#### MS Thesis defence

##### Venue: LH-1, Mathematics Department

The systematic study of determinantal processes began with the work of Macchi (1975), and since then it has appeared in different contexts like random matrix theory (eigenvalues of random matrices), combinatorics (random spanning tree, non-intersecting paths), and physics (fermions, repulsion arising in quantum physics). The defining property of a determinantal process is that its joint intensities are given by determinants, which makes it amenable to explicit computations. One can associate a determinantal process with a finite rank projection on a separable Hilbert space. Let $H$ and K be two finite-dimensional subspaces of a Hilbert space, and $P$ and $Q$ be determinantal processes associated with projections on $H$ and $K$, respectively. Lyons (2003) showed that if $H$ is contained in $K$ then $P$ is stochastically dominated by $Q$. We will give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and provides a unified approach of proving the result in discrete as well as continuous case.

As an application of the above result, we will obtain the stochastic domination between the largest eigenvalues of Wishart matrix ensembles $W(N, N)$ and $W(N-1, N+1)$. It is well known that the largest eigenvalue of Wishart ensemble $W(M, N)$ has the same distribution as the directed last-passage time $G(M, N)$ on $\mathbb{Z}^2$ with the i.i.d. exponential weight. This was recently used by Basu and Ganguly to obtain stochastic domination between $G(N, N)$ and $G(N-1, N+1)$. Similar connections are also known between the largest eigenvalue of the Meixner ensemble and the directed last-passage time on $\mathbb{Z}^2$ with the i.i.d. geometric weight. We prove another stochastic domination result, which combined with Lyons’ result, gives the stochastic domination between the eigenvalue processes of Meixner ensembles $M(N, N)$ and $M(N-1, N+1)$.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 06 Mar 2020