Positive-temperature spin-glass order on the three-dimensional Migdal-Kadanoff lattice

We study the Ising spin glass with independent, identically distributed couplings on the diamond hierarchical lattice with $n$ branches, of effective dimension $1+\log_2 n$; $n=4$ is the Migdal-Kadanoff lattice of $\mathbb{Z}^3$. For $n\ge4$ and every coupling law with a bounded density we prove tha