The Nearest-Neighbor Self-Avoiding Walk with Complex Killing Rates

Abstract

The Green’s function for the nearest-neighbor self-avoiding walk on a hypercubic lattice in d > 2 dimensions is constructed and shown to be analytic for values of the killing rate $ a \in \textbf{C} $ satisfying $ \vert a \vert > \epsilon, \vert \textrm{arg}\,\, a \vert $ \lt; $ 3\pi/4 - b $ with $ \epsilon > 0 $ and 0 \lt; b \lt; $ \pi/4 $. We restrict $ \vert a \vert > \epsilon > 0 $ in order to use the killing rate as an infrared cutoff, which allows us to construct Green’s function using a single scale cluster expansion. The presence of non-real killing introduces complications that we resolve through the use of an appropriate choice of decoupling scheme and a subsidiary expansion. Our methods can be used to control a single momemtum slice in a phase-space expansion.