Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model

Assuming the stated deterministic critical-reference estimates, we prove that the relative second moment of the quenched critical spin correlation in the square-lattice Ising model with independent fair bonds $1\pm \epsilon$ grows as $(\log r{)}^{1/4+o(1)}$ for each sufficiently small fixed ε > 0. The correlation is evaluated at the physical critical temperature, with the thermodynamic limit taken before the disorder moments and the large-distance limit.