Statistical Properties of Strings

TL;DR

This paper numerically studies the statistical properties of cosmic strings using a novel lattice method that avoids boundary condition artifacts, revealing a lower percentage of infinite strings and analyzing the Hagedorn transition.

Contribution

Introduces a new lattice algorithm for string statistics that eliminates boundary condition issues and provides refined measurements of string percolation properties.

Findings

Percentage of infinite strings is 63%, lower than previous 80%.

Identifies the percolation threshold for infinite strings.

Measures critical exponents related to string divergence and correlation length.

Abstract

We investigate numerically the configurational statistics of strings. The algorithm models an ensemble of global $U(1)$ cosmic strings, or equivalently vortices in superfluid $^4$He. We use a new method which avoids the specification of boundary conditions on the lattice. We therefore do not have the artificial distinction between short and long string loops or a `second phase' in the string network statistics associated with strings winding around a toroidal lattice. Our lattice is also tetrahedral, which avoids ambiguities associated with the cubic lattices of previous work. We find that the percentage of infinite string is somewhat lower than on cubic lattices, 63\% instead of 80\%. We also investigate the Hagedorn transition, at which infinite strings percolate, controlling the string density by rendering one of the equilibrium states more probable. We measure the percolation threshold, the critical exponent associated with the divergence of a suitably defined susceptibility of the string loops, and that associated with the divergence of the correlation length.