The Abstract Lace Expansion by Doron Zeilberger

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(Appeared in Adv. Appl. Math. 19(1997), 355-359.)

Written: Dec. 6, 1996.

Mathematical Physicists are a breed apart. They are neither mathematicians nor physicists, and don't know much about either. What they do know a LOT of, is Mathematical Physics.

There are many hidden treasures, most of them yet to be discovered, in Mathematical Physics, that can enrich mathematics and especially combinatorics. Two recent dramatic examples are the Lapointe-Vinet breakthrough about the polynomiality (in the parameter) of the Jack polynomials, and the Izergin-Korepin formula that was used by Kuperberg to give a short proof of the Zeilberger theorem, conjectured by Mills-Robbins-Rumsey, about the number of alternating sign matrices.

Another such treasure is the Brydges-Spencer Lace Expansion, which is generalized and `axiomized' in this paper. I don't have any new applications yet, but I am sure that they would come.

This paper would have not been possible without the very lucid account in Madras and Slade's masterpiece, `The Self Avoiding Walk', Birkhauser. I wish that there would be more books like this, so accessible to the non-specialist!

This paper is dedicated to the memory of Paul Erdos. I am sure that he would include the Lace Expansion in the BOOK.

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