Houston, TX 77005
12:10 p.m. Thursday, Sept. 26, 2013
On Campus | Alumni
The classical (i.e. non-quantum) Ising model in one dimension dates back to the 30's as the earliest and the simplest nontrivial model of magnetism in statistical mechanics. The model treats magnets as being composed of microscopic dipoles whose orientation determines macroscopic magnetic properties of the magnet (in the simplest terms, macroscopic magnetism depends on the statistical alignment of the microscopic magnets). The alignment of the microscopic dipoles (viewed as nodes on a lattice) is typically influenced by (1) temperature, (2) external magnetic field and (3) interaction between neighboring dipoles. The most important quantity of the model is the so-called partition function, as a function of temperature and the external magnetic field. One of the ways to study the model is to study the zeros of the partition function when the magnetic field is complexified. In this case it is known that the zeros always lie on the unit circle. One then asks about the distribution of the zeros as the lattice size (the domain of definition of the model) is taken to infinity (the so-called thermodynamic limit). We show that with appropriate conditions, the problem is equivalent to studying a problem of OPUC with specifically chosen Verblunsky coefficients, where the zeros of the partition function in the thermodynamic limit coincide with the spectrum of the associated CMV matrix (under certain conditions).