Spectrum of SYK model III: Large deviations and concentration of measures

In \cite{FTD1}, we proved the almost sure convergence of eigenvalues of the SYK model, which can be viewed as a type of \emph{law of large numbers} in probability theory; in \cite{FTD2}, we proved that the linear statistic of eigenvalues satisfies the \emph{central limit theorem}. In this article, we continue to study another important theorem in probability theory\,-- the \emph{concentration of measure theorem}, especially for the Gaussian SYK model. We will prove a \emph{large deviation principle} (LDP) for the normalized empirical measure of eigenvalues when $q_n=2$, in which case the eigenvalues can be expressed in term of these of Gaussian random antisymmetric matrices. Such LDP result has its own independent interest in random matrix theory. For general $q_n\geq 3$, we can not prove the LDP, we will prove a concentration of measure theorem by estimating the Lipschitz norm of the Gaussian SYK model.