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 can not 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 \pm 0.015 .