Toronto Probability

The study of critical phenomena in lattice statistical mechanical models such as percolation has a long history in both physics and mathematics. A central problem is to derive the asymptotic behaviour of a model's two-point function at and near a critical point. The talk presents a general theorem providing the asymptotic decay of the solution to certain convolution equations. The theorem applies to the lattice Green function (random walk in any dimension d), the generating function for the number of n-step self-avoiding walks from 0 to x (d at least 5), the probability that 0 and x are connected in a percolation cluster (d at least 15). It gives a precise asymptotic formula for the decay of the two-point function in these three applications, which remains valid both in the subcritical regime (Ornstein-Zernike decay) and also at and near the critical point. This is joint work with Yucheng Liu (arXiv:2605.15545).