Journal article

Lattice Green's functions in all dimensions

Anthony J Guttmann



We give a systematic treatment of lattice Green's functions (LGF) on the d-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality d ≥ 2 for the first three lattices, and for 2 ≤ d ≤ 5 for the hyper-fcc lattice. We show that there is a close connection between the LGF of the d-dimensional hyper-cubic lattice and that of the (d - 1)-dimensional diamond lattice. We give constant-term formulations of LGFs for each of these lattices in all dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the sc, bcc and diamond LGFs and some Ramanujan-type formu..

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University of Melbourne Researchers


Funding Acknowledgements

I would like to thank Geoff Joyce for introducing me to lattice Green's functions exactly 40 years ago when we shared an office, and Larry Glasser with whom I have corresponded and occasionally collaborated for many years. More recently I have benefited from discussions with Gert Almkvist, Richard Brak, David Broadhurst, David Bailey, Jon Borwein, Omar Foda, Ole Warnaar, John Zucker and Wadim Zudilin, and I would like to thank them all. Iwan Jensen and Gary Iliev kindly helped me manipulate some ODEs with Maple. Significant corrections to an earlier draft of the manuscript were made by Gert Almkvist, Omar Foda, Larry Glasser, Bernie Nickel and Wadim Zudilin, for which I am grateful. The information in footnotes 3 and 8 was kindly supplied by Mathew Rogers and Gert Almkvist, respectively. This work is supported by the ARC through a Discovery Grant, and it is a pleasure to acknowledge their support.