TBan-I

Detour: Extreme Value Statistics

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

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) = \int_{-\infty}^E dx \, p(x)}

We define:

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

• First 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 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:

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

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

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

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