Large deviations of zeroes of Gaussian stationary functions
Consider a Gaussian stationary function on the real line (that is, a random function whose distribution is shift-invariant and all its finite marginals have centered multi-normal distribution). What is the probability that it has no zeroes at all in a long interval? What is the probability that it has a significant deficiency or abundance in the number of zeroes? These questions were raised more than 70 years ago, but even modern tools of large deviation theory do not directly apply to answer them. In this talk we will see how real, harmonic and complex analysis shed light on these questions, yielding new results and many open questions.
Based on joint works with O. Feldheim and S. Nitzan (arXiv:1709.00204) and R. Basu, A. Dembo and O. Zeitouni (arXiv:1709.06760).