Fluctuation Bounds for Two Disordered Models

Abstract

We present bounds on the variance of two observables (functions) in disordered models incorporating a large number of independent random variables: linear statistics of eigenvalues of random matrices and the passage time from the origin to a distant vertex in first-passage on the square lattice $\mathbb{Z}^d$, $d>1$. These two models, and the techniques we use to analyze them, are quite different. However, in both cases the nonlinearity of the functions we consider leads to atypical behavior of the fluctuations when compared to simpler models encountered in classical probability, such as i.i.d. sums.

We first discuss the fluctuations of linear statistics of the eigenvalues of large random matrices. Our emphasis is on the connection between the magnitude of these fluctuations and the regularity of the functions used to form the linear statistics. We develop variance bounds and central limit theorems for linear statistics of low-regularity functions of Wigner matrices and invariant ensembles, which improve earlier results even in the case of the classical Gaussian Unitary Ensemble, for which we obtain an optimal result.

In the second part of the thesis, we discuss an extension of the results of Benjamini-Kalai-Schramm and Benaim-Rossignol on sublinear variance for the passage time in first passage percolation. These authors had derived their results under very special hypotheses on the edge-weight distributions. We show how these assumptions can be removed entirely.