High-dimensional Laplace asymptotics up to the concentration threshold

TL;DR

This paper develops explicit asymptotic expansions for high-dimensional Laplace integrals near the concentration threshold, enabling accurate approximations and sampling methods in regimes beyond traditional Gaussian approximations.

Contribution

It extends Laplace asymptotics to the intermediate regime where dimension and inverse temperature are large but not in the Gaussian-approximation limit, providing explicit formulas and bounds.

Findings

Derived asymptotic expansion for log I(λ) with quantitative remainder bounds

Established polynomial transport methods for accurate sampling near the concentration threshold

Provided closed-form approximations for expectations under high-dimensional Gibbs measures

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 $d=o(\lambda)$.