FK–Ising coupling applied to near-critical planar models

We consider the Ising model at its critical temperature with external magnetic field ha^15∕8 on aZ². We give a purely probabilistic proof, using FK methods rather than reflection positivity, that for a=1, the correlation length is ≥const.h^−8∕15 as h↓0. We extend to the a↓0 continuum limit the FK–Ising coupling for all h>0, and obtain tail estimates for the largest renormalized cluster area in a finite domain as well as an upper bound with exponent 1∕8 for the one-arm event. Finally, we show that for a=1, the average magnetization, M(h), in Z² satisfies M(h)∕h^1∕15→ some B∈(0,∞) as h↓0.