Near-critical moment corrections in the bond-diluted Ising model on the Bethe lattice

We study the first corrections to the critical cavity-field distribution of a bond-diluted Ising ferromagnet on a Bethe lattice. At fixed bond occupation above the percolation threshold, the mean-normalized cavity field tends to the branching-process martingale profile. Starting from this known leading limit, we derive a triangular recursion for the first correction to every fixed normalized moment. Strict log-convexity of the critical moments makes the first corrections to moments of order two and higher negative whenever the lattice is diluted. The same expansion determines the next term in the message scale and the relative correction to the spontaneous magnetization. For coordination three, the latter has a closed expression and changes sign as the bond occupation varies. We trace this change to the balance between the message scale and the nonlinear root response, and specify its dependence on the choice of critical distance. Independent exact algebra and finite-coupling distributional iterations test the coefficients and their range of numerical usefulness. All asymptotic statements hold at fixed dilution and fixed moment order; a joint percolation limit is not assumed. Acknowledgments. We acknowledge Gewu Intelligence Lab for providing the collaborative research environment in which this project was developed. W.J. acknowledges support from the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB1680102. This preprint includes the main article, Supplemental Material, their LaTeX sources, and the numerical data and code required for reproduction.