Abstract: 

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, which reveals the values of critical exponents that govern the universal behaviour of the model at and near its critical point.  Proofs are typically available only for dimension two, or above an upper critical dimension where mean-field behaviour is observed.  The talk presents a general theorem providing the asymptotic decay of the solution to convolution equations of a certain sort. It 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 (high enough d). The general theorem 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. It exhibits, in a unified and general way, the crossover from subcritical to critical decay.  This is joint work with Yucheng Liu (Peking University).

Bio-Sketch：
Gordon Slade is a Professor Emeritus in the Department of Mathematics at the University of British Columbia, where he has been a faculty member since 1999. His research lies at the interface of probability theory and mathematical physics, with a particular focus on critical phenomena and phase transitions in statistical mechanics. Professor Slade is widely recognised for his work in the lace expansion and in a rigorous renormalisation group method. He has been elected a Fellow of the Royal Society of London and the Royal Society of Canada, and is the recipient of numerous major awards including the 2010 CRM-Fields-PIMS Prize and the 2018 Jeffery-Williams Prize.