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, whose computation is notoriously difficult. For the Ising model, physicists discovered that above the upper critical dimension d_c, these exponents reduce to the mean-field values, matching those on trees or complete graphs. I will provide an overview of some older and more recent results.

Then, 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, joint with Christophe Garban, Romain Panis and Franco Severo. We will 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 (by the Edwards-Sokal coupling, this probability is the magnetization at the origin), whereas in infinite volume we prove that it decays as n^(-2) for d ≥ 6. You may be wondering at this point why there is a discrepancy in the dimension. To avoid spoiling the surprise, I will provide the answer only during the talk.