We will do an overview of the basic objects of interest in random matrix theory, and some of the main phenomena and results we will try to prove. These include Wigner matrices and the semicircle law, the Gaussian ensembles, the Airy and sine processes, and determinantal point processes.

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We will begin the study of the semicircle law, for real Wigner matrices. For a normalized real Wigner matrix X, we will reduce the proof to the computation of expected traces of powers of X, and at the end remark briefly on ways to reduce assumptions.

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We will recall the statement of Wigner's semicircle law and what was proven last week. We will present two lemmas, how they give us Wigner's semicircle law, and prove the first lemma which involves interesting combinatorial arguments.

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In this talk, we will introduce the notion of Gaussian orthogonal/unitary ensembles. This class of random matrices enjoys many symmetry properties, which greatly simplifies the study of its law. The distribution of its spectral decomposition (eigenvalues and eigenvectors) will be discussed as well as the proof of its derivation.

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In this talk we will take a look at the joint eigenvalue distribution of some Jacobi matrices. Of particular interest to us will be the beta Hermite ensemble and its connection to the previously discussed Gaussian ensemble.

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Today we will start by introducing point processes and their description via joint intensities. We will then define determinantal point processes and show that the eigenvalues of the GUE fall into this category.

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We'll start with an example of a determinantal process motivated by quantum mechanics. Then we'll use this example together with spectral theory of self-adjoint operators to construct a determinantal process with an arbitrary kernel whose eigenvalues belong to [0,1]. Finally, we will see that only such kernels can correspond to determinantal processes. If time permits, we may also discuss some examples and connections to other areas of mathematics, statistics, and maybe even physics.

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The eigenvalues of the GUE follow a determinantal point process. We will see how the asymptotics of its kernel govern the fine-scale structure of the eigenvalues. In particular, I’ll show that the number of eigenvalues in an interval of width \(O(1/\sqrt{n})\) also follows a DPP.

We will introduce Laplace's method and the method of steepest descent as ways to analyze the rate of growth of certain exponential integrals. Then we will use these tools to determine asymptotics of the Hermite wavefunctions in various regimes, and we will use these results to prove that the rescaled GUE eigenvalues DPP kernels converge in the bulk to the sine kernel.

In addition to the sine kernel, the Hermite kernel has a second kernel "hiding inside it", called the Airy kernel. We will scale the Hermite kernel appropriately and show the convergence to the Airy kernel, following a heuristic argument of Tao's (similar to Jake's talk on Mar. 17). This hopefully gentle presentation will be accompanied by a segment on intuition for scaling limits in general, and a short primer on the Airy function.

We will discuss the rigorous calculation deriving the Airy kernel as an edge scaling limit of the Hermite kernel, using some complex analysis and seeing the method of steepest descent in action.

We previously studied the scaling limit of the eigenvalues of the GUE near the edge, and in this next talk we address the question of whether the random operators themselves have a scaling limit in a suitable sense. In fact, we study this question for all beta-Hermite ensembles simultaneously. The existence of these so-called "stochastic Airy operators" was a conjecture of Edelman-Sutton until it was established rigorously by Ramirez-Rider-Virag; in this talk we'll go over the heuristic argument that led to the original conjecture, and we'll give some ideas of the important points of rigorous proof.

In this talk we will study the tail behavior of the distribution of Tracy-Widom(\(\beta\)) distribution, which arises when considering the asymptotics of the larges eigenvalue of the stochastic Airy operator. In order to understand the tail behavior entirely, we will introduce an analogous stochastic differential equation perspective of the problem.

When we replace the normal random variables in the Gaussian unitary ensemble by Brownian motions and look at its eigenvalues, we obtain Dyson Brownian motion. After defining this process, we will introduce its stochastic differential equations and provide a heuristic justification. Additionally we will provide another description of this process in terms of conditional Brownian motions. Finally we will see how the SDEs can be used to give an alternative proof of the density of the eigenvalues of the Gaussian unitary ensemble.

We will do an overview of the results about universality in random matrix theory proved in the last 10-15 years. We will focus on the broad approach of Erdős-Yau et al, which makes use of the quick relaxation of Dyson Brownian motion to equilibirum on local scales. Along the way we will see tools and ideas like local semicircle laws and rigidity estimates for eigenvalues.