Course outline:
(tentative so far, subject to regular updates, based on four 90-min lectures/week)
See the list of relevant papers whose content will be covered in this course. The lecture notes are distributed through this site or directly here
Week 1:
- Introduction to DGFF & its scaling limit (notes for lecture 1)
- definition of DGFF, Green function of SRW (symmetry, positive definiteness)
- reasons for considering d=2
- asymptotic behavior of the Green function
- scaling limit to continuum GFF
- First results: Maximum and intermediate values (notes for lecture 2)
- scaling of the maximum, 1st moment calculation
- Daviaud's level sets, 2nd moment calculation for the Gibbs measure
- level set representation by point measure, main theorem (based on j/w/w O. Louidor)
- link to LQG, definition via martingales (uniqueness and positivity postponed)
- Intermediate level sets: factorization (notes for lecture 3)
- intermezzo on scaling limit of Gibbs-Markov property for DGFF
- 1st and 2nd moment calculations --> non-trivial scaling limit
- enhanced 2nd moment calculation
- factorization to product form
- Intermediate level sets: nailing the limit (notes for lecture 4)
- Gibbs-Markov property in scaling limit
- properties of Z-measures
- characterization of the limit as multiplicative chaos/LQG
- changes required outside 2nd moment regime
Week 2:
- Comparison inequalities for Gaussian processes (notes for lecture 5)
- Kahane's convexity inequality
- Kahane's theory of multiplicative chaos
- comparison of maxima: Slepian, Sudakov-Fernique
- Concentration for the maximum of Gaussian processes (notes for lecture 6)
- Borell-TIS inequality via Gaussian interpolation
- setting and statement of Fernique's majorization bound
- review of background and context (Kolmogorov-Čenstov, Dudley, Talagrand)
- proof by the chaining argument
- consequences for boundedness and continuity
- Connection to Branching Random Walk (notes for lecture 7)
- Dekking-Host argument and subsequential tightness for DGFF maximum
- Upper bound by Branching Random Walk
- Maximum of BRW
- Tightness of the maximum of DGFF (notes for lecture 8)
- Upper tail of DGFF maximum
- Concentric decomposition
- Tightness of the lower tail
Week 3:
- Extremal local extrema (notes for lectures 9-10)
- Separation and size of level sets
- need for structured extremal process and main theorem (based on j/w/w O. Louidor)
- invariance of weak subsequential limits under Dysonization
- Liggett's non-interacting interacting particle system and his "folk theorem"
- Nailing the intensity measure (notes for lecture 10)
- Intermezzo on when law(MP) = law(M) for random measure M implies MP=M a.s. (based on Liggett)
- uniqueness by existence of limit of the maximum
- Gibbs-Markov property and consequences: conformal invariance, characterization
- connection to critical LQG
- Local structure of extremal points (notes for lectures 11-12)
- main result and consequences: supercritical LQG, PD statistics, freezing phenomenon
- concentric decomposition & control via conditional random walk
- intermezzo on Brownian motion above polylogarithmic curves
- Local structure continued
- control of the full extremal process
- local limit theorem for the maximum
- extensions to the Gibbs measure and freezing
Week 4:
- Random walk in DGFF landscape (notes for lecture 13)
- Liouville Brownian Motion (existence in 2nd moment regime)
- random walk in electric field (given by the DGFF)
- main results: subdiffusive hitting-time estimate, effective resistance growth (j/w/w J. Ding, S. Goswami)
- outline of the proofs
- Effective resistance control
- representation via paths and cuts
- electrostatic, distributional and geometric duality
- Russo-Seymour-Welsh type theorem for effective resistance (via Tassion)
- concentration and remaining proofs
- Consequences for random walk driven by DGFF
- subdiffusive behavior of expected exit time
- consequences for path distribution
- conjectures about scaling limit
- Conclusion, open problems and conjectures
- other processes to tackle (and review of existing work)
- conjectures and open problems