Singularities in quantum mechanical integrals

Question

In physics we often look at functions of a complex variable defined by $$\int e^{- \frac{S(x)}{z}}d^nx$$ where the integral is over $\mathbb{R}^n$ and the integral converges for real positive z . There are examples where at least as n goes to infinity (and we take the ratio with a Gaussian integral to get a finite limit) the origin is an accumulation point of singularities. Is there a general name for this type of singularity and/or a classification?