# Improvements for Vachaspati-Vilenkin-type Algorithms for Cosmic String and Disclination Formation

## TL;DR

This paper enhances Monte-Carlo simulations of cosmic string and disclination formation by proposing a tetrakaidekahedral lattice, enabling larger scale simulations without boundary constraints, and introduces efficient search algorithms for defect tracing.

## Contribution

It introduces a new lattice structure and computational methods that improve the scalability and efficiency of simulating defect formation in cosmic strings and liquid crystals.

## Key findings

- Use of tetrakaidekahedral lattice improves simulation accuracy.
- Simulations can be performed on infinite lattices without boundary effects.
- Hash table-based search algorithms significantly reduce defect tracing time.

## Abstract

We derive various consistency requirements for Vachaspati-Vilenkin type Monte-Carlo simulations of cosmic string formation or disclination formation in liquid crystals. We argue for the use of a tetrakaidekahedral lattice in such simulations. We also show that these calculations can be carried out on lattices which are formally infinite, and do not necessitate the specification of any boundary conditions. This way string defects can be traced up to much larger lengths than on finite lattices. The simulations then fall into a more general class of simulations of self-interacting walks, which occupy the underlying lattice very sparsely. An efficient search algorithm is essential. We discuss various search strategies, and demonstrate how to implement hash tables with collision resolution by open addressing. The time to trace a string defect is then proportional only to the string length.