Spectral statistics of non-Hermitian random matrix ensembles

Recently Burkhardt et al. introduced the [Formula: see text]-checkerboard random matrix ensembles, which have a split limiting behavior of the eigenvalues (in the limit all but [Formula: see text] of the eigenvalues are on the order of [Formula: see text] and converge to semi-circular behavior, with the remaining [Formula: see text] of size [Formula: see text] and converging to hollow Gaussian ensembles). We generalize their work to consider non-Hermitian ensembles with complex eigenvalues; instead of a blip new behavior is seen, ranging from multiple satellites to annular rings. These results are based on moment method techniques adapted to the complex plane as well as analysis of singular values.

Ensemble-Based Exigent Analysis. Part I: Estimating Worst-Case Weather-Related Forecast Damage Scenarios

Exigent analysis supplements an ensemble forecast of weather-related damage with a map of the worst-case scenario (WCS), a multivariate confidence bound of the damage. For multivariate Gaussian ensembles, ensemble-based exigent analysis uses a Lagrange multiplier technique to identify the unique maximizing damage map at a given uncertainty level based on the ensemble-estimated covariance of the damage. Exigent analysis is applied to two case studies. First, using ensemble forecasts of 2-m temperature and estimates of the number of inhabitants at each location, exigent analysis is applied to forecast the worst-case heating demand for a large portion of the United States on 8–9 January 2010. The WCS at the 90th percentile results in only 1.26% more heating demand than the ensemble mean. Second, using ensemble forecasts of 2-m temperature and estimates of the number of citrus trees at each location, exigent analysis is applied to forecast the worst-case freeze damage to Florida citrus trees on 11 January 2010. For this case study, the WCS at the 90th percentile damages about 14.2 million trees, about 4.3 times more than the ensemble mean.

A Note on Functional Averages over Gaussian Ensembles

We find a new formula for matrix averages over the Gaussian ensemble. LetHbe ann×nGaussian random matrix with complex, independent, and identically distributed entries of zero mean and unit variance. Given ann×npositive definite matrixAand a continuous functionf:ℝ+→ℝsuch that∫0∞‍e-αt|f(t)|2dt<∞for everyα>0, we find a new formula for the expectation𝔼[Tr(f(HAH*))]. Takingf(x)=log(1+x)gives another formula for the capacity of the MIMO communication channel, and takingf(x)=(1+x)-1gives the MMSE achieved by a linear receiver.