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.

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Statement of the result

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

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

Then

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

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

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

Application

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

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

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

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

for every measurable set A.

References

Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications, Second edition, Applications of Mathematics (New York) 38, New York: Springer-Verlag, xvi+396. ISBN 0-387-98406-2. MR1619036

Category: Statistical mechanics