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 of the random energies ${E_{1},E_{2},\ldots ,E_{M}}$, define

$n(x)=\#\{i\mid E_{\min }<E_{i}<E_{\min }+x\}$

that is, the number of random variables lying above the minimum $E_{\min }$ but less than $E_{\min }+x$. This is itself a random variable. We are interested in its mean value:

${\overline {n(x)}}=\sum _{k=0}^{M-1}k\,{\text{Prob}}[n(x)=k]$

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

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

Step 1: Exact manipulations

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

The starting point is the definition of the key quantity

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

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:

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

Step 2: the Gumbel limit So far, no approximations have been made. To proceed, we use $Q_{M-1}(E)\approx Q_{M}(E)$ and its asymptotics Gumbel form: