Laplace Principle

Asymptotic evaluation of log-integrals of exp(n f) by the supremum of f; a deterministic precursor to Varadhan’s lemma.

Laplace Principle

This lemma is a deterministic prototype for Varadhan's lemma and is closely related to the large deviation principle . It is frequently used to approximate partition functions and free energies via maximization.

Statement

Let $K$ be a compact topological space and let $\mu$ be a finite Borel measure on $K$ such that every nonempty open set has positive $\mu$ -measure. If $f:K\to\mathbb{R}$ is continuous, then

\lim_{n\to\infty}\frac{1}{n}\log\!\int_K e^{n f(x)}\,\mu(dx) ={} \max_{x\in K} f(x).

More generally, if $f$ is upper semicontinuous and bounded above on compact $K$ , the same conclusion holds with $\max$ replaced by $\sup$ .

Key hypotheses and conclusions

Hypotheses

Compactness of $K$ (ensures $f$ attains its maximum when $f$ is continuous).
$\mu$ does not ignore neighborhoods of maximizers (positivity on open sets).
Regularity of $f$ (at least continuity for the simplest version).

Conclusions

Exponential integrals are governed at leading order by the global maximum of $f$ : the integral behaves like $\exp\big(n\max f\big)$ up to subexponential factors.
A discrete analogue: for finitely many numbers $\{a_i\}$ , $\lim_{n\to\infty}\frac1n\log\sum_i e^{n a_i}=\max_i a_i$ .

In statistical mechanics, this principle explains why a partition function (an integral or sum of exponentials) yields a free energy that can often be expressed as a variational supremum. It is the leading-order version of the more refined saddle-point method , which also provides prefactors.