Asymptotic expansion of a Laplace-type integral with a "manifold of maxima"

Question

Moved this here from MathSE because I had no luck there and suspect the question may be harder than I first thought.

Consider the integral $$ I(\alpha)=\int_0^1 dx_1 \int_0^1 dy_1\int_{x_1}^1dx_2\int_{y_1}^1dy_2\,e^{-\alpha(x_2-x_1)(y_2-y_1)} $$ in the limit $\alpha\rightarrow\infty$. To find the asymptotic expansion in this case is not hard: you can evaluate the integral in closed form in terms of Exponential Integral functions and then look up asymptotic expansions, which give $$ I(\alpha)\sim \frac{\log\alpha}{\alpha}. $$ However, I am interested in more complicated versions these kinds of integral where the integrand might be more complicated and therefore I will need to "expand before integrating" (since I cannot evaluate the integral in closed form).

This looks a lot like a problem for Laplace's method, which finds the maximum in the exponential and expands the integrand near it, leading to a Gaussian integral. But my integrands all have the same property: the exponent is maximal not at an isolated point, but on a 1-dimensional subspace of points. Is there a theory of such integrals? I haven't found much. Any texts that mention things like "Laplace's method with a manifold of maxima" that I came across are very technical.

So my questions:

1. Is there a straight-forward derivation of the the asymptotic expansion for the simple integral above using a Laplace-like method, i.e. without evaluating the integral in closed form to start with?

2. Can anyone recommend not-too-technical texts that might help me?

Answer 1

You can perform a linear change of variables so that one of coordinates corresponds to the maximum of the integrand, and then integrate over this direction first ( you will be integrating a constant function, so this is easy!) After this, you will be in the setting where Laplace method will apply.