Wednesday, April 13, 2016 - 2:05pm
1 hour (actually 50 minutes)
We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T). We will pose to discuss analogues of these results in a few other settings, such as zeroes of real-analytic Gaussian functions and winding of planar Gaussian functions, pointing out interesting similarities and differences. For the last part, we consider the "persistence probability" (i.e., the probability that a function has no zeroes at all in some region). Here we present results in the real setting, as even this case is yet to be understood. Based in part on joint works with Jeremiah Buckley and Ohad Feldheim.