Convergence of random holomorphic functions with real zeros, random matrices and the distribution of the zeros of the Riemann zeta function

https://hu-berlin.zoom.us/j/61339297016

The GUE conjecture states that the ordinates of the zeros of the Riemann zeta function on the critical line should behave statistically like eigenvalues of large random matrices: more precisely they should be asymptotically distributed like a sine kernel determinantal point process. In the past two decades, a model has emerged to understand and predict the distribution of values of the Riemann zeta function on the critical line: the characteristic polynomial of random unitary matrices. It has been thought that there should exist a random holomorphic emerging as some scaling limit of the characteristic polynomial. We give a construction of this function and describe its relation to the GUE conjecture. We then show how it naturally appears in ratios in random matrix theory. We then discuss several generalisations of this construction by other authors as well as by J. Najnudel and myself.