  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.
