An introduction to the Ising model
American Mathematical Monthly
Polynomial-time approximation algorithms for the Ising model
SIAM Journal on Computing
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
A more rapidly mixing Markov chain for graph colorings
proceedings of the eighth international conference on Random structures and algorithms
Faster random generation of linear extensions
Discrete Mathematics - Special issue on partial ordered sets
Sampling spin configurations of an Ising system
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Random Structures & Algorithms
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Torpid mixing of simulated tempering on the Potts model
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Slow mixing of glauber dynamics via topological obstructions
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Compact error-resilient computational DNA tiling assemblies
DNA'04 Proceedings of the 10th international conference on DNA computing
| Hi-index | 5.23 |
Algorithms based on Markov chains are ubiquitous across scientific disciplines as they provide a method for extracting statistical information about large, complicated systems. For some self-assembly models, Markov chains can be used to predict both equilibrium and non-equilibrium dynamics. In fact, the efficiency of these self-assembly algorithms can be related to the rate at which simple chains converge to their stationary distribution. We give an overview of the theory of Markov chains and show how many natural chains, including some relevant in the context of self-assembly, undergo a phase transition as a parameter representing temperature is varied in the model. We illustrate this behavior for the non-saturated Ising model in which there are two types of tiles that prefer to be next to other tiles of the same type. Unlike the standard Ising model, we also allow empty spaces that are not occupied by either type of tile. We prove that for a local Markov chain that allows tiles to attach and detach from the lattice, the rate of convergence is fast at high temperature and slow at low temperature.