Martin-L\"of randomness and Galton-Watson processes

TL;DR

This paper characterizes Martin-Löf random closed sets as infinite paths through Galton-Watson trees with a specific survival probability, linking effective Hausdorff dimension to randomness and tree structure.

Contribution

It establishes a precise connection between Martin-Löf randomness, Galton-Watson processes, and effective Hausdorff dimension, providing new criteria for randomness in closed sets.

Findings

Members of Martin-Löf random closed sets correspond to paths in Galton-Watson trees with survival parameter 2/3.

Having effective Hausdorff dimension greater than log2(3/2) is sufficient for membership.

Having effective Hausdorff dimension at least log2(3/2) is necessary for such membership.

Abstract

The members of Martin-L\"of random closed sets under a distribution studied by Barmpalias et al. are exactly the infinite paths through Martin-L\"of random Galton--Watson trees with survival parameter $\frac{2}{3}$. To be such a member, a sufficient condition is to have effective Hausdorff dimension strictly greater than $\gamma=\log_2 \frac{3}{2}$, and a necessary condition is to have effective Hausdorff dimension greater than or equal to $\gamma$.