1. Random Matrices and Statistical Mechanics by Kevin Phillips Advisor: Dr. Michael K.-H. Kiessling

2. Statistical Mechanics: Add to the understanding of the interplay of Random Matrix Theory and Statistical Mechanics. Goals:

3. Understanding the interplay of Random Matrix Theory and Statistical Mechanics. Goals: Random Matrix Theory: For specific U int, U ext and , these are identical! Statistical Mechanics:

4. Classical Gas of Charges

6. The Gibbs Canonical Ensemble Consider an ensemble of systems with the same Hamiltonian each in different states (Q,P) but distributed according to a probability measure which depends only on the Hamiltonian: exponential- Gibbs Measure.

7. The Configuration Gas Integrate out momentum, ask questions only about position! Where:

8. Hamiltonian of Point Charges in 2D For specific U int, U ext and , these are identical! Gibbs Canonical MeasureJoint Eigenvalue Distribution Particle-Particle interaction arising from electric fields. Interaction with positive background.

9. Gibbs Canonical Ensemble : Point Charges in 2D Ok, we have U int and U ext, but what  should we try?

10. Overlap: Random Matrices and 2D Ensembles Matrix TypeWeightPhysical Interpretation? Inverse Temperature Comments Arbitrary Complex Matrices GaussianYes  = 2 Non- Gaussian Yes  = 2 Normal, S-D Quaternion Matrices GaussianYes  = 4 “A ring of truth, but no proof.” Non- Gaussian ??

11. My Project Normal, S-D Quaternion Matrices GaussianYes  = 4 “A ring of truth, but no proof.” 1.) From the previous chart… 2.) Normal Matrices : Real and Complex, Gaussian and non-Gaussian. I Would like to set this in stone.