**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(
*M*P) = law(*M*) for random measure*M*implies*M*P=*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

- Intermezzo on when law(
**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**- 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