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October 10 - 2003

14:15-16:00: Takashi Hara
        Triviality of the hierarchical Ising model in four dimensions.
       
 

In a recent series of papers with Borgs, Chayes, van der Hofstad, and Spencer, dealing with percolation on finite graphs, a new approach to proving convergence of the percolation lace expansion has been developed. In particular, it is proved that a version of the triangle condition for random walk implies the triangle condition for percolation, and hence mean-field critical behaviour. An outline of this new approach will be presented in the talk.

Takashi Hara
         Triviality of the hierarchical Ising model in four dimensions.

One way to constuct quantum field theories is to consider continuum (scaling) limits of lattice spin systems of statistical mechanics.A quantum field theory is called "trivial" if it is equivalent to a free (= generalized gaussian) field theory. It has been long conjectured that continuum (scaling) limits of phi^4 and Ising models are trivial in four dimensions, but no mathematical proof has been given. (In dimensions higher than four, it has been proven that the continuum limits are trivial.) The method of renormaliztion group is expected to be useful for proving the triviality (and/or studying continuum limits in general). In this talk, I first explain the problem and the renormalization group philosophy. Then for the very special case of hierarchical Ising model, I explain how the renormalization group method can be used to prove the triviality (with the aid of computers). This talk is partly based on my joint work with Tetsuya Hattori and Hiroshi Watanabe. (Large part of the talk will be almost identical to my talks at Manbucaba (2001) and Oberwolfach (2002).)