Extrema of the two-dimensional Discrete Gaussian Free Field
arXiv:1712.09972 · doi:10.1007/978-3-030-32011-9
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 (21)
- The Random-Cluster Model
- Convergence in law of the minimum of a branching random walk
- Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
- Recent progress on the Random Conductance Model
- Quantitative stochastic homogenization and large-scale regularity
- Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
- Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
- Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric
- Extreme local extrema of two-dimensional discrete Gaussian free field
- Characterization of invariant measures at the leading edge for competing particle systems
- Convergence of the centered maximum of log-correlated Gaussian fields
- Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field
- Phase coexistence of gradient Gibbs states
- Freezing and decorated Poisson point processes
- Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation
- Extremes of some Gaussian random interfaces
- Exceptional points of two-dimensional random walks at multiples of the cover time
- On level sets of Gaussian fields
- Maximum of the Ginzburg-Landau fields
- Decorated Random Walk Restricted to Stay Below a Curve (Supplement Material)
- Subsequential tightness of the maximum of two dimensional Ginzburg-Landau fields
Cited by in corpus (6)
- Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field
- 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