In the following exercises, we will use the notation from extreme value statistics as introduced in the course.

Exercise 1: The Gaussian case

Let us analyze in detail the case of a Gaussian distribution with zero mean and variance $\sigma ^{2}$ . Using integration by parts, we can write :

P(E)=\int _{-\infty }^{E}{\frac {dx}{\sqrt {2\pi \sigma ^{2}}}}\,e^{-{\frac {x^{2}}{2\sigma ^{2}}}}={\frac {1}{2{\sqrt {\pi }}}}\int _{\frac {E^{2}}{2\sigma ^{2}}}^{\infty }{\frac {dt}{\sqrt {t}}}e^{-t}={\frac {\sigma }{{\sqrt {2\pi }}|E|}}e^{-{\frac {E^{2}}{2\sigma ^{2}}}}-{\frac {1}{4{\sqrt {\pi }}}}\int _{\frac {E^{2}}{2\sigma ^{2}}}^{\infty }{\frac {dt}{t}}e^{-t}

The asymptotic expansion for $E\to -\infty$ is :

P(E)\approx {\frac {\sigma }{{\sqrt {2\pi }}|E|}}e^{-{\frac {E^{2}}{2\sigma ^{2}}}}+O({\frac {e^{-{\frac {E^{2}}{2\sigma ^{2}}}}}{|E|^{2}}})

In general, 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: $\exp(z)\,\exp(-e^{z})$

Exercise 2: The Weakest Link and the Weibull Distribution

Consider a chain of length $L$ subjected to a tensile force $F$ . Define $F_{c}$ as the force required to break the chain. The goal of this exercise is to determine how $F_{c}$ depends on $L$ and to characterize its sample-to-sample fluctuations. Throughout the exercise, you work in the limit of large $L$ .

Let $x_{1},x_{2},\dots ,x_{L}$ denote the strengths of the individual links. Assume that these are positive, identically distributed, and independent random variables. Consider the Gamma distribution with shape parameter $\alpha >0$ and $\Gamma (\alpha )$ the Gamma function:

p(x)={\frac {x^{\alpha -1}}{\Gamma (\alpha )}}e^{-x}

Questions:

Compute the typical value $F_{c}^{typ}$ and discuss its dependence on $L$ .

According to extreme value theory, the probability that the weakest link is smaller than $x$ is

Q_{L}(x)\sim \exp \!{\bigl [}-LP(x){\bigr ]}=\exp \!\!\left[-L\int _{0}^{x}p(t)\,dt\right]

Use the change of variable $z={\frac {x-a_{L}}{b_{L}}}$ with $a_{L}=0$ and $b_{L}=F_{c}^{typ}$ to find an $L$ -independent distribution.

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 $k$ 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 ${\overline {n(x)}}$ , you must sum over $k$ . 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:

{\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 $Q_{M-1}(E)=[1-P(E)]^{M-1}$ .

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