Short Talk: Yucheng Liu

Consider percolation, the Ising model, and the self-avoiding walk on $\mathbb Z^d$. Above their upper critical dimensions, the lace expansion gives a convolution equation for the two-point (correlation) function. We present two simple ways to analyse the convolution equation, to derive the asymptotic behaviour of the critical two-point function. For short-range models, we prove a deconvolution theorem, using only Hölder’s inequality, weak derivatives, and basic Fourier theory, which yields $|x|^{-(d-2)}$ asymptotics for the critical two-point functions. For (spread-out) long-range models, we prove an exact random walk representation of the two-point function, which allows us to derive the asymptotic behaviour via random walk computations. Our methods significantly simplify previous analyses based on the lace expansion. This is based on joint work with Gordon Slade.