LA Probability Forum

UCLA, Math Sciences Room 6627

For torus percolation in high-dimensions the two-point function has a whole critical window (depending on the size of the torus) where it behaves more or less like the critical two-point function in Z^d. This fact can be summarized in a quantitative estimate called the plateau for the torus two-point function. A similar estimate conjecturally holds for the self-avoiding walk two-point function on a torus and partial results have been recently obtained in this direction. In this talk we will focus on plateaux estimates for those two models as well as on the various implications they have : ranging from the torus triangle condition in percolation to the asymptotic number of (weakly) self-avoiding walks on the torus. This is based on joint work with Tom Hutchcroft, Gordon Slade and Jiwoon Park.