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), you have:

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

Step 1: Exact manipulations: You start from the exact expression for the probability 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 k} states in the interval:

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 \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} }

To compute ${\overline {n(x)}}$, you must sum over 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 k} . Use the identity

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 \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:

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

where 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-1}(E) = [1-P(E)]^{M-1}} .

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

${\frac {dQ_{M-1}(E)}{dE}}\,dE\;\sim \;\exp \!\!\left({\frac {E-a_{M}}{b_{M}}}\right)\exp \!\!\left[-\exp \!\!\left({\frac {E-a_{M}}{b_{M}}}\right)\right]{\frac {dE}{b_{M}}}=e^{z}e^{-e^{z}}dz$

where 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 z = (E-a_M)/b_M} .

Compute the integral and verify that you obtain: