In mathematics, Laplace's principle is a basic theorem in large deviations theory which is similar to Varadhan's lemma. It gives an asymptotic expression for the Lebesgue integral of exp(−θφ(x)) over a fixed set A as θ becomes large. Such expressions can be used, for example, in statistical mechanics to determining the limiting behaviour of a system as the temperature tends to absolute zero.
Let A be a Lebesgue-measurable subset of d-dimensional Euclidean space R<sup>d</sup> and let φ : R<sup>d</sup> â R be a measurable function with
Then
where ess inf denotes the essential infimum. Heuristically, this may be read as saying that for large θ,
The Laplace principle can be applied to the family of probability measures P<sub>θ</sub> given by
to give an asymptotic expression for the probability of some event A as θ becomes large. For example, if X is a standard normally distributed random variable on R, then
for every measurable set A.