A Markov chain is a random process in which each transition to a new state is determined in a "memoryless" manner that only depends on the current state. Markov chains are ubiquitous objects in probability theory that permeate through both pure and applied mathematics. When one runs an irreducible Markov chain for a long time, the distribution of the states will converge to a stationary distribution. This course will develop tools for computing this stationary distribution and estimating how quickly the Markov chain converges to it.
Meeting times: T & TH 10:30-11:45 AM
Contact Information:
Office: SC 235
Email: colindefant@gmail.com
Office Hours: T 12:00-1:00 PM (this might change).
Problem Set 3 TBA
Problem Set 4 TBA
Problem Set 5 TBA
Potential Sources for Final Projects
(You are welcome to choose something that is not from this list!)
Proof of Aldous' Spectral Gap Conjecture by F. Caputo, T. M. Liggett, and T. Richthammer
Fast Mixing in Sparse Random Ising Models by K. Liu, S. Mohanty, A. Rajaraman, and D. X. Wu.
Lifting Markov Chains to Speed up Mixing by F. Chen, L. Lovasz, and I. Pak.
Characterization of Cutoff for Reversible Markov Chains by R. Basu, J. Hermon, and Y. Peres.
Cutoff for Non-negatively Curved Markov Chains by J. Salez.
Cutoff Phenomenon for the Asymmetric Simple Exclusion Process and the Biased Card Shuffling by C. Labbe and H. Lacoin.
Limit Profile for Projections of Random Walks on Groups by E. Nestoridi and S. Olesker-Taylor.
The S_k Shuffle Block Dynamics by E. Nestoridi, A. Priestley, and D. Schmid.
The Limit Profile of Star Transpositions by E. Nestoridi.
Cutoff on Trees is Rare by N. Gantert, E. Nestoridi, and D. Schmid
Cutoff for the Cyclic Adjacent Transposition Shuffle by E. Nestoridi and D. Nam.
Cutoff for Random to Random Card Shuffle by M. Bernstein and E. Nestoridi.
A Markov Chain on Permutations which Projects to the PASEP by S. Corteel and L. Williams.
Tableaux Combinatorics for the Asymmetric Exclusion Process by S. Corteel and L. Williams.
The Inhomogeneous t-PushTASEP and Macdonald Polynomials by A. Ayyer, J. Martin, and L. Williams.
The Shape of a Random Affine Weyl Group Element and Random Core Partitions by T. Lam.
Correlations in the Multispecies TASEP and a Conjecture by Lam by A. Ayyer and S. Linusson.
The Oriented Swap Process by O. Angel, A. Holroyd, and D. Romik.
Permuton Limit of a Generalization of the Mallows and k-Card-Minimum Models by J. Jasinska and B. Rath.
Grothendieck Shenanigans: Permutons from Pipe Dreams via Integrable Probability by A. Morales, G. Panova, L. Petrov, and D. Yeliussizov.