Determinantal Processes and Stochastic Domination

In this thesis we explore the stochastic domination in determinantal pro- cesses. Lyons (2003) showed that if K1 ≤ K2 are two finite rank projection kernels and P1, P2 are determinantal measures associated with them, then P2 stochastically dominates P1, written P1 ≺ P2, that is for every increas- ing event A we have P1(A) ≤ P2(A). We give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and also 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 domi- nation between the largest eigenvalue of Wishart matrix ensemble 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 Z2 with i.i.d. exponential weights. We, thus, obtain stochastic domination between G(n, n) and G(n − 1, n + 1)— answering a question of R. Basu and S. Ganguly.

We also prove another stochastic domination result which combined with the Lyons’ result gives the stochastic domination between the largest eigenvalues of Meixner ensemble M (n, n) and M (n − 1, n + 1). It is also known that the largest eigenvalue of the Meixner ensemble M(m,n) has the same distribution as the directed last passage time G(m, n) on Z2 with i.i.d. geometric weights, which in turn proves that the directed last pas- sage time (with i.i.d. geometric weights) G(n, n) stochastically dominates G(n−1,n+1).