Convergence and coupling for spin glasses and hard spheres

Area of Science:

• Statistical mechanics
• Computational physics
• Markov chain theory

Background:

• Markov chains are fundamental to simulating complex systems.
• Exact sampling from equilibrium distributions is computationally challenging.
• Existing methods struggle with low temperatures and high densities in models like spin glasses and hard spheres.

Purpose of the Study:

• To introduce and analyze a novel local-patch algorithm for Markov chain analysis.
• To establish general relations between convergence and coupling of Markov chains.
• To demonstrate the algorithm's effectiveness for statistical-mechanics models.

Main Methods:

• Developing general relations between transfer matrices for Markov chain convergence and coupling.
• Applying a local-patch algorithm to compute rigorous upper bounds for coupling time.
• Utilizing the "coupling from the past" protocol for exact sampling.
• Adapting the algorithm for spin glasses and hard-sphere models.

Main Results:

• The local-patch algorithm outperforms previous exact-sampling methods for 2D and 3D spin glasses at lower temperatures.
• Variants of the algorithm show potential for reaching the 3D spin-glass transition temperature.
• The algorithm successfully samples hard-sphere models, including 2D hard disks, at higher densities than previously possible.

Conclusions:

• The local-patch algorithm offers a significant advancement in exact sampling for complex statistical-mechanics models.
• This method enhances the ability to simulate systems at challenging low temperatures and high densities.
• The algorithm provides a powerful tool for exploring equilibrium distributions in various physical systems.