Laplace principle (large deviations theory)

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.

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

∫ A e − φ ( x ) d x < ∞ . {\displaystyle \int _{A}e^{-\varphi (x)}\,dx<\infty .}

Then

lim θ → ∞ 1 θ log ⁡ ∫ A e − θ φ ( x ) d x = − e s s i n f x ∈ A ⁡ φ ( x ) , {\displaystyle \lim _{\theta \to \infty }{\frac {1}{\theta }}\log \int _{A}e^{-\theta \varphi (x)}\,dx=-\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,

∫ A e − θ φ ( x ) d x ≈ exp ⁡ ( − θ e s s i n f x ∈ A ⁡ φ ( x ) ) . {\displaystyle \int _{A}e^{-\theta \varphi (x)}\,dx\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

P θ ( A ) = ( ∫ A e − θ φ ( x ) d x ) / ( ∫ R d e − θ φ ( y ) d y ) {\displaystyle \mathbf {P} _{\theta }(A)=\left(\int _{A}e^{-\theta \varphi (x)}\,dx\right){\bigg /}\left(\int _{\mathbf {R} ^{d}}e^{-\theta \varphi (y)}\,dy\right)}

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

lim ε ↓ 0 ε log ⁡ P [ ε X ∈ A ] = − e s s i n f x ∈ A ⁡ x 2 2 {\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.

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