Exact recovery in block spin Ising models at the critical line. (English) Zbl 1439.62254

Summary: We show how to exactly reconstruct the block structure at the critical line in the so-called Ising block model. This model was recently re-introduced by Q. Berthet et al. [Ann. Stat. 47, No. 4, 1805–1834 (2019; Zbl 1420.62268)]. There the authors show how to exactly reconstruct blocks away from the critical line and they give an upper and a lower bound on the number of observations one needs; thereby they establish a minimax optimal rate (up to constants). Our technique relies on a combination of their methods with fluctuation results obtained in [M. Löwe and K. Schubert, Electron. Commun. Probab. 23, Paper No. 53, 12 p. (2018; Zbl 1402.60027)]. The latter are extended to the full critical regime. We find that the number of necessary observations depends on whether the interaction parameter between two blocks is positive or negative: In the first case, there are about \(N\log N\) observations required to exactly recover the block structure, while in the latter case \(\sqrt{N}\log N\) observations suffice.


62P35 Applications of statistics to physics
62K10 Statistical block designs
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
80A10 Classical and relativistic thermodynamics
Full Text: DOI arXiv Euclid


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