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On a generalization of the Switching Lemma for the Transverse Ising Model

Mathematical 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, Fernandez). 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.