The saddle point approximation

In this appendix we are going to see how the saddle point approximation works in general. Let us define the class of integrals:

$$I_{N}=\int e^{Nf(x)}dx\quad \quad N\gg 1$$

$$I_{N}=\int e^{Nf(x)}dx\quad \quad N\gg 1$$

Suppose that $f(x)$$f(x)$ has a unique maximum at $x=x^{*}$$x=x^{*}$ and that ${\textstyle \lim _{|x|\to \infty }f(x)=-\infty }$${\textstyle \lim _{|x|\to \infty }f(x)=-\infty }$. Then, expanding $f$$f$ around $x^{*}$$x^{*}$ we have:

$$f(x)=f(x^{*})+{\frac {(x-x^{*})^{2}}{2}}f''(x^{*})+{\frac {(x-x^{*})^{3}}{6}}f'''(x^{*})+\cdots$$

$$f(x)=f(x^{*})+{\frac {(x-x^{*})^{2}}{2}}f''(x^{*})+{\frac {(x-x^{*})^{3}}{6}}f'''(x^{*})+\cdots$$

(and obviously $f'(x^{*})=0$$f'(x^{*})=0$ because $x^{*}$$x^{*}$ is a maximum). Setting $z=x-x^{*}$$z=x-x^{*}$ we can write, stopping the expansion at the third order:

$$I_{N}\approx e^{Nf(x^{*})}\int e^{N{\frac {z^{2}}{2}}f''(x^{*})+N{\frac {z^{3}}{6}}f'''(x^{*})}dz$$

$$I_{N}\approx e^{Nf(x^{*})}\int e^{N{\frac {z^{2}}{2}}f''(x^{*})+N{\frac {z^{3}}{6}}f'''(x^{*})}dz$$

Calling $y=z{\sqrt {N}}$$y=z{\sqrt {N}}$ and remembering that ${\textstyle f''(x^{*})<0}$${\textstyle f''(x^{*})<0}$ since $x^{*}$$x^{*}$ is a maximum, we have:

$$I_{N}=e^{Nf(x^{*})}{\frac {1}{\sqrt {N}}}\int e^{-{\frac {y^{2}}{2}}|f''(x^{*})|+{\frac {y^{2}}{6{\sqrt {N}}}}f'''(x^{*})}dy$$

$$I_{N}=e^{Nf(x^{*})}{\frac {1}{\sqrt {N}}}\int e^{-{\frac {y^{2}}{2}}|f''(x^{*})|+{\frac {y^{2}}{6{\sqrt {N}}}}f'''(x^{*})}dy$$

Therefore for very large $N$$N$ the term proportional to $f'''(x^{*})$$f'''(x^{*})$ in the exponential (like all the following terms of the complete expansion) is negligible, so:

$$I_{N}\approx e^{Nf(x^{*})}{\frac {1}{\sqrt {N}}}\int e^{-{\frac {y^{2}}{2}}|f''(x^{*})|}dy$$

$$I_{N}\approx e^{Nf(x^{*})}{\frac {1}{\sqrt {N}}}\int e^{-{\frac {y^{2}}{2}}|f''(x^{*})|}dy$$

and computing the Gaussian integral:

$$I_{N}\approx {\sqrt {\frac {2\pi }{N|f''(x^{*})|}}}\cdot e^{Nf(x^{*})}$$

$$I_{N}\approx {\sqrt {\frac {2\pi }{N|f''(x^{*})|}}}\cdot e^{Nf(x^{*})}$$

Therefore we see that the saddle point approximation essentially states that an integral of the form $I_{N}$$I_{N}$ can be approximated, provided that $N$$N$ is large, with the value of the integrand calculated at its maximum (up to a multiplicative factor).