TBan-I

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

Exercise 1: The Gumbel Distribution

Let's go back to the end of Lecture 1. In the Gaussian case, expand $A(E)$ around $a_{M}$:

$Q_{M}(E)\sim \exp \left(-MP(E)\right)\sim \exp \left(-Me^{A(a_{M})+A'(a_{M})\cdot (E-a_{M})}\right)$

Show that by setting

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

you find

$\exp(A(a_{M}))={\frac {1}{M}}\qquad {\text{and}}\qquad Q_{M}(E)\equiv {\text{Prob}}(E_{\min }>E)\sim \exp \!\!\left[-\exp \!\!\left({\frac {E-a_{M}}{b_{M}}}\right)\right]$

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

Exercise 2: The Weakest Link and the Weibull Distribution

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

Let 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 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 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 \alpha > 0} and $\Gamma (\alpha )$ the Gamma function:

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

Questions:

• Compute the typical 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 F_c^{ typ}} and discuss its dependence on $L$.

• According to extreme value theory, the probability that the weakest link is smaller 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 x} is

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_L(x) \sim \exp\!\bigl[-L P(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 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 a_L = 0} and 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 b_L = F_c^{typ}} to find an $L$-independent distribution.

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:

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

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 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:

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

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 $Q_{M-1}(E)\approx Q_{M}(E)$ 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 $E\approx a_{M}$, 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 }