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
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.