The near-critical random bond FK-percolation model

Abstract: We study FK-percolation where the edge parameters are chosen as independent random variables in the near-critical regime. We show that if these parameters satisfy a natural centering condition around the critical point, then the quenched model typically exhibits critical behaviour at scales much larger than the deterministic characteristic length. More precisely, in a box of size $N$, if the homogeneous model with deterministic edge parameter $p$ looks critical in the regime $|p-p_c|\le \textrm W$, then the quenched model with random edge parameters $\mathbf p$ that typically satisfy $|\mathbf p-p_c|\le \textrm W^{1/3}$ looks critical, assuming some conjectured inequality on critical exponents, and up to logarithmic corrections. We also treat the special case of Bernoulli percolation, where we show that if one first samples non-degenerate independent random edge parameters centered around $\frac12$, and then a percolation configuration on these edges, the quenched model almost surely looks critical at large scales.