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Mean-field Bound on the 1-arm Exponent for Ising Ferromagnets in High Dimensions

By Akira Sakai, Hokkaido University

The Ising model is a statistical-mechanical model for magnets.  It is now known that, if the spin-spin coupling is non-negative and reflection-positive, then it exhibits a continuous phase transition.  In particular, the critical 1-spin expectation at the center of a ball of
radius $r$ vanishes as $r$ goes to infinity.  It is believed to decay in powers of $r$, with an exponent $\rho$ called the 1-arm exponent.  Presumably this exponent takes on the mean-field value 1 in high dimensions, but the best possible bound so far is $(d-2)/2$, due to a
hyperscaling inequality.

I will show how we achieve the mean-field bound on the Ising 1-arm exponent, i.e., $\rho\le1$.  

This talk is based on my ongoing project with Satoshi Handa and Markus Heydenreich.