This thesis studies the asymptotic behavior of two canonical ensembles: the fluctuations of the linear statistics of $\beta$-ensemble and the normalizability of the Gibbs measure associated with nonlinear Schrödinger equation (NLS). Both models exhibit a formal distribution of the form: $e^{-\beta \mathcal{H(x)}}dx$. The underlying heuristic idea for both models involves expanding the Hamiltonian around its minimizer $x_0$: $\begin{align*} \mathcal{H}(x) \approx \mathcal{H}(x_0)+ \langle{x-x_0, \nabla^2 \mathcal{H}(x_0) (x-x_0)}\rangle, \end{align*}$ which leads to an approximation of the models by $e^{\langle{x-x_0, \nabla^2 \mathcal{H}(x_0) (x-x_0)\rangle}}dx$. However, the rigorous executions of these ideas differ and are more intricate. In Chapter 1, the fluctuation of the linear statistics for $\beta$-ensembles is illustrated through a novel covariance formula and central limit theorem (CLT). The result is realized by a random walk representation and a homogenization argument assisted with a derivative heat kernel estimate. In Chapter 2, we prove the normalizability of Gibbs measure associated with radial focusing nonlinear Schrödinger equation (NLS) on the 2-dimensional disc $\mathbb D$, at critical mass threshold. The result is proved by meticulously expanding the Hamiltonian at the solution manifold and performing a spectrum analysis for the operator related to the quadratic form of $\nabla^2\mathcal{H}$.