TBan-I

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

Exercise 1: The Gumbel Distribution

In the spirit of the central limit theorem, you look for a scaling form:

$E_{\min }=a_{M}+b_{M}z$

The constants $a_{M}$ and $b_{M}$ absorb the dependence on $M$, while the random variable $z$ is distributed according to a probability distribution that does not depend on $M$.

In the Gaussian case, expand $A(E)$ around $a_{M}$:

By setting

$a_{M}=E_{\min }^{\text{typ}}=-\sigma {\sqrt {2\log M}}+\ldots \qquad {\text{and}}\qquad b_{M}={\frac {1}{A'(a_{M})}}={\frac {\sigma }{\sqrt {2\log M}}}$

you find

Therefore, the variable $z=(E-a_{M})/b_{M}$ is distributed according to an M-independent distribution.

It is possible to generalize this result and classify the scaling forms into the Gumbel universality class:

• Characteristics:
  • Applies when the tails of $p(E)$ decay faster than any power law.
  • Examples: the Gaussian case discussed here or exponential distributions $p(E)=\exp(E)\quad {\text{with}}\quad E\in (-\infty ,0)$.

• Scaling Form:

$P(z)=\exp(z)\,\exp(-e^{z})$

Exercise 3: number of states above the minimum

Definition 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 n(x) } :Given a realization of the random 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, 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 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}} but less than 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}+x} . This is itself a random variable. We are interested in its mean value:

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 \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:

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 \overline{n(x)} = e^{x/b_M}-1 }

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

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

To compute 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 \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

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

to arrive at the form:

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 \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{d Q_{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:

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 \frac{d Q_{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} .

The main contribution to the integral comes from the region near 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 \approx a_M} , 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 P(E) \approx e^{(E-a_M)/b_M}/M} .

Compute the integral and verify that you obtain:

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 \overline{n(x)} = \bigl(e^{x/b_M}-1\bigr) \int_{-\infty}^{\infty} dz \, e^{2z - e^z} = e^{x/b_M}-1 }