Statistical analysis method for the worldvolume hybrid Monte Carlo algorithm

We discuss the statistical analysis method for the worldvolume hybrid Monte Carlo (WV-HMC) algorithm [M. Fukuma and N. Matsumoto, Prog. Theor. Exp. Phys. 2021, 023B08 (2021)], which was recently introduced to substantially reduce the computational cost of the tempered Lefschetz thimble method. In the WV-HMC algorithm, the configuration space is a continuous accumulation (worldvolume) of deformed integration surfaces, and sample averages are considered for various subregions in the worldvolume. We prove that, if a sample in the worldvolume is generated as a Markov chain, then the subsample in the subregion can also be regarded as a Markov chain. This ensures the application of the standard statistical techniques to the WV-HMC algorithm. We particularly investigate the autocorrelation times for the Markov chains in various subregions, and find that there is a linear relation between the probability of being in a subregion and the autocorrelation time for the corresponding subsample. We numerically confirm this scaling law for a chiral random matrix model.