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

Abstract

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 that at sufficiently low temperature $T>0$, uniformly in the level of the lattice, the poles are ordered, every spin has Edwards-Anderson parameter $1-O(T)$ when the poles are fixed, and the overlap of two free-boundary replicas has second moment $1-O(T)$. For standard Gaussian couplings, $n=3,4,5,7,8$ and every $T\le T^*(n)=4/15,\,6/7,\,6/5,\,7/4,\,2$ ($0.89$ to $0.98$ times numerical estimates of $T_c$), a computer-assisted proof gives pole order and, as the level tends to infinity, site-overlap order and the blue-cluster statements below; likewise for every larger $n$ at these temperatures. Previously, spin-glass order on these lattices with frustrated independent couplings had been proved only for more than $10^{18}$ branches (Collet and Eckmann, 1984). The key step is growth of the effective coupling under the exact renormalization map: for $n\ge4$, by a one-step contraction, via Rogozin's inequality, of a density bound on an event of high probability; near $T_c$, by interval-arithmetic bounds in the peakedness order. A transfer theorem following the ancestor-chain argument of Collet and Eckmann turns this growth into bulk statements. In the Chayes-Machta-Redner representation the same input gives a blue cluster that contains both poles with probability tending to one and whose expected density is bounded away from zero, while the largest other blue cluster has density tending to zero in probability and in $L^1$; hence blue-cluster density imbalance. To our knowledge this is the first proof of such an imbalance for a spin glass on a graph other than the complete graph.