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Ruelle mentions ( http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B94%5D.pdf ) Lee and Yang' s "circle theorem", which comes from statistical mechanics and shall have not yet explored connections with zeta functions and Weil conjectures. Does one know now more about that? (Thanks to Alexandre Eremenko for the hint to that interesting article in an other MO thread)

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The introduction to a Borcea and Branden paper (arxiv.org/abs/0809.3087, 2009) indicates interesting connections with Lee/Yang, but suggests little more has been done with connections to the Weil conjectures (citing the same Ruelle article you have here). –  Benjamin Dickman Jul 6 '13 at 22:13
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Perhaps you could add a little context to the question: "I have called this beautiful result a failure because, while it has important applications in physics, it remains at this time isolated in mathematics. One might think of a connection with zeta functions (and the Weil conjectures); the idea of such a connection is not absurd, as our second example will show. But the miracle has not happened: one still does not know what to do with the circle theorem." As you can see, this speculation is based on wishful thinking more than experimental evidence or substantial analogies. –  S. Carnahan Jul 6 '13 at 22:44

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I do not know about any connection with Weil's conjectures, and this should be considered an extended comment rather than an answer.

Actually Ruelle was wrong when he said that this result "remains isolated" in mathematics. A new proof which fits very well into "manstream" mathematics was given in 1981 in MR0623156 (83c:82008) Lieb, Elliott H.; Sokal, Alan D. A general Lee-Yang theorem for one-component and multicomponent ferromagnets. Comm. Math. Phys. 80 (1981), no. 2, 153–179.

It uses a result of Takagi (early 20-s century) on the distribution of zeros of polynomials, which does not seem to be related to physics. Subsequent papers of A. Sokal further explore this idea.

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