Statistical properties of worm algorithms for two dimensional frustrated Ising models

We study the distribution of lengths and other statistical properties of worms constructed by Monte Carlo worm algorithms in the power-law three-sublattice ordered phase of frustrated triangular and kagome lattice Ising antiferromagnets. Viewing each step of the worm construction as a position increment (step) of a random walker, we demonstrate that the persistence exponent $\theta$ and the dynamical exponent $z$ of this random walk depend only on the universal power-law exponents of the underlying critical phase, and not on the details of the worm algorithm or the microscopic Hamiltonian. Further, we argue that the detailed balance criterion obeyed by such worm algorithms and the power-law correlations of the underlying equilibrium system together give rise to two related properties of this random walk: First, the steps of the walk are expected to be power-law correlated in time. Second, the position distribution of the walker relative to its starting point is given by the equilibrium position distribution of a particle in an attractive logarithmic central potential of strength $\eta_m$, where $\eta_m$ is the universal power-law exponent of the equilibrium defect-antidefect correlation function of the underlying spin system. We derive a scaling relation, $z = (2-\eta_m)/(1-\theta)$, that allows us to express the dynamical exponent $z(\eta_m)$ of this process in terms of its persistence exponent $\theta(\eta_m)$. Our measurements of $z(\eta_m)$ and $\theta(\eta_m)$ are consistent with this relation over a range of values of the universal equilibrium exponent $\eta_m$, and yield subdiffusive ($z>2$) values of $z$ in the entire range. Thus we demonstrate that the worms represent a discrete-time realization of a fractional Brownian motion characterized by these properties.