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Sampling Algorithms for Lattice Fields

Lattice Monte Carlo replaces path-integral averages by samples from a finite-regulator measure. Correctness requires a transition that preserves that measure and reaches its relevant support; efficiency depends on how quickly the observable of interest decorrelates. This chapter develops both questions separately.

Enter this chapter

Section titled “Enter this chapter”

Begin with Markov-Chain Sampling of Lattice Fields to define the target measure and the transition kernel that should preserve it.

Topics

Section titled “Topics”
  1. Markov-Chain Sampling of Lattice Fields — Tests invariance, support coverage, and equilibration.
  2. Local, Cluster, and Global Update Families — Chooses updates appropriate to the action’s structure.
  3. Hybrid Monte Carlo and Symplectic Molecular Dynamics — Combines reversible molecular dynamics with an acceptance correction.
  4. Pseudofermions, Determinant Ratios, and Solver Bias — Tracks positive determinant representations and solver approximations.
  5. Algorithm Validation and Ensemble Provenance — Checks the implemented transition against exact benchmarks.
  6. Critical Slowing Down, Topological Freezing, and Algorithmic Acceleration — Measures slow modes before claiming acceleration.

How the ideas fit together

Section titled “How the ideas fit together”

In HMC, numerical integration error is corrected by a Metropolis step only when the proposal satisfies the required reversibility and volume-preservation conditions. High acceptance alone does not verify those properties. Nor does a correct stationary distribution imply rapid sampling: a chain may remain trapped in slowly changing topological sectors. The relevant efficiency measure therefore includes autocorrelation and the cost of obtaining independent information about the target observable. Duane et al. 1987, pp. 216–222.

Where to go next

Section titled “Where to go next”

Continue to Statistical Inference and Error Budgets to measure how correlations change the uncertainty of a sample mean.

References

Section titled “References”
  • Duane, S., Kennedy, A. D., Pendleton, B. J., and Roweth, D. (1987). “Hybrid Monte Carlo.” Physics Letters B 195(2), 216–222. doi:10.1016/0370-2693(87)91197-X.