Stochastic domination in beta ensembles

We give a stochastic comparison and ordering of the largest eigenvalues, with parameter $\beta$, for Hermite $\beta$-ensembles and Laguerre $\beta$-ensembles. Although stochastic comparison results are well known in Laguerre ensembles (for $\beta=1,2,4$) using the last passage percolation models, our results are novel even for $\beta=1,2,4$, in Hermite ensembles. Taking limit, we recover a stochastic domination result for Tracy-Widom distributions obtained in (Pedreira, 2022). Using this, we also obtain a result on the signs of means of Tracy-Widom distributions. Our methods also provide stochastic domination results for spiked beta ensembles as well. We compare ordering of all the eigenvalues collectively, with $\beta$ as a parameter, by proving ordering of the moments of Hermite and Laguerre $\beta$-ensembles. In order to generalize the stochastic domination results of (Pedreira, 2022) to higher order analogues of Tracy-Widom distributions, we study tail estimates of these distributions. We show that the description of these distributions as eigenvalues of a stochastic operator is inconsistent with the known tail estimates. As a result, we disprove a conjecture of Krishnapur, Rider and Vir\'ag (Comm. Pure Appl. Math., 2017) for $\beta=2$ and $k=1$.