The Continuous‐Time Lace Expansion. Issue 11 (4th September 2021)

• Record Type: Journal Article
• Authors: Brydges, David; Helmuth, Tyler; Holmes, Mark
• Is Part Of: Communications on pure and applied mathematics. Volume 74:Issue 11(2021)
• Page Start: 2251
• Page End: 2309
• Publication Date: 2021-09-04
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.22021
• Languages: English
• ISSNs: 0010-3640

Abstract

Abstract : 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 g ϕ 4 model on ℤ d when n =1, 2, and prove that the critical Green's function G ν c x is asymptotically a multiple of x 2 − 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.