Effect of scale on long-range random graphs and chromosomal inversions

TL;DR

This paper investigates how the size of local interactions influences the emergence of giant components in long-range random graphs and relates these findings to chromosomal inversions and genetic processes.

Contribution

It establishes thresholds for the appearance of giant components in circle-based percolation models and connects these thresholds to phase transitions in related random transposition processes.

Findings

Giant component appears when L(n)  (\,log n)^2

No giant component when L  \,log n

Phase transition in random transpositions depends on L(n) tending to infinity

Abstract

We consider bond percolation on $n$ vertices on a circle where edges are permitted between vertices whose spacing is at most some number L=L(n). We show that the resulting random graph gets a giant component when $L\gg(\log n)^2$ (when the mean degree exceeds 1) but not when $L\ll\log n$. The proof uses comparisons to branching random walks. We also consider a related process of random transpositions of $n$ particles on a circle, where transpositions only occur again if the spacing is at most $L$. Then the process exhibits the mean-field behavior described by Berestycki and Durrett if and only if L(n) tends to infinity, no matter how slowly. Thus there are regimes where the random graph has no giant component but the random walk nevertheless has a phase transition. We discuss possible relevance of these results for a dataset coming from D. repleta and D. melanogaster and for the typical length of chromosomal inversions.