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Replica symmetry breaking at arbitrary depth

P. M. Aronow, Patrick Lopatto

Palomar Registrypalomar/PALOMAR-2026-10-11-000017Oct 11, 2026math.PRscore 0

A Lean formalization of open phases with every prescribed finite replica symmetry breaking depth in mixed even-spin Ising models at zero external field and positive temperature. The first selected result constructs strictly increasing even degrees beginning at four and a nonempty open set of positive effective covariance coefficients, with an optimizing probability measure having exactly k+1 support points. The independent statement uses concrete Gaussian Cole–Hopf recursions and comparison against all finite schemes at every competing depth. The substantive development constructs minimizers…

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