The global elliptic law, sand clock density and <i>V</i> -law. 40 years of the <i>G</i> -elliptic law

Abstract We prove the Global Elliptic Law for random matrices <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">Ξ</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msubsup> <m:mi>ξ</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {\Xi_{n}=\{\xi_{ij}^{(n)}\}} of the general form, i.e. their diagonal entries <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>ξ</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> </m:math> {\xi_{ii}^{(n)}} have nonzero expectations and the pairs of the entries <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>ξ</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mi>ξ</m:mi> <m:mrow> <m:mi>j</m:mi> <m:mo>⁢</m:mo> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(\xi_{ij}^{(n)},\xi_{ji}^{(n)})} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>≠</m:mo> <m:mi>j</m:mi> </m:mrow> </m:math> {i\neq j} have nonzero covariances. In this case the global elliptic law means that the support of the accompanying spectral density of eigenvalues of matrix <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Ξ</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {\Xi_{n}} looks like the picture of several galaxies made by telescope. The proof of the Global Elliptic Law based on generalized Lindeberg’s condition, Victoria transform, respect method and new limit theorems.