I am trying to compute the asymptotics on $t$ of the following integral: \begin{equation} I(t)=\int_{\mathbb{R}^{n}}e^{-|\lambda|^{2}/2t}\prod_{i<j}\left( e^{\lambda_{j}/t}-e^{\lambda_{i}/t} \right)d\lambda. \end{equation} According to the paper I am reading, it suffices to consider the leading nonzero term in the product $\prod_{i<j}\left( e^{\lambda_{j}/t}-e^{\lambda_{i}/t} \right)$. Given that this claim is true, this leads to an easier computation of the integral. However, I do not understand why the claim is correct. According to the paper, once $t$ is fixed, if $|\lambda|^{2}$ is much larger than $t$, the exponential term in the integral will decay exponentially fast, and in particular
"$\lambda_{i}/t$ can be assumed to be $O(t^{1/2-\epsilon})$".
I don't see why this last statement is true, nor how it implies the claim. I have tried splitting the integral in two, one with the nonzero leading term of the integrand and other with the rest, but I have not succeeded in proving that this second integral does not contribute to the asymptotics. I also have doubts regarding the quoted statement; if we are considering asymptotics on $t$, is it "legal" to study the behaviour of $\lambda_{i}/t$? Thank you for your help.