“Mwer viel einst zu verkünden hat, schweigt viel in sich hinein. Wer einst den Blitz zu zünden hat, muß lange Wolke sein.” ― Friedrich Nietzsche

Schramm-Loewner Evolutions (2023)

References:[1].Schramm–Loewner Evolution, Antti Kemppainen [2].Schramm–Loewner Evolution, Jason Miller [3].Schramm-Loewner Evolution, G. Lawler [4].Lectures on Schramm–Loewner Evolution, N. Berestycki & J.R. Norris

Ⅰ. Preliminary: Conformal Mapping, BM, Harmonic Measure and Green Function

Ⅱ. Mapping-out Functions: Compact Hulls and Capacity

Ⅲ. Chordal Loewner Theory: Loewner Transform and Differential Equation

Ⅳ. SLE: Properties, Bessel Flows, Hitting probability, Phase Transition and Conformal Transform

Ⅴ. SLE(6): Locality

Ⅵ. SLE(8/3): Brownian Excursions and Restriction

Ⅶ. SLE(4) and Continumm Gaussian Free Fields

Random Interlacements (2023)

Reference:[1].An Introduction to Random Interlacements, A. Drewitz , B. Ráth , A. Sapozhnikov [2]. Vacacnt Set of Random Interlacements and Percolation, A. Sznitman [3].Lectures on the Poisson Process, Gunter Last, Mathew Penrose [4]. On scaling limits and Brownian interlacements, Alain-Sol Sznitman [5].Percolative Properties of Brownian Interlacements and Its Vacant Set, Xinyi Li

Ⅰ. General Theory of Poisson Process

Ⅱ. Discrete Time RW Random Interlacement: Existence, Properties and Stochastic Domination

Ⅲ. Discrete Time RW Random Interlacement Point Process: Construction and Properties

Ⅳ. Percolation of Vacant Sets on Discrete Time RW Random Interlacements

Ⅴ. Continuous Time Brownian Random Interlacement: Brownian Capacity, Construction and Properties, Occupation Time and its Scaling Limit

Bernoulli Percolation in Z^d (2024)

References: [1].Percolation, Geoffrey Grimmett [2].Percolation Theory, Vincent Tassion [3].Introduction to percolation theory, Hugo Duminil-Copin

Ⅰ. Basic Settings: Bernoulli Bond Percolation and Site Percolation

Ⅱ. Basic Techniques: FKG Inequality, BK Inequality, Russo’s Formula and Reliability Theory

Ⅲ. Properties of Percolation: Egordicity, Uniqueness of ∞-Cluster, Conti. Zones and Conti. of θ(p)

Ⅳ. Subcritical Percolation: Exponential Decay and Correlation Length

Ⅴ. Renormalization and Subcritical Decay in Volum

Ⅵ. Percoaltion in Z^2: Critical Point, RSW Theory and Supercritical Behavior

Conformal Invariance of 2D Lattice Models (2023)

References:[1]Random Planar Curves and Schramm-Loewner Evolutions, Wendelin Werner[2] Lectures on two-dimensional critical percolation, Wendelin Werner [3] Conformal invariance of lattice models, Hugo Duminil-Copin, Stanislav Smirnov [4]Percolation Theory, Kim Christensen

Mostly base on the lectures given by Hao WU, YMSC

Ⅰ.Bernoulli Bond Percolation: Phase Transition

Ⅱ.Bernoulli Site Percolation: Critical Values, Cardy’s Formula

Ⅲ.Random Cluster Model: Phase Transition and Self-Dual Points

Ⅵ.FK-Ising Model: Discrete Complex Anaysis, Conformal Invariance

Ⅴ.Ising Model: Critical Value, Conformal Invaraince

Loops and Occupation Time (2023)

Reference:[1]. Topics in Occupation Times and Gaussian Free Fields, A. Sznitman [2].Lecture notes on the Gaussian Free Field, W. Werner, E. Powell

Ⅰ. Basic Potential Theory

Ⅱ. Isomorphism Theorems

Ⅲ. Continuous Time RW Loops: Rooted Loops, Pointed Loops and Unrooted Loops

Ⅳ. Poisson Gas of Continuous Time RW Loops: Occupation Fields, Representation Formula and Connection with Random Interlacements

Ⅴ. Discrete Time RW Loops and its Poisson Ensembles: Properties, Excursion Decomposition and Occupation Fields

Discrete Gaussian Free Fields (2023)

References:[1].Statistical Mechanics of Lattice Systems:a Concrete Mathematical Introduction, Chap.8, Friedli, S. and Velenik, Y.[2].Lecture notes on the Gaussian Free Field, W. Werner, E. Powell

Ⅰ. Discrete Gaussian Free Fields: Properties, Thermodynamic Limits, Variation on Markovian Property and Determinant of Laplacian

Ⅱ. Random Spanning Trees and GFFs: Wilson’s Algorithm and Occupation Fields