of the Ising model and other classical spin systems. Some inequalities are “geometric”
in the sense that they involve the sites of the spins. One important inequality is the
Simon-Lieb-Aizenman-Rivasseau inequality that provides an upper bound for
two-spin correlations in terms of a “separating set”. Another inequality is due to Messager
and Miracle-Solé and it shows that correlations decrease when the sites are more distant.
The new result is that these inequalities also hold in the (spin 1/2) quantum XY model.

The proofs make use of a representation by Gallavotti as an Ising model with
additional plaquette interactions, and also new probabilistic representations 
involving “temporal directed random currents” and dimers.

The talk is based on two ongoing collaborations: with Saksida and Sohinger, and with
Heeney, Lis, and Ryan.