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+log2 n; n=4 is the Migdal-Kadanoff lattice of ℤ3. For n≥4 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≤ T*(n)=4/15, 6/7, 6/5, 7/4, 2 ($0.89$ to $0.98$ times numerical estimates of Tc), 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 1018 branches (Collet and Eckmann, 1984). The key step is growth of the effective coupling under the exact renormalization map: for n≥4, by a one-step contraction, via Rogozin's inequality, of a density bound on an event of high probability; near Tc, 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 L1; 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.
Related
- Existence and Absence of Spin-Glass Phases on Migdal--Kadanoff Hierarchical Lattices: A Complete Classification
- Hardy inequalities on the positive integer lattice: a dimensional transition
- Existence of positive Solutions to Hénon equation on integer Lattice graph and Half-Space
- Lattice points in high-dimensional ℓq balls with small radii
- Classification of Finite-Dimensional Simple Novikov Algebras and Their Irreducible Bimodules in Positive Characteristic
BibTeX
@misc{pei2026positivetemperature,
title = {{Positive-temperature spin-glass order on the three-dimensional Migdal-Kadanoff lattice}},
author = {Yan Ru Pei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38220},
eprint = {2609.38220},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38220},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive-temperature spin-glass order on the three-dimensional Migdal-Kadanoff lattice
AU - Yan Ru Pei
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.38220
UR - https://arxiv.org/abs/2609.38220
AB - 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 ℤ^3. For n≥4 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≤ 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≥4, 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.
ER -