High-dimensional Laplace asymptotics up to the concentration threshold
Alexander Katsevich, Anya Katsevich
Abstract
We study high-dimensional Laplace-type integrals of the form $I(λ):=\int_{\mathbb R^d} g(x)e^{-λf(x)}dx$ in the regime where $d$ and $λ$ are both large. Until now, rigorous bounds for Laplace expansions in growing dimension have been restricted to the "Gaussian-approximation" regime, known to hold when $d^2/λ\to0$. This excludes many practically relevant regimes, including those arising in physics and modern high-dimensional statistics, which operate beyond this threshold while still satisfying the concentration condition $d/λ\to0$. Here, we close this gap. We develop an explicit asymptotic expansion for $\log I(λ)$ with quantitative remainder bounds that remain valid throughout this intermediate region, arbitrarily close to the concentration threshold $d/λ\to0$. Fix any $L\ge1$ and suppose $g(0)=1$. Assume that, in a neighborhood of the minimizer of $f$, the operator norms of the derivatives of $f$ and $g$ are bounded independently of $d$ and $λ$ through orders $2(L+1)$ and $2L$, respectively. Assuming also some mild global growth conditions on $f$ and $g$, we prove that $$ \log I(λ)=\sum_{k=1}^{L-1} b_k(f,g)λ^{-k}+O(d^{L+1}/λ^L),\quad d^{L+1}/λ^L\to0, $$ and that the coefficients satisfy $b_k(f,g)= O(d^{k+1})$. Moreover, the coefficients $b_k(f,g)$ coincide with those arising from the formal cumulant-based expansion of $\log I(λ)$. The proof is constructive and proceeds via explicit polynomial changes of variables that iteratively "quadratize" the exponent while controlling Jacobian effects, thereby avoiding heavy Gaussian concentration machinery. We illustrate the expansion on two representative examples.