Paper Info
Title
The Ising Model on Trees: Boundary Conditions and Mixing Time
Abstract
We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics for the Ising model. Specifically, we show that the mixing time on an n-vertex regular tree with (+) boundary remains 0(n log n) at all temperatures (in contrast to the free boundary case, where the mixing time is not bounded by any fixed polynomial at low temperatures). We also show that this bound continues to hold in the presence of an arbitrary external field. Our results are actually stronger, and provide tight bounds on the log-Sobolev constant and the spectral gap of the dynamics. In addition, our methods yield simpler proofs and stronger results for the mixing time in the regime where it is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hard constraints like the antiferromagnetic Potts model at zero temperature (colorings) and the hard-core model (independent sets).
Year
DOI
Venue
2003
10.1109/SFCS.2003.1238235
FOCS
Keywords
Field
DocType
Ising model,Markov processes,Potts model,antiferromagnetism,computational complexity,graph colouring,trees (mathematics),Glauber dynamics,Ising model,antiferromagnetic Potts model,boundary condition,colorings,hard-core model,independent sets,log-Sobolev constant,mixing time,spectral gap,trees,zero temperature
Boundary value problem,Discrete mathematics,Glauber,Combinatorics,Polynomial,Ising model,Spectral gap,Time complexity,Potts model,Mathematics,Bounded function
Conference
ISSN
ISBN
Citations 
0272-5428
0-7695-2040-5
9
PageRank 
References 
Authors
1.23
11
3
Name
Order
Citations
PageRank
Fabio Martinelli121221.91
Alistair Sinclair21506308.40
Dror Weitz325819.56