Improved rapid phase ordering for Ising dynamics on random regular graphs
Abstract
We study Glauber dynamics for the Ising model on random d-regular graphs with n vertices at low temperature, in the regime β≥ C/d where C,d are sufficiently large constants. In this regime, the full mixing time is exponential in n, because of the bottleneck between the two dominant phases: configurations with a majority of plus spins and configurations with a majority of minus spins. We show that, for sufficiently large d, if the dynamics is initialized with an excess of at least εC,d n plus spins over minus spins, then it quasi-equilibrates in time O(log n) to the Gibbs measure conditioned on positive magnetization. In addition, the required bias εC,d can be chosen so that εC,d→ 0 as d,C→∞. Previous bound of \cite{GSS25+} in the Ising and large-d setting required β≳ (log d)/d. A key difficulty is that, even from the all-plus initialization, one must control the propagation of negative information in spacetime in a low-temperature model with strong local correlations. We address this by developing a new bootstrap approach to proving subcriticality of disagreement clusters under the grand coupling.
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BibTeX
@misc{li2026improved,
title = {{Improved rapid phase ordering for Ising dynamics on random regular graphs}},
author = {Zhangsong Li and Allan Sly},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27673},
eprint = {2610.27673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27673},
}
RIS
TY - UNPB
TI - Improved rapid phase ordering for Ising dynamics on random regular graphs
AU - Zhangsong Li
AU - Allan Sly
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27673
UR - https://arxiv.org/abs/2610.27673
AB - We study Glauber dynamics for the Ising model on random d-regular graphs with n vertices at low temperature, in the regime β≥ C/d where C,d are sufficiently large constants. In this regime, the full mixing time is exponential in n, because of the bottleneck between the two dominant phases: configurations with a majority of plus spins and configurations with a majority of minus spins. We show that, for sufficiently large d, if the dynamics is initialized with an excess of at least ε_C,d n plus spins over minus spins, then it quasi-equilibrates in time O(log n) to the Gibbs measure conditioned on positive magnetization. In addition, the required bias ε_C,d can be chosen so that ε_C,d→ 0 as d,C→∞. Previous bound of \cite{GSS25+} in the Ising and large-d setting required β≳ (log d)/d. A key difficulty is that, even from the all-plus initialization, one must control the propagation of negative information in spacetime in a low-temperature model with strong local correlations. We address this by developing a new bootstrap approach to proving subcriticality of disagreement clusters under the grand coupling.
ER -