|Date: Thursday, March 10, 2022
Location: (Hybrid) 2058 East Hall (11:30 AM to 1:30 PM)
Title: Dissertation Defense: Free Energy and Overlaps of a Spherical Spin Glass with an External Field
Abstract: The defense will be a hybrid format.
2058 East Hall
Meeting ID: 913 5138 1080
Since they were first developed in the 1970s, spin glass models have captured the attention of mathematicians, physicists, computer scientists, statisticians, biologists, economists, and others because of their intriguing probabilistic properties. This thesis analyzes the 2-spin spherical Sherrington Kirkpatrick (SSK) spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field.
My talk will begin with a brief introduction spin glasses in general and the SSK model in particular. I will describe the distribution of spins (i.e. the geometry of the Gibbs measure) and how the spin distribution changes in the presence of an external field. I will then discuss the transitional case in which the strength of the external field goes to zero as the dimension of the spin variable grows. I will present results for overlaps with the external field, with the ground state and with a replica, focusing on what each of these overlaps tells us about the distribution of spins. I will also discuss an application of these results to magnetic susceptibility. Finally, I will provide a brief overview of our methods, including a contour integral representation of the partition function as well as random matrix techniques.
The analysis throughout this thesis focuses on the SSK model with external field strength h ∼ N −α for 0 < α < 1. This analysis reveals that the free energy exhibits a transition at α = 1/6 in the low temperature case but at α = 1/4 at the high temperature case. Furthermore, the overlaps do not exhibit any transition in the high temperature case, but exhibit two transitions in the low temperature case, at α = 1/6 and α = 1/2. These scalings are referred to as the mesoscopic and microscopic external field respectively. In the final chapter of the thesis, I present a more detailed analysis of the overlaps in the microscopic setting and their application to magnetic susceptibility.
Lizbee's advisor is Jinho Baik.
Speaker: Elizabeth Collins-Woofin