H 1

We are interested in the asymptotic behavior of the cumulative distribution $P(E)$ in the left tail $E\to -\mathrm{\infty }$, since the minimum is controlled by the regime where $MP(E)=O(1)$.

Starting from the Gaussian distribution with zero mean and variance ${\sigma }^2$, we write the cumulative as

$P(E)=\underset{-\mathrm{\infty }}{\overset{E}{\int }}\frac{dx}{\sqrt{2\pi {\sigma }^2}}{e}^{-x^2/2{\sigma }^2}.$

Using integration by parts (or equivalently the change of variable $t=x^2/2{\sigma }^2$), one finds

$P(E)=\frac{\sigma }{\sqrt{2\pi }|E|}{e}^{-E^2/2{\sigma }^2}-\frac{1}{4\sqrt{\pi }}\underset{E^2/2{\sigma }^2}{\overset{\mathrm{\infty }}{\int }}\frac{dt}{t}{e}^{-t}.$

For $E\to -\mathrm{\infty }$, the second term is subleading, and the cumulative admits the asymptotic expansion

$P(E)=\frac{\sigma }{\sqrt{2\pi }|E|}{e}^{-E^2/2{\sigma }^2}[1+O\left(\frac{1}{E^2}\right)].$

This result can be written in the form

$P(E)=\exp (A(E)),A(E)=-\frac{E^2}{2{\sigma }^2}-\log \left(\frac{\sqrt{2\pi }|E|}{\sigma }\right)+\dots ,$

which is the expression used in the course to derive the scaling form of the minimum.