TBan-I

Nei seguente esercizio useremo le notazioni della statistica dei valori estremi usate nel corso.

Exercise 3: number of states above the minimum

Definition of $n(x)$:

Given a realization, $n(x)$ is defined as the number of random variables above the minimum $E_{\min }$ such that their value is smaller than $E_{\min }+x$. This quantity is a random variable, and we are interested in its average value:

${\overline {n(x)}}=\sum _{k}k\,{\text{Prob}}[n(x)=k]$

The Final goal is to show that for large 'M', when the extremes are described by the Gumbel distribution :

${\overline {n(x)}}=e^{x/b_{M}}-1$

The key relation for this quantity is:

${\text{Prob}}[n(x)=k]=M{\binom {M-1}{k}}\int _{-\infty }^{\infty }dE\;p(E)[P(E+x)-P(E)]^{k}\left(1-P(E+x)\right)^{M-k-1}$

We use the following identity to sum over $k$:

$\sum _{k=0}^{M-1}k{\binom {M-1}{k}}(A-B)^{k}B^{M-1-k}=(A-B){\frac {d}{dA}}\sum _{k=0}^{M-1}{\binom {M-1}{k}}(A-B)^{k}B^{M-1-k}=(M-1)(A-B)A^{M-2}$

to arrive at the form:

${\overline {n(x)}}=M(M-1)\int _{-\infty }^{\infty }dE\;p(E)\left[P(E+x)-P(E)\right](1-P(E))^{M-2}=M\int _{-\infty }^{\infty }dE\;\left[P(E+x)-P(E)\right]{\frac {dQ_{M-1}(E)}{dE}}$