Universality and critical phenomena in string defect statistics

The idea of biased symmetries to avoid or alleviate cosmological problems caused by the appearance of some topological defects is familiar in the context of domain walls, where the defect statistics lend themselves naturally to a percolation theory description, and for cosmic strings, where the proportion of infinite strings can be varied or disappear entirely depending on the bias in the symmetry. In this paper we measure the initial configurational statistics of a network of string defects after a symmetry-breaking phase transition with initial bias in the symmetry of the ground state. Using an improved algorithm, which is useful for a more general class of self-interacting walks on an infinite lattice, we extend the work [in M. Hindmarsh and K. Strobl, Nucl. Phys. B 437, 471 (1995)] to better statistics and a different ground-state manifold, namely, ${\mathrm{RP}}^2$, and explore various different discretizations. Within the statistical errors, the critical exponents of the Hagedorn transition are found to be quite possibly universal and identical to the critical exponents of three-dimensional bond or site percolation. This improves our understanding of the percolation theory description of defect statistics after a biased phase transition. We also find strong evidence that the existence of infinite strings in the Vachaspati-Vilenkin algorithm [T. Vachaspati and A. Vilenkin, Phys. Rev. D 30, 2036 (1984)] is generic to all (string-bearing) vacuum manifolds, all discretizations thereof, and all regular three-dimensional lattices.