We examine simulations of the formation of domain walls, cosmic strings, and monopoles on a cubic lattice, in which the topological defects are assumed to lie at the zeros of a piecewise constant 1, 2, or 3 component Gaussian random field, respectively. We derive analytic expressions for the corresponding topological defect densities in the continuum limit and show that they fail to agree with simulation results, even when the fields are smoothed on small scales to eliminate lattice effects. We demonstrate that this discrepancy, which is related to a classic geometric fallacy, is due to the anisotropy of the cubic lattice, which cannot be eliminated by smoothing. This problem can be resolved by linearly interpolating the field values on the lattice, which gives results in good agreement with the continuum predictions. We use this procedure to obtain a lattice-free estimate (for Gaussian smoothing) of the fraction of the total length of string in the form of infinite strings: $f_{\infty }=0.716\mathrm{}\pm \mathrm{0.015.}$

• Received 30 September 1997

DOI:https://doi.org/10.1103/PhysRevD.58.103501