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On a generalization of the Switching Lemma for the Transverse Ising ModelMathematical Physics & Probability
|Speaker: ||Nick Crawford, UC Berkeley|
|Location: ||2112 MSB|
|Start time: ||Wed, Oct 29 2008, 4:10PM|
In the 1980's M. Aizenman and coauthors proved a number of long
standing conjectures for the Ising model on $Z^d$: 1) A Gaussian
Scaling Limit for Ising (and Ising-type) models on $Z^d$, $d>4$ at $0
$ external field (due to Aizenman), 2) A study of ferromagnetism for
long range Ising models (Aizenman and R. Fernandez), 3) The
Sharpness of the Magnetic Transition on Z^d (Aizenman, D. Barsky,
All of these results rely on a graphical representation for the Ising
model called the Random Current Representation, along with a
combinatorial technique called the "Switching Lemma" (both introduced
by Aizenman in the proving the first result above).
In this talk, I will explain generalizations of these two techniques
to the Transverse Ising Model, a simple quantum mechanical
generalization of the Ising model. As a first application, we prove
that if the external field is nonzero for the quantum model, then
truncated correlations decay exponentially.
Based on joint work with D. Ioffe.