Statistical Properties of Random Banded Matrices: Analytical Results

Abstract

Recently it was realized that there exist important classes of Random Matrices (RM)- that of banded ¹ and sparse ² type — that are appreciably different in their prop erties from the classical Gaussian Ensembles of RM³ studied by Wigner, Dyson, Mehta and others. We will call the matrix Random Banded (RBM) if its entries H _ij are random variables with zero mean value and variance $\overline {{{\left| {{H_{ij}}} \right|}^2}} = A(i,j)$ decaying to zero when the distance r = |i - j| from the main diagonal tends to infinity. Matrices of such a type found recently numerous applications in Solid State Physics and especially in the study of Quantum Chaos Phenomena (see papers^1,4,5 and references therein). Without much loss of generality one can put A(i, j) = a(|i-j|) and in the most studied case the “shapefunction” a(r) is exponentially (or faster) decaying when r exceeds some typical scale b (called the “bandwidth”).