Abstract: The goal of this talk is to study the two-point function (i.e. point-to-point connection probabilities) in the subcritical infinite-range Bernoulli percolation with exponentially decaying coupling constants. I will start by reviewing classical results on the two-point function and its associated correlation length in the nearest neighbour case. Then, I will explain why, contrary to expectations, the classical picture does not necessarily hold in the infinite-range case with exponentially decaying couplings constants: the correlation length might be non-analytic in the subcritical regime and the usual OZ-asymptotics might not hold, even in one-dimensional setting. I will explain necessary and sufficient conditions for this to occur, and describe the precise asymptotics of the two point-function in the subcritical regime.
Although I will focus on Bernoulli percolation, the results hold much more generally (in particular for FK percolation).
This is based on joint work with Dmitry Ioffe, Sébastien Ott and Yvan Velenik.