Journal article

The Continuous-Time Lace Expansion

David Brydges, Tyler Helmuth, Mark Holmes



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 m..

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University of Melbourne Researchers


Awarded by EPSRC

Awarded by Australian Research Council

Funding Acknowledgements

The authors thank Gordon Slade for various helpful comments regarding a previous version of this manuscript. TH thanks Gady Kozma for a helpful discussion. TH and DB thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Scaling limits, rough paths, quantum field theory" when work on this paper was undertaken. DB is grateful for partial support by the Simons Foundation during the same programme and thanks the University of Virginia for hospitality during 2016 and 2017 during this collaboration. This work was supported by EPSRC grants LNAG/036 and RG91310. TH held positions at UC Berkeley and the University of Bristol while this project was being carried out, and was supported by EPSRC Grant EP/P003656/1 and an NSERC PDF. MH is supported by the grant Future Fellowship FT160100166 from the Australian Research Council.