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Graduate Seminar WS 2021/22:

Branching Brownian motion and log-correlated fields

Prof.Dr. Anton Bovier, Dr. Adrien Schertzer

Time and place: Tuesdays, 14h ct ???

 

Prelimiary meeting to assign the talks of the seminar: Thursday, July 22, 2021, at 18:00, via ZOOM

Here is a provisional list of talks

 

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. 

Literature:

Anton Bovier, Gaussian processes on trees. Cambridge University Press, 2016. (main source)

Louis-Pierre Arguin, Extrema of log-correlated random variables: Principles and Examples, https://arxiv.org/abs/1601.00582, 2016.

Arguin, Louis-Pierre.Belius, David, Bourgade, Paul. Maximum of the Characteristic Polynomial of Random Unitary Matrices.arXiv:1511.07399, 2015.

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

Marek Biskup. Extrema of the two-dimensional Discrete Gaussian Free Field. Lecture Notes, 2017