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In [[mathematics]], '''Laplace's principle''' is a basic [[theorem]] in [[large deviations theory]], similar to [[Varadhan's lemma]]. It gives an [[Asymptotic analysis|asymptotic expression]] for the [[Lebesgue integration|Lebesgue integral]] of exp(&minus;''&theta;&phi;''(''x'')) over a fixed set ''A'' as ''&theta;'' 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 measure|Lebesgue-measurable]] [[subset]] of ''d''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''d''</sup> and let ''&phi;''&nbsp;:&nbsp;'''R'''<sup>''d''</sup>&nbsp;→&nbsp;'''R''' be a [[measurable function]] with

:<math>\int_A e^{- \varphi(x)} \, \mathrm{d} x < + \infty.</math>

Then

:<math>\lim_{\theta \to + \infty} \frac1{\theta} \log \int_{A} e^{- \theta \varphi(x)} \, \mathrm{d} x = - \mathop{\mathrm{ess \, inf}}_{x \in A} \varphi(x),</math>

where ess inf denotes the [[essential infimum]]. Heuristically, this may be read as saying that for large ''&theta;'',

:<math>\int_{A} e^{- \theta \varphi(x)} \, \mathrm{d} x \approx \exp \left( - \theta \mathop{\mathrm{ess \, inf}}_{x \in A} \varphi(x) \right).</math>

==Application==

The Laplace principle can be applied to the family of [[probability measure]]s '''P'''<sub>''&theta;''</sub> given by

:<math>\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)</math>

to give an asymptotic expression for the probability of some set/event ''A'' as ''&theta;'' becomes large. For example, if ''X'' is a standard [[normal distribution|normally distributed]] [[random variable]] on '''R''', then

:<math>\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}</math>

for every measurable set ''A''.

==References==

* {{cite book
| last= Dembo
| first = Amir
| coauthors = Zeitouni, Ofer
| title = Large deviations techniques and applications
| series = Applications of Mathematics (New York) 38
| edition = Second edition
| publisher = Springer-Verlag
| location = New York
| year = 1998
| pages = xvi+396
| isbn = 0-387-98406-2
}} {{MathSciNet|id=1619036}}

[[Category:Asymptotic analysis]]
[[Category:Large deviations theory]]
[[Category:Probability theorems]]
[[Category:Statistical mechanics]]
[[Category:Mathematical principles]]
[[Category:Theorems in analysis]]