Extrema of the two-dimensional Discrete Gaussian Free Field

Abstract

These lecture notes offer a gentle introduction to the two-dimensional Discrete Gaussian Free Field with particular attention paid to the scaling limits of the level sets at heights proportional to the absolute maximum. The bulk of the text is based on recent joint papers with O. Louidor and with J. Ding and S. Goswami. Still, new proofs of the tightness and distributional convergence of the centered DGFF maximum are presented that by-pass the use of the modified Branching Random Walk. The text contains a wealth of instructive exercises and a list of open questions and conjectures for future research.

235 pages, 35 figures, some 140 exercises, a number of open problems. Version to appear in the school proceedings, small corrections to chapters 5&6

References in corpus (8)

• Quantitative stochastic homogenization and large-scale regularity
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Characterization of invariant measures at the leading edge for competing particle systems
• Freezing and decorated Poisson point processes
• Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation
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Cited by in corpus (5)

• Subsequential scaling limits for Liouville graph distance
• Exceptional points of two-dimensional random walks at multiples of the cover time
• Lower Gaussian heat kernel bounds for the Random Conductance Model in a degenerate ergodic environment
• Properties of the gradient squared of the discrete Gaussian free field
• A limit law for the most favorite point of simple random walk on a regular tree