Lectures

Topics: Recap of Markov Chains, Distance to Equilibrium, Relaxation and Mixing Times, The Cutoff Phenomenon.

Ref:  Ch. 1 of Salez Notes [1].

Topics: Random Walks on groups, Distinguishing Statistics, Couplings, Coalescence times, Applications: Random walk on the cycle, Ehrenfest Model.

Ref:  Ch. 2 of Salez Notes [1].

Topics: Spectral Properties of transition kernels, Wilson's method

Ref:   Ch. 3 of Salez Notes [1].

Topics: Limit profiles of random walk in a cycle and hypercube [Notes]

Ref:   Ch. 3 of Salez Notes [1],  Nestoridi - Olesker-Taylor paper [5],  Ceccherini-Silberstein et al. [4].

Topics: Geometric Techniques - Volume, degree, diameter; Conductance

Ref:   Ch. 4 of Salez Notes [1]

Topics: Curvature, Application - Phase transition in Curie-Weiss model

Ref:   Ch. 4 of Salez Notes [1]

Topics: 

1. Carne Varopoulos bound

2. Dirichlet Form and Poincare constant

Remarks: An observation from the lecture regarding reversibility in the Carne Varopoulos bound

Ref:   Ch. 4 and 5 of Salez Notes [1]

Topics:  Variational Techniques: Poincare Bound, Cheeger inequalities, Comparison Principle

Ref:   Ch. 5 of Salez Notes [1]

Topics: Congestion and Distinguished Paths

Ref:   Ch. 5 of Salez Notes [1]

Topics: Localization Schemes - Overview and toy example.

Ref: Sec. 1 and Sec. 2 of Chen and Eldan [5]

Topics: Localization schemes -  Definition, coordinate-by-coordinate localization, Associated Markov chain.

Ref: Sec. 2 of Chen and Eldan [5]

Topics: Localization Schemes -  Coordinate-by-coordinate localization to Glauber dynamics; Linear-tilt localization.

Ref: Sec. 2 of Chen and Eldan [5]

Topics: Localization Schemes -  More on Linear-tilt localization and the Spectral Gap of an associated Markov Chain.

Ref: Sec. 2 of Chen and Eldan [5]

Topics: Localization Schemes -  Approximate Conservation of Variance for k-Glauber Dynamics, Part 1.

Ref: Sec. 3 of Chen and Eldan [5]

Topics: Localization Schemes -  Approximate Conservation of Variance for k-Glauber Dynamics, Part 2. Spectral Independence, Theorem of ALOG

Ref: Sec. 3 of Chen and Eldan [5]

Topics: Localization Schemes - Annealing and Optimal mixing for Ising Models with Interaction matrix operator norm less than 1/2.

Ref: Sec. 4-5 of Chen and Eldan [5]

Topics: Entropy factorization via curvature - Motivation, General framework setup, Main result (statement)

Ref: Sec. 1 of Caputo and Salez [3]

Topics: Proof of Theorem 2 (Main result, Step 3 in [3]) assuming variance contraction (Step 2 in [3]) and Lipschitz contraction (Step 1 in [3]).

Ref: Sec. 2 of Caputo and Salez [3]

Topics:  Applications of Entropy factorization on Block Dynamics of Card Shuffling and Down-up walk on n-subsets.

Ref: Sec. 3 of Caputo and Salez [3]

Topics:  Completed the Proof of Theorem 2 i.e. Variance contraction (Step 2 in [3]) and Lipschitz contraction (Step 1 in [3]).

Ref: Sec. 3 of Caputo and Salez [3]

Topics:  Consequences of Entropy Factorization: Peres-Tetali Conjecture and Shearer's inequality, Strong Data Processing Inequality, An application of Brascamp-Lieb inequality in Permanent bound.

Ref: Remarks 2 and 3 of Caputo and Salez [3], Chapter 3 exercise 18 [9], Theorem 7.2 [2]