Ongoing trimester: Spin systems and phases of matter
Mathematics
Extract one-arm exponent in FK models from the convergence of height functions to GFF
byXia Jiaming
Abstract:We consider FK models with \(q\) in \([0,4]\) on the square lattice and the whole plane. We assume the convergence of height functions to GFF and in particular we assume that we know the variance \(\sigma^2\) of the GFF. Then, we sketch an approach to get the exponent \(\alpha_1\) describing the probability of having a primal crossing of an annulus. The basis for this approach is the BKW coupling relating the height function to the interface loops of FK. We show that by choosing appropriate test functions (viewed as placing charges on the plane), we can get relations between \(\sigma^2\), \(\alpha_1\), and a factor accounting for local concentration of small interface loops.