A

Classical references for the saddle-point methods are [5] and [15]. Let Ω be a bounded domain on Rn, S : Ω → R, f : Ω → R and λ > 0 be a large positive parameter. The Laplace method consists in studying the asymptotics as λ→∞ of the multi-dimensional Laplace integrals

F (λ) = ∫

f(x)e−λS(x)dx

Let S and f be smooth functions and we assume that the function S has a minimum only at one interior non-degenerate critical point x0 ∈ Ω:

∇xS(x0) = 0 , ∇2xS(x0) > 0 , x0 ∈ Ω

x0 is called the saddle-point. Then in the neighborhood of x0 the function S has the following Taylor expansion

S(x) = S(x0) + 1 2

(x− x0)†∇2xS(x0)(x− x0) + o((x− x0)3)

As λ→∞, the main contribution of the integral comes from the neighborhood of x0. Replacing the function f by its value at x0, we obtain a Gaussian integral where the integration over x can be performed

F (λ) ≈ f(x0)e−λS(x0) ∫

≈ f(x0)e−λS(x0) ∫ Rn e−

One gets the leading asymptotics of the integral as λ→∞

F (λ) ∼ f(x0)e−λS(x0) (

2pi λ

More generally, doing a Taylor expansion at the n-th order for S (resp. n− 2 order for f) around x = x0, we obtain

F (λ) ∼ e−λS(x0) (

2pi λ

akλ −k

k of the f and S at the point x0. For example, at the first-order (in one dimension), we find

F (λ) ∼ √

2pi λS′′(c)

e−λS(x0) (f(x0)

− 1 λ

( − f

′′(x0) 2S′′(x0)

+ f(x0)S(4)(x0)

8S′′(x0)2 + f ′(x0)S(3)(x0)

2S′′(x0)2 − 5 (S

′′′(x0)) 2 f(x0)

24S′′(x0)3

))