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Register a new userExtrema of log-correlated random variables: Principles and Examples
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Louis-Pierre Arguin
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These notes were written for the mini-course Extrema of log-correlated random variables: Principles and Examples at the Introductory School held in January 2015 at the Centre International de Rencontres Mathematiques in Marseille. There have been many advances in the understanding of the high values of log-correlated random fields from the physics and mathematics perspectives in recent years. These fields admit correlations that decay approximately like the logarithm of the inverse of the distance between index points. Examples include branching random walks and the two-dimensional Gaussian free field. In this paper, we review the properties of such fields and survey the progress in describing the statistics of their extremes. The branching random walk is used as a guiding example to prove the correct leading and subleading order of the maximum following the multiscale refinement of the second moment method of Kistler. The approach sheds light on a conjecture of Fyodorov, Hiary & Keating on the maximum of the Riemann zeta function on an interval of the critical line and of the characteristic polynomial of random unitary matrices.
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We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).
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