Graduation Year
2023
Document Type
Dissertation
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
Razvan Teodorescu, Ph.D.
Co-Major Professor
Seung-Yeop Lee, Ph.D.
Committee Member
Dmitry Khavinson, Ph.D.
Committee Member
Evguenii Rakhmanov, Ph.D.
Committee Member
Sherwin Kouchekian, Ph.D.
Committee Member
David Rabson, Ph.D.
Keywords
Random Matrix Theory, Orthogonal Polynomials, Statistical Mechanics, Riemann-Hilbert, 2D Quantum Gravity, The Ising Model
Abstract
The 2D Ising model has played an important role in the theory of phase transitions, as one of only ahandful of exactly solvable models in statistical mechanics. The original model, introduced in the 1920s, has a rich mathematical structure. It thus came as a pleasant surprise when physicists studying matrix models of 2D gravity found that, coupled to quantum gravity, the planar Ising model still had an elegant solution. The methods used by V. Kazakov and his collaborators involved the method of orthogonal polynomials. However, these methods were formal, and no direct analytic derivation of the phase transition has been described in the literature since the original paper of V. Kazakov in 1986. In this work, we present a rigorous proof of Kazakov’s results, using steepest descent analysis for biorthogonal polynomials. We are able to calculate the genus 0 partition function, and we also find that the phase transition is described by the string equation of a 3rd order reduction of the KP hierarchy, in agreement with the predictions of G. Moore, M. Douglas, and their collaborators. This is part of a forthcoming paper with Maurice Duits and Seung-Yeop Lee.
Scholar Commons Citation
Hayford, Nathan, "Matrix Models of 2D Critical Phenomena" (2023). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/10438
