Random Matrix Ensembles with no Quadratic Term

Question

A large class of interesting physical problems (for example, 2D quantum gravity, matrix glass models etc.) are related to random matrix integrals with potential energy functions that have a leading Gaussian term followed by non-Gaussian interactions: \begin{equation} Z = \int dM e^{-V(M)} \quad V(M) = \text{Tr}M^2 + \sum_i c_i \text{Tr} M^i + \text{Tr} (M Q) \end{equation} In the formula above, $M$ is an N by N Hermitian matrix, and $Q$ is some fixed matrix coupled to $M$. When the coefficients $c_i$ have appropriate scalings wrt N, we can do Feynman diagram type perturbation theory, by Taylor expanding the exponential of the nonquadratic terms. However, sometimes we run into trickier cases where no quadratic term is present. For example, consider the following integral: \begin{equation} Z = \int dM e^{-\frac{1}{N^{n-1}} \text{Tr} M^n + i \text{Tr} M Q} \end{equation} I tried searching online for techniques to deal with integrals of this form, but didn't have any luck. If such techniques exist, could someone point me to the appropriate references? Even if it is not exactly solvable, any approximate result would be appreciated too.