Alladi Ramakrishnan Hall

On Links Between the Random Operator and Random Matrix Theories

Prof Leonid Pastur

Department of Low Temperature Physics, Kharkov, Ukrain

We present several families of selfadjoint ergodic operators for which we
prove that their Integrated Density of States converges weakly to a certain measure as the
parameter indexing operators of a given family tends to infinity. We identify the limiting
measure with the infinite size limit of the Normalized Counting Measure of eigenvalues of
certain random matrices. We then give an informal discussion of these
results as possible indications of the presence of the continuous spectrum
of the random ergodic operators belonging to considered families for
sufficiently large values of the indexing parameters.