Saddle-Point (Laplace) Method in One Dimension

Refined asymptotics for integrals of exp(n f(x)) near a nondegenerate maximizer, giving Gaussian prefactors beyond the Laplace principle.

Saddle-Point (Laplace) Method in One Dimension

This lemma refines the Laplace principle by identifying the leading prefactor when the maximum is attained at a nondegenerate interior point.

Statement (one-dimensional nondegenerate maximum)

Let $f:[a,b]\to\mathbb{R}$ be twice continuously differentiable and suppose:

$f$ has a unique global maximum at an interior point $x_0\in(a,b)$ ,
$f''(x_0)<0$ (nondegenerate maximum),
$g:[a,b]\to\mathbb{R}$ is continuous with $g(x_0)\neq 0$ .

Define

I_n = \int_a^b e^{n f(x)}\, g(x)\,dx.

Then, as $n\to\infty$ ,

I_n ={} e^{n f(x_0)}\, g(x_0)\, \sqrt{\frac{2\pi}{n\,|f''(x_0)|}}\, \big(1+o(1)\big).

Under higher smoothness assumptions on $f$ and $g$ , one can obtain a full asymptotic expansion in powers of $1/n$ .

Key hypotheses and conclusions

Hypotheses

A unique interior maximizer $x_0$ for $f$ .
Nondegeneracy: $f''(x_0)<0$ .
Mild regularity of $g$ and $f$ near $x_0$ .

Conclusions

The integral is asymptotically Gaussian around $x_0$ after Taylor expansion of $f$ : the leading exponential growth is $e^{n f(x_0)}$ , and the subexponential prefactor is of order $n^{-1/2}$ .
This yields a quantitative refinement of the Laplace principle: $\frac1n\log I_n \to f(x_0)$ .

In statistical mechanics, saddle-point estimates justify mean-field and variational approximations of partition functions (for example, after introducing an order parameter or via Hubbard–Stratonovich transforms) and they often explain the appearance of Gaussian fluctuations around equilibrium points.