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Field theoretic cluster expansions and the Brydges-Kennedy forest sum formula

Tuesday 8th April 2008 - 11:30 to 12:30
INI Seminar Room 1

In the first half of this talk, we will discuss cluster expansions in the context of quantum field theory, using the simple example of a weak phi^4 perturbation of a Gaussian measure with massive, i.e., rapidly decaying covariance, on a lattice. The first methods of this kind, introduced during the early seventies in the groundbreaking work of Glimm-Jaffe-Spencer, were very involved and partly accounted for the reputation of inaccessibility of constructive field theory among mathematicians. Over the years many improvements were introduced, and the state-of-the-art for these cluster expansions has reached a much simpler formulation thanks to the Brydges-Kennedy forest sum formula and its variations. The second half of the talk will focus on the combinatorics of this identity. We will sketch several methods of proof for this algebraic identity, and doing so, we will point out its connection with Moebius inversion for partition lattices or subarrangements of the A_n hyperplane arrangement.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons