in all the \(z_i\), we can replace the first row with \(1,z_1,z_1^{2},\cdots ,z_1^{N-1}\) and multiply the result by \(N\); \(z_1\) can be eliminated from the other rows by subtracting a suitable multiple of the first row. The resulting determinant is symmetric in the \(N-1\) variables \(z_2,z_3,\cdots ,z_{N}\); therefore we replace the second row with \(z_2^{*},z_2^{*}z_2,\cdots ,z_2^{*}z_2^{N-1}\) and multiply the result by \(N-1\). The process can be repeated and we get \[ \begin{aligned} E_{N_c}(\alpha)&=CN!\int_{\lvert z_i \rvert \geqslant \alpha}^{} \left\{ \prod_{i}^{} \mathrm{d}x_i\mathrm{d}y_i\right\}\exp \left( -\sum_{i}^{N} \lvert z_i \rvert ^{2} \right) \\ &\times\det \begin{bmatrix} 1 & z_1 & \cdots & z_1^{N-1} \\ z_2^{*} & z_2^{*}z_2 & \cdots & z_2^{*}z_2^{N-1} \\ \vdots & & &\vdots \\ z_{N}^{*N-1} & z_{N}^{*N-1}z_{N} & \cdots & z_{N}^{*N-1}z_{N}^{N-1} \\ \end{bmatrix} \end{aligned} \]

By changing to polar coordinates and performing the angular integrations first we see that \[ \int_{\lvert z \rvert \geqslant \alpha}^{} \mathrm{e}^{-\lvert z \rvert ^{2}} z^{*j}z^{k} \mathrm{d}x\mathrm{d}y=\pi \delta_{jk} \Gamma(j+1,\alpha^{2}), \tag{2.24} \]

so that \[ E_{N_c}(\alpha)=CN!\pi^{N}\prod_{j=1}^{N} \Gamma(j,\alpha^{2}), \tag{2.25} \]

where \(\Gamma(j,\alpha^{2})\) is the incomplete gamma function \[ \Gamma(j,\alpha^{2})=\int_{\alpha^{2}}^{\infty} \mathrm{e}^{-x} x^{j-1} \mathrm{d}x=\Gamma(j)\mathrm{e}^{-\alpha^{2}} \sum_{l=0}^{j-1} \frac{\alpha^{2l}}{l!}. \tag{2.26} \]

Since \(E_{N_c}(0)=1\), the constant \(C\) can be determined from (3.1.23) as \[ C^{-1}=\pi^{N}\prod_{j=1}^{N} j!, \tag{2.27} \]

and therefore \[ E_{N_c}(\alpha)=\prod_{j=1}^{N} \left( \mathrm{e}^{-\alpha^{2}} \sum_{l=0}^{j-1} \frac{\alpha^{2l}}{l!} \right). \tag{2.28} \]

We will use the spectral law with symmetric functions of the form \[ F(z_1,\cdots ,z_n)=\sum_{i_1,\cdots i_k\ \text{distinct}}^{} f(z_{i_1})\cdots f(z_{i_k}). \]

\(k\)-points correlations

Let \(z\in \mathbb{C}\mapsto \gamma(z)=\pi^{-1}\mathrm{e}^{-\lvert z \rvert ^{2}}\) be the density of the standard Gaussian \(\mathcal{N}(0,\frac{1}{2}I_2)\) on \(\mathbb{C}\). For every \(1\leqslant k\leqslant n\), the \(k\)-point correlation is \[ \varphi_{n,k}(z_1,\cdots ,z_k):=\int_{\mathbb{C}^{n-k}}^{} \varphi_{n}(z_1,\cdots ,z_n) \mathrm{d}z_{k+1}^{r}\mathrm{d}z_{k+1}^{i}\cdots \mathrm{d}z_{n}^{r}\mathrm{d}z_n^{i}. \tag{2.29} \]

Theorem (Dyson, 1970) Let \(K(x,y)\) be a function with complex values, such that \[ \bar{K}(x,y)=K(y,x), \tag{2.30} \]

Assume that \[ \int K(x,y)K(y,z) \mathrm{d}y=K(x,z), \tag{2.31} \]

Let \([K(z_i,z_j)]_{N}\) denote the \(N\times N\) matrix with its \((i,j)\) element equal to \(K(z_i,z_j)\). Then \[ \int_{}^{} \det[K(z_i,z_j)]_{N} \mathrm{d}z_{N}^{r}\mathrm{d}z_{N}^{i}=(c-N+1)\det[K(z_i,z_j)]_{N-1}, \tag{2.32} \]

where \[ c=\int_{}^{} K(z,z) \mathrm{d}z^{r}\mathrm{d}z^{i} \tag{2.33} \]

Note that (2.30) and (2.31) mean that the linear operator defined by the kernel \(K(x,y)\) is a projector, and the constant \(c\) is its trace (hence a nonnegative integer).

proof From the definition of the determinant, \[ \det[K(z_i,z_j)]_{N}=\sum_{P}^{} \sigma(P)\prod_{1}^{l} (K(\xi,\eta)K(\eta,\zeta)\cdots K(\theta,\xi)), \tag{2.34} \]

where the permutation \(P\) consists of \(l\) cycles of the form \((\xi \rightarrow \eta \rightarrow \zeta \rightarrow\cdots \rightarrow \theta\rightarrow \xi)\). Now the index \(N\) occurs somewhere. - \(N\) forms a cycle by itself, and \(K(z_{N},z_{N})\) gives on integration the scalar constant \(c\); the remaining factor is by definition \(\det[K(z_i,z_j)]_{N-1}\); - \(N\) occurs in a longer cycle and integration on \(x_{N}\) reduces the length of this cycle by one. This can happen in \((N-1)\) ways since the index \(N\) can be inserted between any two indices of the cyclic sequence \(1,2,\cdots ,N-1\). Alos the resulting permutation over \(N-1\) indices has an opposite sign. The remaining expression is by definition \(\det[K(z_i,z_j)]_{N-1}\).

Adding the two contributions we get the result.

Theorem (\(k\)-points correlations) For every \(1\leqslant k\leqslant n\), \[ \varphi_{n,k}(z_1,\cdots ,z_k)=\frac{(n-k)!}{n!}\gamma(z_1)\cdots \gamma(z_k)\det [K(z_i,z_j)]_{1\leqslant i,j\leqslant k} \tag{2.35} \]

where \[ K(z_i,z_j):=\sum_{l=0}^{n-1} \frac{(z_iz_j^{*})^{l}}{l!}=\sum_{l=0}^{n-1} H_l(z_i)H_l(z_j)^{*} \tag{2.36} \]

with \[ H_{l}(z):=\frac{1}{\sqrt{l!}}z^{l}. \tag{2.37} \]

In particular, by taking \(k=n\) we get \[ \varphi_{n,n}=\varphi_{n}(z_1,\cdots ,z_n)=\frac{1}{n!}\gamma(z_1)\cdots \gamma(z_n)\det[K(z_i,z_j)]_{1\leqslant i,j\leqslant n}. \]

proof

It's easy to check the \(k=n\) situation \[ \varphi_{n}(z_1,\cdots ,z_n)=\frac{\pi^{-n}}{n!}\exp \left( -\sum_{i=1}^{n} \lvert z_i \rvert ^{2} \right) \det[K(z_i,z_j)]_{1\leqslant i,j\leqslant n}. \]

By the previous theorem, setting \(K(x,y)=\sum_{l=0}^{n-1} H_l(x)H_l(y)^{*}\), we get

\[ \int_{}^{} \mathrm{e}^{-\lvert y \rvert ^{2}} K(x,y)K(y,z) \mathrm{d}y^{r}\mathrm{d}y^{i}=\pi K(x,z) \]

and \[ \int_{}^{} \mathrm{e}^{-\lvert z \rvert ^{2}} K(z,z) \mathrm{d}z^{r}\mathrm{d}z^{i}=n\pi \]

So we have \[ \begin{aligned} \varphi_{n,n-1}&=\int_{\mathbb{C}}^{} \frac{\pi^{-n}}{n!}\exp \left( -\sum_{i=1}^{n} \lvert z_i \rvert ^{2} \right) \det[K(z_i,z_j)]_{1\leqslant i,j\leqslant n} \mathrm{d}z_n^{r}\mathrm{d}z_n^{i} \\ &=\frac{\pi^{-n+1}}{n!}\exp \left( -\sum_{i=1}^{n-1} \lvert z_i \rvert ^{2} \right) \det[K(z_i,z_j)]_{1\leqslant i,j\leqslant n-1} \\ \end{aligned} \]

By induction, (2.35) holds.

Reference

1. Charles Bordenave and Djalil Chafaï, The circular law ↩︎
2. R. A. Horn and Ch. R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1994, Corrected reprint of the 1991 original. ↩︎
3. Madan Lal Mehta, Random matrices, third ed. Pure and Applied Mathematics, vol.142, Acad. Press, 2004. ↩︎