The logarithmic potential is a classical object of potential theory intimately connected with the two dimensional Laplacian. It appears also in free probability theory via the free entropy, and in partial differential equations e.g. Patlak-Keller-Segel models. This post concerns only it usage for the spectra of non Hermitian random matrices.
Let \( {\mathcal{P}(\mathbb{C})} \) be the set of probability measures on \( {\mathbb{C}} \) which integrate \( {\log\left|\cdot\right|} \) in a neighborhood of infinity. For every \( {\mu\in\mathcal{P}(\mathbb{C})} \), the logarithmic potential \( {U_\mu} \) of \( {\mu} \) on \( {\mathbb{C}} \) is the function
\[ U_\mu:\mathbb{C}\rightarrow(-\infty,+\infty] \]
defined for every \( {z\in\mathbb{C}} \) by
\[ U_\mu(z)=-\int_{\mathbb{C}}\!\log|z-z’|\,\mu(dz’) =-(\log\left|\cdot\right|*\mu)(z). \ \ \ \ \ (1) \]
This integral in well defined in \( {(-\infty,+\infty]} \) thanks to the assumption \( {\mu\in\mathcal{P}(\mathbb{C})} \). Note that \( {U_\mu(z)=+\infty} \) if \( {\mu} \) has an atom at point \( {z} \). For the circular law \( {\mathcal{U}_\sigma} \) of density
\[ (\sigma^2\pi)^{-1}\mathbf{1}_{\{z\in\mathbb{C}:|z|\leq\sigma\}} \]
we have, for all \( {z\in\mathbb{C}} \), the remarkable formula
\[ U_{\mathcal{U}_\sigma}(z)= \begin{cases} \log(\sigma)-\log|z\sigma^{-1}| & \text{if } |z|>\sigma, \\ \log(\sigma)-\frac{1}{2}(|z\sigma^{-1}|^2-1) & \text{if } |z|\leq\sigma, \end{cases} \ \ \ \ \ (2) \]
see e.g. the book of Saff and Totik. Let \( {\mathcal{D}'(\mathbb{C})} \) be the set of Schwartz-Sobolev distributions (generalized functions). Since \( {\log\left|\cdot\right|} \) is Lebesgue locally integrable on \( {\mathbb{C}} \), one can check by using the Fubini theorem that \( {U_\mu} \) is Lebesgue locally integrable on \( {\mathbb{C}} \). In particular,
\[ U_\mu<\infty \]
Lebesgue almost everywhere (a.e.), and
\[ U_\mu\in\mathcal{D}'(\mathbb{C}). \]
Since \( {\log\left|\cdot\right|} \) is the fundamental solution of the Laplace equation in \( {\mathbb{C}} \), we have, in \( {\mathcal{D}'(\mathbb{C})} \),
\[ \Delta U_\mu=-2\pi\mu. \ \ \ \ \ (3) \]
In other words, for every smooth and compactly supported “test function” \( {\varphi:\mathbb{C}\rightarrow\mathbb{R}} \),
\[ \int_{\mathbb{C}}\!\Delta\varphi(z)U_\mu(z)\,dz =-2\pi\int_{\mathbb{C}}\!\varphi(z)\,\mu(dz). \]
The identity (3) means that \( {\mu\mapsto U_\mu} \) in the distributional Green potential of the Laplacian in \( {\mathbb{C}} \). The function \( {U_\mu} \) contains enough information to recover \( {\mu} \):
Lemma 1 (Unicity) For every \( {\mu,\nu\in\mathcal{P}(\mathbb{C})} \), if \( {U_\mu=U_\nu} \) a.e. then \( {\mu=\nu} \).
Proof: Since \( {U_\mu=U_\nu} \) in \( {\mathcal{D}'(\mathbb{C})} \), we get \( {\Delta U_\mu=\Delta U_\nu} \) in \( {\mathcal{D}'(\mathbb{C})} \). Now (3) gives \( {\mu=\nu} \) in \( {\mathcal{D}'(\mathbb{C})} \), and thus \( {\mu=\nu} \) as measures since \( {\mu} \) and \( {\nu} \) are Radon measures. ☐
Let \( {A} \) be an \( {n\times n} \) complex matrix. We define the discrete probability measure on \( {\mathbb{C}} \)
\[ \mu_A:=\frac{1}{n}\sum_{k=1}^n\delta_{\lambda_k(A)} \]
where \( {\lambda_1(A),\ldots,\lambda_n(A)} \) are the eigenvalues of \( {A} \), i.e. the roots in \( {\mathbb{C}} \) of its characteristic polynomial \( {P_A(z):=\det(A-zI)} \). We also define the discrete probability measure on \( {[0,\infty)} \)
\[ \nu_A:=\frac{1}{n}\sum_{k=1}^n\delta_{s_k(A)} \]
where \( {s_1(A),\ldots,s_n(A)} \) are the singular values of \( {A} \), i.e. the eigenvalues of the positive semidefinite Hermitian matrix \( {\sqrt{AA^*}} \). We have now
\[ U_{\mu_A}(z) =-\int_{\mathbb{C}}\!\log\left| z’-z\right|\,\mu_A(dz’) =-\frac{1}{n}\log\left|\det(A-zI)\right| =-\frac{1}{n}\log\left| P_A(z)\right| \]
for every \( {z\in\mathbb{C}\setminus\{\lambda_1(A),\ldots,\lambda_n(A)\}} \). We have also the alternative expression
\[ U_{\mu_A}(z) =-\frac{1}{n}\log\det(\sqrt{(A-zI)(A-zI)^*}) =-\int_0^\infty\!\log(t)\,\nu_{A-zI}(dt). \ \ \ \ \ (4) \]
The identity (4) bridges the eigenvalues with the singular values, and is at the heart of the following lemma, which allows to deduce the convergence of \( {\mu_A} \) from the one of \( {\nu_{A-zI}} \). The strength of this Hermitization lies in the fact that in contrary to the eigenvalues, one can control the singular values with the entries of the matrix. The price payed here is the introduction of the auxiliary variable \( {z} \) and the uniform integrability. We recall that on a Borel measurable space \( {(E,\mathcal{E})} \), we say that a Borel function \( {f:E\rightarrow\mathbb{R}} \) is uniformly integrable for a sequence of probability measures \( {(\eta_n)_{n\geq1}} \) on \( {E} \) when
\[ \lim_{t\rightarrow\infty}\varlimsup_{n\rightarrow\infty}\int_{\{|f|>t\}}\!|f|\,d\eta_n=0. \ \ \ \ \ (5) \]
We will use this property as follows: if \( {(\eta_n)_{n\geq1}} \) converges weakly to \( {\eta} \) and \( {f} \) is continuous and uniformly integrable for \( {(\eta_n)_{n\geq1}} \) then \( {f} \) is \( {\eta} \)-integrable and \( {\lim_{n\rightarrow\infty}\int\!f\,d\eta_n=\int\!f\,\eta} \). The idea of using Hermitization goes back at least to Girko. However, theorem 2 and lemma 3 below are inspired from the approach of Tao and Vu.
Theorem 2 (Girko Hermitization) Let \( {(A_n)_{n\geq1}} \) be a sequence of complex random matrices where \( {A_n} \) is \( {n\times n} \) for every \( {n\geq1} \), defined on a common probability space. Suppose that for a.a. \( {z\in\mathbb{C}} \), there exists a probability measure \( {\nu_z} \) on \( {[0,\infty)} \) such that a.s.
- (i) \( {(\nu_{A_n-zI})_{n\geq1}} \) converges weakly to \( {\nu_z} \) as \( {n\rightarrow\infty} \)
- (ii) \( {\log(\cdot)} \) is uniformly integrable for \( {\left(\nu_{A_n-zI}\right)_{n\geq1}} \)
Then there exists a probability measure \( {\mu\in\mathcal{P}(\mathbb{C})} \) such that
- (j) a.s. \( {(\mu_{A_n})_{n\geq1}} \) converges weakly to \( {\mu} \) as \( {n\rightarrow\infty} \)
- (jj) for a.a. \( {z\in\mathbb{C}} \),
\[ U_\mu(z)=-\int_0^\infty\!\log(t)\,\nu_z(dt). \]
Moreover, if \( {(A_n)_{n\geq1}} \) is deterministic, then the statements hold without the “a.s.”
Proof: Let \( {z} \) and \( {\omega} \) be such that (i-ii) hold. For every \( {1\leq k\leq n} \), define
\[ a_{n,k}:=|\lambda_k(A_n-zI)| \quad\text{and}\quad b_{n,k}:=s_k(A_n-zI) \]
and set \( {\nu:=\nu_z} \). Note that \( {\mu_{A_n-zI}=\mu_{A_n}*\delta_{-z}} \). Thanks to the Weyl inequalities and to the assumptions (i-ii), one can use lemma 3 below, which gives that \( {(\mu_{A_n})_{n\geq1}} \) is tight, that \( {\log\left| z-\cdot\right|} \) is uniformly integrable for \( {(\mu_{A_n})_{n\geq1}} \), and that
\[ \lim_{n\rightarrow\infty}U_{\mu_{A_n}}(z)=-\int_0^\infty\!\log(t)\,\nu_z(dt)=:U(z). \]
Consequently, a.s. \( {\mu\in\mathcal{P}(\mathbb{C})} \) and \( {U_\mu=U} \) a.e. for every adherence value \( {\mu} \) of \( {(\mu_{A_n})_{n\geq1}} \). Now, since \( {U} \) does not depend on \( {\mu} \), by lemma 1, a.s. \( {\left(\mu_{A_n}\right)_{n\geq1}} \) has a unique adherence value \( {\mu} \), and since \( {(\mu_n)_{n\geq1}} \) is tight, \( {(\mu_{A_n})_{n\geq1}} \) converges weakly to \( {\mu} \) by the Prokhorov theorem. Finally, by (3), \( {\mu} \) is deterministic since \( {U} \) is deterministic, and (j-jj) hold. ☐
The following lemma is in a way the skeleton of the Girko Hermitization of theorem 2. It states essentially a propagation of a uniform logarithmic integrability for a couple of triangular arrays, provided that a logarithmic majorization holds between the arrays. See arXiv:0808.1502v2 for a proof.
Lemma 3 (Logarithmic majorization and uniform integrability) Let \( {(a_{n,k})_{1\leq k\leq n}} \) and \( {(b_{n,k})_{1\leq k\leq n}} \) be two triangular arrays in \( {[0,\infty)} \). Define the discrete probability measures
\[ \mu_n:=\frac{1}{n}\sum_{k=1}^n\delta_{a_{n,k}} \quad\text{and}\quad \nu_n:=\frac{1}{n}\sum_{k=1}^n\delta_{b_{n,k}}. \]
If the following properties hold
- (i) \( {a_{n,1}\geq\cdots\geq a_{n,n}} \) and \( {b_{n,1}\geq\cdots\geq b_{n,n}} \) for \( {n\gg1} \),
- (ii) \( {\prod_{i=1}^k a_{n,i} \leq \prod_{i=1}^k b_{n,i}} \) for every \( {1\leq k\leq n} \) for \( {n\gg1} \),
- (iii) \( {\prod_{i=k}^n b_{n,i} \leq \prod_{i=k}^n a_{n,i}} \) for every \( {1\leq k\leq n} \) for \( {n\gg1} \),
- (iv) \( {(\nu_n)_{n\geq1}} \) converges weakly to some probability measure \( {\nu} \) as \( {n\rightarrow\infty} \),
- (v) \( {\log(\cdot)} \) is uniformly integrable for \( {(\nu_n)_{n\geq1}} \),
then
- (j) \( {(\mu_n)_{n\geq1}} \) is tight,
- (jj) \( {\log(\cdot)} \) is uniformly integrable for \( {(\mu_n)_{n\geq1}} \),
- (jjj) we have, as \( {n\rightarrow\infty} \),
\[ \int_0^\infty\!\log(t)\,\mu_n(dt) =\int_0^\infty\!\log(t)\,\nu_n(dt)\rightarrow\int_0^\infty\!\log(t)\,\nu(dt), \]
and in particular, for every adherence value \( {\mu} \) of \( {(\mu_n)_{n\geq1}} \),
\[ \int_0^\infty\!\log(t)\,\mu(dt)=\int_0^\infty\!\log(t)\,\nu(dt). \]
The logarithmic potential is related to the Cauchy-Stieltjes transform of \( {\mu} \) via
\[ S_\mu(z) :=\int_{\mathbb{C}}\!\frac{1}{z’-z}\,\mu(dz’) =(\partial_x-i\partial_y)U_\mu(z) \quad\text{and thus}\quad (\partial_x+i\partial_y)S_\mu=-2\pi\mu \]
in \( {\mathcal{D}'(\mathbb{C})} \). The term “logarithmic potential” comes from the fact that \( {U_\mu} \) is the electrostatic potential of \( {\mu} \) viewed as a distribution of charges in \( {\mathbb{C}\equiv\mathbb{R}^2} \). The logarithmic energy
\[ \mathcal{E}(\mu) :=\int_{\mathbb{C}}\!U_\mu(z)\,\mu(dz) =-\int_{\mathbb{C}}\int_{\mathbb{C}}\!\log\left| z-z’\right|\,\mu(dz)\mu(dz’) \]
is up to a sign the Voiculescu free entropy of \( {\mu} \) in free probability theory. The circular law \( {\mathcal{U}_\sigma} \) minimizes \( {\mu\mapsto\mathcal{E}(\mu)} \) under a second moment constraint. In the spirit of (4) and beyond matrices, the Brown spectral measure of a nonnormal bounded operator \( {a} \) is
\[ \mu_a:=(-4\pi)^{-1}\Delta\int_0^\infty\!\log(t)\,\nu_{a-zI}(dt) \]
where \( {\nu_{a-zI}} \) is the spectral distribution of the self-adjoint operator \( {(a-zI)(a-zI)^*} \). Due to the logarithm, the Brown spectral measure \( {\mu_a} \) depends discontinuously on the \( {*} \)-moments of \( {a.} \) For random matrices, this problem is circumvented in the Girko Hermitization by requiring a uniform integrability, which turns out to be a.s. satisfied for random matrices with i.i.d. entries with finite positive variance.