An efficient algorithm to generate large random uncorrelated Euclidean distances: the random link model

TL;DR

This paper introduces an efficient algorithm for generating large, uncorrelated Euclidean distances in high-dimensional spaces, enabling scalable analysis of random link models with reduced memory requirements.

Contribution

The authors develop techniques leveraging pseudo-random generators to simulate large random link systems with linear memory, maintaining quadratic time complexity.

Findings

Memory usage reduced to O(N) for large systems

Maintains O(N^2) time complexity

Enables scalable simulations of high-dimensional random distances

Abstract

A disordered medium is often constructed by $N$ points independently and identically distributed in a $d$-dimensional hyperspace. Characteristics related to the statistics of this system is known as the random point problem. As $d \to \infty$, the distances between two points become independent random variables, leading to its mean field description: the random link model. While the numerical treatment of large random point problems pose no major difficulty, the same is not true for large random link systems due to Euclidean restrictions. Exploring the deterministic nature of the congruential pseudo-random number generators, we present techniques which allow the consideration of models with memory consumption of order O(N), instead of $O(N^2)$ in a naive implementation but with the same time dependence $O(N^2)$.