Nonvanishing of complexified connection probabilities

Determine whether, for each subcritical FK-percolation parameter p and sufficiently large connection distance, the complex extensions of the finite-distance connection probabilities remain nonzero throughout some common complex neighbourhood of p.

Background

A direct approach to complexifying the mass would require taking logarithms of complex extensions of finite-distance connection probabilities. Such a logarithm can be defined uniformly only if these complexified probabilities do not vanish in a common neighbourhood of the real subcritical parameter. The authors do not know whether this nonvanishing property holds and note that cancellations may produce zeros arbitrarily close to the real parameter; Appendix A.4 gives a sequence of local events illustrating related accumulating zeros.