The Continuous‐Time Lace Expansion

• Published in: Communications on pure and applied mathematics Vol. 74; no. 11; pp. 2251 - 2309
• Main Authors: Brydges, David, Helmuth, Tyler, Holmes, Mark
• Format: Journal Article
• Language: English
• Published: Melbourne John Wiley & Sons Australia, Ltd 01.11.2021; John Wiley and Sons, Limited
• Subjects: Coupling; Green's functions; Lace; Random walk
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.22021

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 gϕ4 model on ℤd when n=1,2, and prove that the critical Green's function Gνcx is asymptotically a multiple of x2−d when d≥5 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.