Graduate Seminar WS 2019/20:
Log-correlated processes: from branching random walks to the Riemann zeta function
Prof.Dr. Anton Bovier, Maximillian Fels
Time and place: Tuesdays, 16h ct in Room 0.011
The analysis of extreme values of stochastic processes has for a long time been an important theme of applied probability. While much of the classical theory concerned independent random variables or the identification of conditions under which the extremes of a processs are well described by those of iid variables, more recently, a class of processes where correlations just begin to matter has emerged as a new universality class. They are called log-correlated processes since the covariance decays essentially like the logarithm of the distance. The classical example of such a process is Branching Brownian Motion (BBM) and its close relative, the Branching Random Walk (BRW). In these examples, the underlying tree structure helps in the precise anlysis of the laws of the maxxima and the extremal process. But many examples without expleicit trees structure have been shown to fall in the same class: the Gaussian Free Field (GFF), cover times of random walks, and, quite curiously, certain features of the properties of the (randomised) Riemann zeta fuction on the critical line. In the seminar we will look at techniques to analysis such problems and a number of examples. As a guideline we use the lecture notes by Louis-Pierre Arguin, but we will need several original papers to fill in the details.
Literature:
Louis-Pierre Arguin, Extrema of log-correlated random variables: Principles and Examples, https://arxiv.org/abs/1601.00582, 2016.
Adam Harper, A note on the maximum of the Riemann zeta function, and log-correlated random variables, arXiv:1304.0677
Adam Harper. The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwi\l\l, and Soundararajan].arXiv:1904.08204. 2019
Louis-Pierre Arguin, David Belius, and Adam Harper. Maxima of a randomized Riemann zeta function, and branching random walks. arXiv:1506.00629, 2015
Anton Bovier, Extreme values of random processes, Lecture Notes.
Anton Bovier, Gaussian processes on trees. Cambridge University Press, 2016.
Arguin, Louis-Pierre.Belius, David, Bourgade, Paul. Maximum of the Characteristic Polynomial of Random Unitary Matrices.arXiv:1511.07399, 2015.
Marek Biskup. Extrema of the two-dimensional Discrete Gaussian Free Field. Lecture Notes, 2017