Uniform covariance bounds for constrained Ising replicas
Fix $n \ge 1$, a real symmetric matrix $J \in \mathbb R^{n \times n}$ and a partition $\mathcal A$ of $[n]$. For $N \ge 2$ write a configuration of $N$ replicas as $\sigma = (\sigma(1), \dots, \sigma(N)) \in (\{-1, 1\}^n)^N$. Choose integers $M_A$, $A \in \mathcal A$, such that $$\Omega_{N,M} = \Bigl\{ \sigma : \sum_{i=1}^N \sum_{j \in A} \sigma_j(i) = M_A \text{ for every } A \in \mathcal A \Bigr\}$$ is nonempty, and for an external field $v \in \mathbb R^{Nn}$ let $$\nu_{N,M,v}(\sigma) = \frac{\mathbf 1_{\Omega_{N,M}}(\sigma)}{Z_{N,M,v}} \exp\Bigl\{ \frac12 \sum_{i=1}^N \sigma(i)^{\mathsf T} J \sigma(i) + \sum_{i=1}^N \sum_{j=1}^n v_{ij} \sigma_j(i) \Bigr\}.$$ Caputo and Morellini proved that when $J$ is positive definite with largest eigenvalue $\lambda_{\max}(J) < 1/2$ and the only constraint is on the total magnetization ($\mathcal A = \{[n]\}$), the covariance matrix of all $Nn$ spins has largest eigenvalue at most $2/(1 - 2\lambda_{\max}(J))$, for every external field. They also noted that equivalence-of-ensembles estimates give a bound uniform in $N$ for each fixed external field.
Problem. Is there a finite constant $C(J, \mathcal A)$ such that $$\bigl\| \operatorname{Cov}_{\nu_{N,M,v}}(\sigma) \bigr\|_{\mathrm{op}} \le C(J, \mathcal A)$$ for every $N \ge 2$, every $M$ with $\Omega_{N,M}$ nonempty and every $v \in \mathbb R^{Nn}$? The covariance is that of all $Nn$ spins, with the Euclidean operator norm. The constant may grow with $J$ and $n$ but must not depend on $N$, $M$ or $v$.
Source: Caputo and Morellini (2025), §4.5, the conjecture on tilted canonical Ising measures. Stated by AIM — Open Applied Problems with the constrained measure written out from their (1.23)–(1.24), uniformly also in the magnetizations.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, canonical ensembles, covariance bounds, Ising model
References
@misc{caputo2025,
author = {Caputo, Pietro and Morellini, Mario},
title = {Kac's program and relative entropy decay for nonlinear spin-exchange dynamics},
year = {2025},
eprint = {2511.05223},
archivePrefix = {arXiv},
}
@misc{aim-ising-replica-covariance,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniform covariance bounds for constrained Ising replicas}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/576-ising-replica-covariance.md},
note = {Entry 576 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove that the covariance of finitely many Ising replicas conditioned on fixed magnetizations has a bound independent of the number of replicas, the conserved totals and all replica-dependent external fields, for every fixed interaction matrix. The constraints may conserve the spin totals in each part of an arbitrary partition. This proves the covariance conjecture of Caputo and Morellini. The …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/ising_replica_covariance.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00756, "Uniform covariance bounds for constrained Ising replicas", clopen.solutions.