Path Coupling for Ising and Potts Models

We study the convergence behavior of Markov chains for the Ising and Potts models on graphs with n vertices and maximum degree &Delta.; In the case of Glauber dynamics for the Potts model with q colors, we use path coupling to establish rapid mixing (in order nlogn steps) in the high temperature regime where the inverse temperature β is less than log({1+2/Δ}/{1+Δ}). For the Ising model on the complete graph, a well known argument based on path coupling shows that Glauber dynamics has fast mixing. In contrast, for random walk Metropolis chains for the Ising model on the complete graph, we show that a similar approach using path coupling fails to establish a contraction, and therefore does not establish mixing time bounds.

Author

• Gao, Zhikun

Language

• en

Keyword

• Potts model
• Path coupling
• Metropolis chains
• Ising model
• Glauber dynamics

Date created

• 2014

Type of Work

• Master's Thesis

License

• All rights reserved

Rights statement

• In Copyright

GW Unit

• Statistics

Degree

• M.S.

Advisor

• Barta, Winfried

Committee Member(s)

• Mahmoud, Hosam

Persistent URL

•  https://scholarspace.library.gwu.edu/etd/p5547r64d