High-dimensional Laplace asymptotics up to the concentration threshold

Abstract

We study high-dimensional Laplace-type integrals $I(\lambda):=(\lambda/2\pi)^{d/2}\int_{\mathbb R^d} g(x)e^{-\lambda f(x)}dx$ in the regime where both $d$ and $\lambda$ are large. Existing rigorous Laplace-expansion results in growing dimension are largely confined to the "Gaussian-approximation" regime $d^2/\lambda\to0$, which excludes many practically relevant settings that lie beyond this threshold but still satisfy the concentration condition $d/\lambda\to0$. We close this gap by deriving an explicit asymptotic expansion for $\log I(\lambda)$ with quantitative remainder bounds that remain valid throughout this intermediate region, arbitrarily close to the concentration threshold. Fix $L\ge1$ and assume that, in a neighborhood of the global minimizer of $f$, the operator norms of derivatives of $f$ and $g$ are bounded independently of $d,\lambda$ up to orders $2L+2$ and $2L$, respectively. Assuming also some mild global growth conditions, we prove $$\log I(\lambda)=\sum_{k=1}^{L-1} b_k(f,g)\lambda^{-k}+O(d^{L+1}/\lambda^L), \qquad d^{L+1}/\lambda^L\to0,$$ with coefficients satisfying $b_k(f,g)=O(d^{k+1})$. Moreover, the $b_k(f,g)$ coincide with the coefficients from the formal cumulant expansion of $\log I(\lambda)$. We also study computation for concentrating densities $\pi(x)\propto e^{-\lambda f(x)}$. For smooth observables $g$, our expansion yields closed-form, analytic approximations of $\mathbb E_{X\sim\pi}[g(X)]$. For sampling, we construct explicit polynomial transports $x_L$ such that $\pi_L:=(x_L)_\# N(0,\lambda^{-1}I_d)$ satisfies $\mathrm{TV}(\pi,\pi_L)\lesssim d^{L+1}/\lambda^L$ for $L=1,2,3,\dots$, yielding an accurate procedure arbitrarily close to the concentration threshold $d=o(\lambda)$.