Gaussain Related Intergration Trick
January 27, 2021
Some math tricks related to gaussain intergrations.
Saddle point integration
Target integration function: \(I= \int dx \exp{[N\phi(x)]}\), \(N\) is large. we can expand \(\phi(x)\) at its maximum point\(\phi'(x_{max})=0\) :
\[\begin{aligned} I &= \int dx \exp{[N\phi(x_{max})-N\frac{1}{2}\phi''(x_{max})(x-x_{max})^2]} \\ &= e^{N\phi(x_{max})}\cdot\underbrace{\int dx \exp{[-N\frac{1}{2}\phi''(x_{max})(x-x_{max})^2]}}_{\text{gaussian integration}} \\ &= e^{N\phi(x_{max})}\sqrt{\frac{2\pi}{N\phi''(x_{max})}} \end{aligned}\]One important application of saddle point integration is Stirling Approximation. We have the \(\Gamma\) function formThis can be achieved by differentiate the equation \(\int_{0}^{\infty} d x e^{-\alpha x}=\frac{1}{\alpha}\) for \(N\) times with respect to \(\alpha\) and set \(\alpha=1\) : $$N!=\int_{0}^{\infty} d x x^{N+1} e^{-x}$$ Try yourself with saddle point integration trick to prove that: $$\log N !=N \log N-N+\frac{1}{2} \log (2 \pi N)$$
Linearize term
This trick appears in the replica trick to solve the Gardner’s capacity:
\[e^{-\frac{q}{2}(\sum_au_a)^2} = \int \frac{dt}{\sqrt{2\pi}}e^{-t^2/2+it\sqrt{q}\sum_{a=1}^nu_a}\]