Abstract: 

In this talk, I will focus on behavior of the Ising model in high dimensions (d ≥ 4). Widom proposed that thermodynamic quantities follow power laws governed by critical exponents, and above the upper critical dimension d_c = 4, these exponents reduce to the mean-field values (matching those on trees or complete graphs). I will talk about a recent work about the so-called one-arm event (the origin connects to distance n) in the FK-Ising model, 

We observe that this exponent depends on the boundary condition: for wired boundary conditions, we prove that this probability decays up to constants as n^(-1) for d ≥ 4, whereas in infinite volume we prove that it decays as n^(-2) for d ≥ 6, but not for d = 4, 5.