The purpose of this thesis is to derive three results in probability theory. The first is a proof that small powers of the normalized absolute characteristic polynomial of a random matrix sampled from either the Gaussian Orthogonal or Symplectic Ensemble converges in law to a Gaussian multiplicative chaos measure. The method to demonstrate this is the establishment of a new asymptotic relation between the fractional moments of the absolute characteristic polynomials of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles. The second result is to demonstrate a transitional regime for the fluctuations of the ground-state energy in the spherical Sherrington-Kirkpatrick model as one applies a weak magnetic field. To achieve this, we achieve stronger rates for the convergence of certain tridiagonal matrices to their operator limits. The final result is a derivation of a formula for the ground-state energy of the pure spherical bipartite spin glass model (for $p,q$ large), as well a result showing concentration for the complexity of low-lying critical points this model in this model, on an exponential scale.

Creator

• Kivimae, Pax

DOI

• https://doi.org/10.21985/n2-3tr2-ca20

Subject

• Mathematics

Language

• en

Alternate Identifier

• etdadmin_upload_923657
• http://dissertations.umi.com/northwestern:16191

Keyword

• Probability
• Random Matrices

Date created

• 2022-01-01

Resource type

• Dissertation

Rights statement

• In Copyright