The Continuous-Time Lace Expansion

We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion applies when the self-interaction is a sufficiently nice function of the local time of a continuous-time random walk. As a special case we obtain a continuous-time lace expansion for a class of spin systems that admit continuous-time random walk representations. We apply our lace expansion to the n-component (Formula presented.) model on (Formula presented.) when n=1,2, and prove that the critical Green's function (Formula presented.) is asymptotically a multiple of (Formula presented.) when (Formula presented.) and the coupling is weak. As another application of our method, we establish the analogous result for the lattice Edwards model at weak coupling. © 2021 Wiley Periodicals LLC.