# Laplace principle (large deviations theory)

In mathematics, Laplace's principle is a basic theorem in large deviations theory, 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.

## Statement of the result

Let A be a Lebesgue-measurable subset of d-dimensional Euclidean space Rd and let φ : Rd → R be a measurable function with

${\displaystyle \int _{A}e^{-\varphi (x)}\,\mathrm {d} x<+\infty .}$

Then

${\displaystyle \lim _{\theta \to +\infty }{\frac {1}{\theta }}\log \int _{A}e^{-\theta \varphi (x)}\,\mathrm {d} x=-{\mathop {\mathrm {ess\,inf} }}_{x\in A}\varphi (x),}$

where ess inf denotes the essential infimum. Heuristically, this may be read as saying that for large θ,

${\displaystyle \int _{A}e^{-\theta \varphi (x)}\,\mathrm {d} x\approx \exp \left(-\theta {\mathop {\mathrm {ess\,inf} }}_{x\in A}\varphi (x)\right).}$

## Application

The Laplace principle can be applied to the family of probability measures Pθ given by

${\displaystyle \mathbf {P} _{\theta }(A)=\left(\int _{A}e^{-\theta \varphi (x)}\,\mathrm {d} x\right){\Big /}\left(\int _{\mathbf {R} ^{d}}e^{-\theta \varphi (y)}\,\mathrm {d} y\right)}$

to give an asymptotic expression for the probability of some set/event A as θ becomes large. For example, if X is a standard normally distributed random variable on R, then

${\displaystyle \lim _{\varepsilon \downarrow 0}\varepsilon \log \mathbf {P} {\big [}{\sqrt {\varepsilon }}X\in A{\big ]}=-{\mathop {\mathrm {ess\,inf} }}_{x\in A}{\frac {x^{2}}{2}}}$

for every measurable set A.