Kazhdan-Lusztig polynomials and drift configurations

The coefficients of the Kazhdan-Lusztig polynomials $P_{v,w}(q)$ are nonnegative integers that are upper semicontinuous on Bruhat order. Conjecturally, the same properties hold for $h$-polynomials $H_{v,w}(q)$ of local rings of Schubert varieties. This suggests a parallel between the two families of polynomials. We prove our conjectures for Grassmannians, and more generally, covexillary Schubert varieties in complete flag varieties, by deriving a combinatorial formula for $H_{v,w}(q)$. We introduce \emph{drift configurations} to formulate a new and compatible combinatorial rule for $P_{v,w}(q)$. From our rules we deduce, for these cases, the coefficient-wise inequality $P_{v,w}(q)\preceq H_{v,w}(q)$.Comment: 26 pages. To appear in Algebra & Number Theor

Central Limit Theorem for Linear Statistics of Eigenvalues of Band Random Matrices

We prove the Central Limit Theorem for linear statistics of the eigenvalues of band random matrices provided $\sqrt{n} \ll b_n \ll n$ and test functions are sufficiently smooth.Comment: minor revision; to appear in Random Matrices: Theory and Application