In the physics literature, it is well understood that critical statistical mechanics models on Z^d behave like their mean-field counterpart for sufficiently high d. The dimension d_c at which such a “trivial” behaviour arises is called the upper-critical dimension. The Ising model, arguably the most classical statistical mechanics model, is known to satisfy d_c=4 (even though this is not fully proved rigorously). In this talk, we consider a percolation model known as the FK-Ising model, whose connectivity probabilities are directly related to Ising correlations. For this model, we study the event that the origin is connected to distance n (the so-called one-arm event). We prove that this event exhibits different critical exponents in dimensions d=4, d=5 and d\geq6. We also compute the corresponding exponent for the Ising model’s magnetisation, which is the same for all d\geq4. These results corroborate a  prediction made by theoretical physicists that the FK-Ising model has critical dimensions d_c=6, which is (perhaps surprisingly) different than the Ising model’s d_c=4.

Based on joint works with Diederik van Engelenburg, Christophe Garban and Romain Panis.