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:

$\text{Prob}[n(x)=k]=M(\genfrac{}{}{0ex}{}{M-1}{k})\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dEp(E)[P(E+x)-P(E){]}^k{(1-P(E+x))}^{M-k-1}$

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

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

to arrive at the form:

$\overline{n(x)}=M(M-1)\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dEp(E)[P(E+x)-P(E)](1-P(E){)}^{M-2}=M\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dE[P(E+x)-P(E)]\frac{d{Q}_{M-1}(E)}{dE}$

Gumbel limit So far, no approximations have been made. To proceed, we use $Q_{M-1}(E)\approx Q_M(E)$ and its asymptotics: