H 1

Exercises: Extreme Value Statistics

Exercise 1: Gaussian tails and Gumbel scaling

We are interested in the asymptotic behavior of the cumulative distribution $P(E)$ in the left tail $E\to -\mathrm{\infty }$, since the minimum is controlled by the regime where $MP(E)=O(1)$.

Starting from the Gaussian distribution with zero mean and variance ${\sigma }^2$, we write the cumulative as

$P(E)=\underset{-\mathrm{\infty }}{\overset{E}{\int }}\frac{dx}{\sqrt{2\pi {\sigma }^2}}{e}^{-x^2/2{\sigma }^2}.$

Using integration by parts (or equivalently the change of variable $t=x^2/2{\sigma }^2$), one finds

$P(E)=\frac{\sigma }{\sqrt{2\pi }|E|}{e}^{-E^2/2{\sigma }^2}-\frac{1}{4\sqrt{\pi }}\underset{E^2/2{\sigma }^2}{\overset{\mathrm{\infty }}{\int }}\frac{dt}{t}{e}^{-t}.$

For $E\to -\mathrm{\infty }$, the second term is subleading, and the cumulative admits the asymptotic expansion

$P(E)=\frac{\sigma }{\sqrt{2\pi }|E|}{e}^{-E^2/2{\sigma }^2}[1+O\left(\frac{1}{E^2}\right)].$

This result can be written in the form

$P(E)=\exp (A(E)),A(E)=-\frac{E^2}{2{\sigma }^2}-\log \left(\frac{\sqrt{2\pi }|E|}{\sigma }\right)+\dots ,$

which is the expression used in the course to derive the scaling form of the minimum.

Using Gaussian variables, we analyzed a situation where the minimum is controlled by the far left tail of the distribution. As a consequence, the natural centering constant was the typical minimum $a_M=E_{\min }^{\mathrm{t}\mathrm{y}\mathrm{p}}$.

We now consider a qualitatively different case, where the random variables are bounded from below. In this situation, the minimum is controlled by the behavior of the distribution close to the edge of its support.

This will lead to a different choice of scaling parameters: the centering constant is fixed by the lower bound of the support, while the scale of fluctuations is set by the typical minimum itself.

Exercise 2: Weakest-link statistics and the Weibull law

In this exercise, we consider a situation that is qualitatively different from the Gaussian case. Here, the random variables are bounded from below, and the minimum is controlled by the behavior of the distribution close to the edge of its support.

This will naturally lead to a different choice of scaling parameters $a_M$ and $b_M$.

Consider a chain of length $L$ subjected to a tensile force $F$. The chain breaks when its weakest link breaks. We denote by $F_c$ the force required to break the chain.

Let $x_1,x_2,\dots ,x_L$ be the breaking strengths of the individual links. Assume that they are independent, identically distributed, and strictly positive random variables.

Throughout the exercise, we work in the limit of large $L$.

The strength of each link is drawn from a Gamma distribution with shape parameter $\alpha >0$:

$p(x)=\frac{x^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}{e}^{-x},x\ge 0.$

Questions:

• Compute the typical breaking force $F_c^{\mathrm{t}\mathrm{y}\mathrm{p}}$ of the chain and determine its dependence on $L$. (Hint: use the condition $LP(x)=O(1)$.)

• The breaking force of the chain is equal to the minimum of the $L$ random variables. According to extreme value statistics, its distribution satisfies

$Q_L(x)\equiv \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}(F_c>x)\sim \exp [-LP(x)],P(x)=\underset{0}{\overset{x}{\int }}p(t)dt.$

Show that the appropriate scaling form is obtained by introducing $a_L=0$ and $b_L=F_c^{\mathrm{t}\mathrm{y}\mathrm{p}}$, and defining the rescaled variable

$z=\frac{x-a_L}{b_L}.$

Determine the limiting, $L$-independent distribution of $z$, and identify the corresponding universality class.

Exercise 3: Number of states close to the minimum

In this exercise, we study a further consequence of extreme value statistics by characterizing not only the minimum energy, but also the number of states lying close to it. This quantity will play an important role in the discussion of the Random Energy Model.

Definition of $n(x)$: Given a realization of the random energies $\{E_1,E_2,\dots ,E_M\}$, we define

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

that is, the number of states lying within an energy window $x$ above the minimum. This is a random variable. We are interested in its disorder average,

$\overline{n(x)}=\sum _{k=0}^{M-1}k\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}[n(x)=k].$

Final goal: Show that, in the large-$M$ limit, provided the minimum belongs to the Gumbel universality class,

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

This result depends only on the scaling parameter $b_M$ introduced in the extreme value analysis.

Step 1: Exact manipulations The probability that exactly $k$ states fall in the interval $(E_{\min },E_{\min }+x)$ can be written exactly as

$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}[n(x)=k]=M(\genfrac{}{}{0ex}{}{M-1}{k})\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dEp(E)[P(E+x)-P(E){]}^k[1-P(E+x){]}^{M-k-1}.$

Using the identity

$\sum _{k=0}^{M-1}k(\genfrac{}{}{0ex}{}{M-1}{k})(A-B{)}^k{B}^{M-1-k}=(M-1)(A-B)A^{M-2},$

one obtains

$\overline{n(x)}=M(M-1)\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dEp(E)[P(E+x)-P(E)](1-P(E){)}^{M-2}=-M\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dE[P(E+x)-P(E)]\frac{d{Q}_{M-1}(E)}{dE},$

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

Step 2: Gumbel limit From now on, we assume that the extreme value statistics of the energies is described by the Gumbel distribution. So far, no approximations have been made.

For large $M$, we may use $Q_{M-1}(E)\simeq Q_M(E)$ and its Gumbel scaling form,

$-\frac{d{Q}_M(E)}{dE}dE\sim \exp \left(\frac{E-a_M}{b_M}\right)\exp [-\exp \left(\frac{E-a_M}{b_M}\right)]\frac{dE}{b_M}.$

Introducing the scaling variable $z=(E-a_M)/b_M$, the main contribution to the integral comes from the region $E\approx a_M$, where $P(E)\approx e^{(E-a_M)/b_M}/M$.

Compute the integral and show that

$\overline{n(x)}=(e^{x/b_M}-1)\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dz{e}^{2z-e^z}=e^{x/b_M}-1.$