Asymptotics of Fixed Point Distributions for Inexact Monte Carlo Algorithms

TL;DR

This paper develops a method to determine the equilibrium distribution of inexact Markov Chain Monte Carlo algorithms, accounting for errors from numerical integration and non-commuting operators, with applications to lattice QCD.

Contribution

It introduces a general approach to find fixed point distributions for inexact MCMC algorithms considering operator non-commutativity and numerical errors.

Findings

Derived explicit error formulas using Baker-Campbell-Hausdorff formula.

Analyzed fixed point distributions for inexact Hybrid algorithms.

Applied the method to lattice QCD algorithms.

Abstract

We introduce a simple general method for finding the equilibrium distribution for a class of widely used inexact Markov Chain Monte Carlo algorithms. The explicit error due to the non-commutivity of the updating operators when numerically integrating Hamilton's equations can be derived using the Baker-Campbell-Hausdorff formula. This error is manifest in the conservation of a ``shadow'' Hamiltonian that lies close to the desired Hamiltonian. The fixed point distribution of inexact Hybrid algorithms may then be derived taking into account that the fixed point of the momentum heatbath and that of the molecular dynamics do not coincide exactly. We perform this derivation for various inexact algorithms used for lattice QCD calculations.