Beyond the Bochner Method: Optimal Spectral Gaps for Block Dynamics of Spin Systems like the O(N) Model
Abstract
Let μ be a probability distribution on a spin system with n sites and possibly infinite spin spaces. We prove local-to-global spectral-gap bounds for block Glauber dynamics. Given a symmetric, entrywise nonnegative matrix A with zero diagonal and λmax(A)<1 encoding local spectral bounds for a spin system, we derive global spectral-gap bounds for block dynamics on its marginal distributions. These bounds are sharp: the Gaussian system with covariance (I−A)−1 attains equality for every choice of disjoint blocks. Our proof extends the Bochner method through an abstract inequality for orthogonal projections on Hilbert spaces. As an application, we prove that Glauber dynamics for the O(N) model, with N≥ 2, arbitrary external fields, and couplings of either sign, has spectral gap at least (1−|Jabs|op/N)/n, where J is the interaction matrix and Jabs=(|Jij|). We also obtain spectral bounds for swap walks and local spectral expansion of partite simplicial complexes. For complexes formed from disjoint groups of ranks in random flags of subspaces over 𝔽q, we obtain a one-sided local spectral expansion bound of 2√q/(q+1). This bound is independent of the number and sizes of each group, and is nontrivial for every prime power q. Finally, for random forests F of any finite graph with arbitrary positive edge weights, we prove that Pr(e,f∈ F)≤ 4/3 Pr(e∈ F)Pr(f∈ F) for distinct edges e,f, improving the factor 2 supplied by general log-concavity bounds for matroid independent sets. The argument combines our extended Bochner method with the analysis of an auxiliary real stable polynomial.
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BibTeX
@misc{leake2026beyond,
title = {{Beyond the Bochner Method: Optimal Spectral Gaps for Block Dynamics of Spin Systems like the $O(N)$ Model}},
author = {Jonathan Leake and Anqi Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05228},
eprint = {2611.05228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05228},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond the Bochner Method: Optimal Spectral Gaps for Block Dynamics of Spin Systems like the O(N) Model
AU - Jonathan Leake
AU - Anqi Li
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05228
UR - https://arxiv.org/abs/2611.05228
AB - Let μ be a probability distribution on a spin system with n sites and possibly infinite spin spaces. We prove local-to-global spectral-gap bounds for block Glauber dynamics. Given a symmetric, entrywise nonnegative matrix A with zero diagonal and λ_max(A)<1 encoding local spectral bounds for a spin system, we derive global spectral-gap bounds for block dynamics on its marginal distributions. These bounds are sharp: the Gaussian system with covariance (I−A)^−1 attains equality for every choice of disjoint blocks. Our proof extends the Bochner method through an abstract inequality for orthogonal projections on Hilbert spaces. As an application, we prove that Glauber dynamics for the O(N) model, with N≥ 2, arbitrary external fields, and couplings of either sign, has spectral gap at least (1−|J_abs|_op/N)/n, where J is the interaction matrix and J_abs=(|J_ij|). We also obtain spectral bounds for swap walks and local spectral expansion of partite simplicial complexes. For complexes formed from disjoint groups of ranks in random flags of subspaces over 𝔽_q, we obtain a one-sided local spectral expansion bound of 2√q/(q+1). This bound is independent of the number and sizes of each group, and is nontrivial for every prime power q. Finally, for random forests F of any finite graph with arbitrary positive edge weights, we prove that Pr(e,f∈ F)≤ 4/3 Pr(e∈ F)Pr(f∈ F) for distinct edges e,f, improving the factor 2 supplied by general log-concavity bounds for matroid independent sets. The argument combines our extended Bochner method with the analysis of an auxiliary real stable polynomial.
ER -