Let $M_n$ denote a random symmetric $(n\times n)$-matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take value $-1$ and $1$ with probability $1/2$). Improving the earlier result by Costello, Tao, and Vu [4], we show that $M_n$ is nonsingular with probability $1-O\left(n^{-C}\right)$ for any positive constant $C$. The proof uses an inverse Littlewood–Offord result for quadratic forms, which is of interest of its own.

We prove spectral and dynamical localization for the multidimensional random displacement model near the bottom of its spectrum by showing that the approach through multiscale analysis is applicable. In particular, we show that a previously known Lifshitz tail bound can be extended to our setting and prove a new Wegner estimate. A key tool is given by a quantitative form of a property of a related single-site Neumann problem which can be described as “bubbles tend to the corners.”

For $4\nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces, and we calculate the second integral cohomology group of the level $L$ subgroup of the mapping class group. (In a previous paper, the author determined this rationally.) This entails calculating the abelianization of the level $L$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of ${Sp}_{2g}(\mathbb{Z}/L)$ with coefficients in the adjoint representation.

In this paper we prove the fundamental lemma for Deligne–Lusztig functions. Namely, for every Deligne–Lusztig function $\varphi$ on a $p$-adic group $G$ we write down an explicit linear combination ${\varphi }^H$ of Deligne–Lusztig functions on an endoscopic group $H$ such that $\varphi$ and ${\varphi }^H$ have “matching orbital integrals.” In particular, we prove a conjecture of Kottwitz. More precisely, we do it under some mild restriction on $p$.