On perturbations of Ising models

The paper proves that, for d ≥ 2, (i) high-temperature uniqueness for the nearest-neighbor Ising model persists under small d-th order decreasing perturbations, and (ii) low-temperature phase coexistence persists under small symmetric d-th order decreasing perturbations. Part (i) is established via a variational-norm formulation of Dobrushin’s criterion, showing ||Φ_L||_var = 4dL and uniqueness when 2dL + |||Ψ||| < 1, because ||Ψ||_var ≤ 2|||Ψ|||, exactly as used in the candidate solution’s Dobrushin argument (paper: Proposition 2.1, Theorem 2.2, Corollary 2.3; ). Part (ii) is proved by reducing general symmetric d-th order decreasing perturbations to ones that are zero on non-path-connected sets (rectangular grouping Ψ → Ψ̃ with ‖Ψ̃‖ ≤ |||Ψ|||), then showing positive magnetization under + boundary conditions for large L via a Peierls-type counting of x-enclosing sets and a weight ratio bound (Theorem 3.1, Proposition 3.2, equation (3.1), Proposition 3.3, and the counting lemma; ). The candidate solution presents a standard alternative: Dobrushin uniqueness with the same smallness threshold, and a direct Peierls contour energy bound ΔE ≥ (2L − C_d|||Ψ|||)|∂V| leading to two Gibbs/equilibrium states for symmetric Ψ. This matches the paper’s conclusion, though the paper’s low-temperature proof proceeds via an explicit δ-map and counting of enclosing sets rather than explicit contour weights. Minor note: the candidate’s Lipschitz constant for Bernoulli single-site conditionals uses 1/2 where the tight bound is 1/4, but constants are anyway absorbed in the coarse norm bounds and do not affect the qualitative conclusion.