The continuous-time lace expansion

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 $g|\phi {|}^4$ model on ${\mathbb{Z}}^d$ when $n=1,2$, and prove that the critical Green's function $G_{{\nu }_c}(x)$ is asymptotically a multiple of $|x{|}^{2-d}$ when $d\ge 5$ at weak coupling. As another application of our method we establish the analogous result for the lattice Edwards model at weak coupling.

Final version

References in corpus (5)

• Marginal triviality of the scaling limits of critical 4D Ising and ${\varphi }_4^4$ models
• Introduction to a renormalisation group method
• Application of the lace expansion to the ${\phi }^4$ model
• Loop-weighted Walk
• The geometry of random walk isomorphism theorems

Cited by in corpus (2)

• The geometry of random walk isomorphism theorems
• Kotani's Theorem for the Fourier Transform