Tridiagonal random matrix: Gaussian fluctuations and deviations

This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrix. Under quite general assumptions, we prove that the traces are approximately normal distributed. Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth-death Markov kernel, the random birth-death $Q$ matrix and the $\beta$-Hermite ensemble. Furthermore, under some independent and identically distributed condition, we also prove the large deviation principle as well as the moderate deviation principle for the traces.