THE EIGENFUNCTIONS OF THE TRANSFER OPERATOR FOR THE DYSON MODEL IN A FIELD

The paper’s Theorem A states exactly the three claims the candidate solution addresses—(i) uniqueness of the equilibrium state and the eigenprobability, (ii) absolute continuity µ+ ≪ ν and existence of an L1(ν) eigenfunction, and (iii) nonexistence of any continuous version of dµ+/dν—under Dobrushin uniqueness, e.g., the strong-field bound |h| > 2βζ(α) + log(4βζ(α)) for α ∈ (3/2, 2] (the paper even writes this bound explicitly) . The proof in the paper proceeds via: (a) establishing DUC from a strong-field variant (geometric bound via Georgii’s Example 8.13) and then uniqueness of Gibbs measures, including on the half-line; identifying the unique eigenprobability via Cioletti–Lopes–Stadlbauer ; (b) constructing intermediate interactions Ψ(k), writing telescoping Radon–Nikodym factors, and proving uniform integrability using a de la Vallée–Poussin t log t control derived from Gaussian concentration under DUC and l2-summability of cross-bond influences (the α > 3/2 threshold) to conclude µ ≪ ν(0) and hence µ+ ≪ ν with dµ+/dν an L1 eigenfunction ; and (c) showing that any continuous version of dµ+/dν would force boundedness on cylinders, contradicted by a lower bound that diverges as the cylinder length grows, obtained using GKS/FGK positivity, Dobrushin comparison, and explicit polynomial-decay estimates (including a Jaffard-type bound on the Dobrushin resolvent used earlier), yielding essential discontinuity at 1Z+ and thus no continuous eigenfunction . The candidate solution follows the same structure: it invokes the same DUC strong-field condition, the same half-line Ruelle–DLR correspondence for eigenprobabilities, the same intermediate-interaction path with telescoping RN factors and uniform integrability (using the α > 3/2 l2-threshold), and the same positivity/decay machinery to force divergence on plus-cylinders. Differences are stylistic (e.g., the candidate sketches a harmonic-type divergence while the paper proves divergence via a precise double sum). Substantively, they match.