TBan-I

Detour: Extreme Value Statistics

Consider the $M$ energies $E_1,\dots ,E_M$ as independent and identically distributed (i.i.d.) random variables drawn from a distribution $p(E)$. It is useful to introduce the cumulative probability of finding an energy smaller than E

$P(E)=\underset{-\mathrm{\infty }}{\overset{E}{\int }}dxp(x)$

We define:

$E_{\min }=\min (E_1,\dots ,E_M)$

Our goal is to compute the cumulative distribution:

$Q_M(E)\equiv \text{Prob}(E_{\min }>E)$

for large $M$. To achieve this, we rely on three key relations:

• First relation:

$Q_M(E)=(1-P(E){)}^M$

This relation is exact but depends on M and the precise form of $p(E)$. However, in the large M limit, a universal behavior emerges.

• Second relation:

$P(E_{\min }^{\text{typ}})=1/M$

This is an estimation of the typical value of the minimum. It is a crucial relation that will be used frequently in this context.

• Third relation

$Q_M(E)=e^{M\log (1-P(E))}\sim \exp (-MP(E))$

This is an approximation valid  for large M and around the typical value of the minimum energy.