Empirical spectral measure

The empirical spectral measure μ_H of H is defined by

$${\displaystyle \mu _{H}(A)={\frac {1}{n}}\,\#\left\{{\text{eigenvalues of }}H{\text{ in }}A\right\}=N_{1_{A},H},\quad A\subset \mathbb {R} .}$$

Usually, the limit of ${\displaystyle \mu _{H}}$ is a deterministic measure; this is a particular case of self-averaging. The cumulative distribution function of the limiting measure is called the integrated density of states and is denoted N(λ). If the integrated density of states is differentiable, its derivative is called the density of states and is denoted ρ(λ).

The limit of the empirical spectral measure for Wigner matrices was described by Eugene Wigner; see Wigner semicircle distribution and Wigner surmise. As far as sample covariance matrices are concerned, a theory was developed by Marčenko and Pastur.^[36][37]

The limit of the empirical spectral measure of invariant matrix ensembles is described by a certain integral equation which arises from potential theory.^[38]

Fluctuations

For the linear statistics N_f,H = n⁻¹ Σ f(λ_j), one is also interested in the fluctuations about ∫ f(λ) dN(λ). For many classes of random matrices, a central limit theorem of the form

$${\displaystyle {\frac {N_{f,H}-\int f(\lambda )\,dN(\lambda )}{\sigma _{f,n}}}{\overset {D}{\longrightarrow }}N(0,1)}$$

is known.^[39][40]

The variational problem for the unitary ensembles

Consider the measure

${\displaystyle \mathrm {d} \mu _{N}(\mu )={\frac {1}{{\widetilde {Z}}_{N}}}e^{-H_{N}(\lambda )}\mathrm {d} \lambda ,\qquad H_{N}(\lambda )=-\sum \limits _{j\neq k}\ln |\lambda _{j}-\lambda _{k}|+N\sum \limits _{j=1}^{N}Q(\lambda _{j}),}$

where ${\displaystyle Q(M)}$ is the potential of the ensemble and let ${\displaystyle \nu }$ be the empirical spectral measure.

We can rewrite ${\displaystyle H_{N}(\lambda )}$ with ${\displaystyle \nu }$ as

${\displaystyle H_{N}(\lambda )=N^{2}\left[-\int \int _{x\neq y}\ln |x-y|\mathrm {d} \nu (x)\mathrm {d} \nu (y)+\int Q(x)\mathrm {d} \nu (x)\right],}$

the probability measure is now of the form

${\displaystyle \mathrm {d} \mu _{N}(\mu )={\frac {1}{{\widetilde {Z}}_{N}}}e^{-N^{2}I_{Q}(\nu )}\mathrm {d} \lambda ,}$

where ${\displaystyle I_{Q}(\nu )}$ is the above functional inside the squared brackets.

Let now

${\displaystyle M_{1}(\mathbb {R} )=\left\{\nu$ :\nu \geq 0,\ \int _{\mathbb {R} }\mathrm {d} \nu =1\right\}}

be the space of one-dimensional probability measures and consider the minimizer

${\displaystyle E_{Q}=\inf \limits _{\nu \in M_{1}(\mathbb {R} )}-\int \int _{x\neq y}\ln |x-y|\mathrm {d} \nu (x)\mathrm {d} \nu (y)+\int Q(x)\mathrm {d} \nu (x).}$

For ${\displaystyle E_{Q}}$ there exists a unique equilibrium measure ${\displaystyle \nu _{Q}}$ through the Euler-Lagrange variational conditions for some real constant ${\displaystyle l}$

${\displaystyle 2\int _{\mathbb {R} }\log |x-y|\mathrm {d} \nu (y)-Q(x)=l,\quad x\in J}$

${\displaystyle 2\int _{\mathbb {R} }\log |x-y|\mathrm {d} \nu (y)-Q(x)\leq l,\quad x\in \mathbb {R} \setminus J}$

where ${\displaystyle J=\bigcup \limits _{j=1}^{q}[a_{j},b_{j}]}$ is the support of the measure and

${\displaystyle q=-\left({\frac {Q'(x)}{2}}\right)^{2}+\int {\frac {Q'(x)-Q'(y)}{x-y}}\mathrm {d} \nu _{Q}(y)}$.

The equilibrium measure ${\displaystyle \nu _{Q}}$ has the following Radon–Nikodym density

${\displaystyle {\frac {\mathrm {d} \nu _{Q}(x)}{\mathrm {d} x}}={\frac {1}{\pi }}{\sqrt {q(x)}}.}$^[41]

Bulk statistics

Formally, fix ${\displaystyle \lambda _{0}}$ in the interior of the support of ${\displaystyle N(\lambda )}$. Then consider the point process

$${\displaystyle \Xi (\lambda _{0})=\sum _{j}\delta {\Big (}{\cdot }-n\rho (\lambda _{0})(\lambda _{j}-\lambda _{0}){\Big )}~,}$$

where ${\displaystyle \lambda _{j}}$ are the eigenvalues of the random matrix.

The point process ${\displaystyle \Xi (\lambda _{0})}$ captures the statistical properties of eigenvalues in the vicinity of ${\displaystyle \lambda _{0}}$. For the Gaussian ensembles, the limit of ${\displaystyle \Xi (\lambda _{0})}$ is known;^[2] thus, for GUE it is a determinantal point process with the kernel

$${\displaystyle K(x,y)={\frac {\sin \pi (x-y)}{\pi (x-y)}}}$$

(the sine kernel).

The universality principle postulates that the limit of ${\displaystyle \Xi (\lambda _{0})}$ as ${\displaystyle n\to \infty }$ should depend only on the symmetry class of the random matrix (and neither on the specific model of random matrices nor on ${\displaystyle \lambda _{0}}$). Rigorous proofs of universality are known for invariant matrix ensembles^[42][43] and Wigner matrices.^[44][45]

Edge statistics

Local laws

^[46] The typical statement of the Wigner semicircular law is equivalent to the following statement: For each fixed interval ${\displaystyle [\lambda _{0}-\Delta \lambda ,\lambda _{0}+\Delta \lambda ]}$ centered at a point ${\displaystyle \lambda _{0}}$, as ${\displaystyle N}$, the number of dimensions of the gaussian ensemble increases, the proportion of the eigenvalues falling within the interval converges to ${\displaystyle \int _{[\lambda _{0}-\Delta \lambda ,\lambda _{0}+\Delta \lambda ]}\rho (t)dt}$, where ${\displaystyle \rho (t)}$ is the density of the semicircular distribution.

If ${\displaystyle \Delta \lambda }$ can be allowed to decrease as ${\displaystyle N}$ increases, then we obtain strictly stronger theorems, named "local laws".