I have encountered an integral of the form:
$$ I = \int_{\mathbb{R}^N} \mathrm{d}^N x \, e^{- N f_N(x)} \,. $$
I am interested in both the case where $N$ is large but finite and the case where $N \rightarrow \infty$. The function $f_N(x)$ depends on $N$ in the sense that it depends on $x$, and $N$ is the dimension of $x$ (i.e. $x \in \mathbb{R}^N)$.
I am wondering if the method of steepest descent/Laplace's method may be applied to this integral, generally speaking (i.e. without further characterizing the function $f_N(x)$). In the standard treatment of this method, the dimension of the integral is held fixed as a parameter in the exponent is taken to be large. The difference is that here the dimension of the integral also becomes large in this limit.
