Nick Ercolani: Random Toeplitz Operators and Algebraic Curves

Toeplitz operators are a natural test class for studying questions in spectral theory stemming from their connection to constant coefficient difference operators. Classical studies have focused on the self-adjoint case,  but recent motivations from numerical linear algebra in noisy systems have motivated both numerical and analytical studies of the non-self-adjoint cases. Together with John Peca-Medlin and Jonathan Ramalheira-Tsu we have been studying a particular subclass of non-self-adjoint Toeplitz operators, with Laurent polynomial symbol, whose asymptotic spectrum nevertheless remains real. Analytical methods of Schmidt and Spencer enable one to understand the spectral densities through an algebraic curve associated to the polynomial symbol. Building on this we have opened a new area of investigation into the ensemble of our operator class with random symbol. This brings together and relates ideas from random matrix theory and random polynomial theory. In particular, we are able to describe in some detail the intensity measures and spacing measures of our systems. This has a number of potential applications which we hope to be able to mention; chiefly, a seemingly enigmatic connection to Boolean probability.