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

$P(E)=\int _{-\infty }^{E}dx\,p(x)$

We define:

$E_{\min }=\min(E_{1},\dots ,E_{M})$

Our goal is to compute the cumulative distribution:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_M(E) \equiv \text{Prob}(E_{\min} > E)}

for large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E)} . However, in the large M limit, a universal behavior emerges.

• Second relation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

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

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