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

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 $\{a_i\}$, $\lim_{n\to\infty}\frac1n\log\sum_i e^{n a_i}=\max_i a_i$.

Proof idea / significance

Let $M=\max_K f$. For the upper bound, $\int_K e^{n f}\,d\mu\le e^{nM}\mu(K)$ gives $\limsup \le M$. For the lower bound, choose $x_\star\in K$ with $f(x_\star)=M$ and a neighborhood $U$ where $f\ge M-\varepsilon$. Then $\int_K e^{n f}\,d\mu\ge \int_U e^{n f}\,d\mu\ge \mu(U)\,e^{n(M-\varepsilon)}$, so $\liminf \ge M-\varepsilon$ and let $\varepsilon\downarrow 0$.

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.