LBan-1

Overview

This lesson is structured in three parts:

• Self-averaging and disorder in statistical systems

Disordered systems are characterized by a random energy landscape, however, in the thermodynamic limit, physical observables become deterministic. This property, known as self-averaging, does not always hold for the partition function which is the quantity that we can compute. When it holds the annealed average $\ln {\overline {Z}}$ and the quenched average $\ln Z$ coincides otherwiese we have

$\ln Z<\ln {\overline {Z}}$

• The Random Energy Model

We study the Random Energy Model (REM) introduced by Bernard Derrida. In this model at each configuration is assigned an independent energy drawn from a Gaussian distribution of extensive variance. The model exhibits a freezing transition at a critical temperature​, below which the free energy becomes dominated by the lowest energy states.

• Extreme value statistics and saddle-point analysis

The results obtained from a saddle-point approximation can be recovered using the tools of extreme value statistics.

Part I

Random energy landascape

In a system with $N$ degrees of freedom, the number of configurations grows exponentially with $N$. For simplicity, consider Ising spins that take two values, $\sigma _{i}=\pm 1$, located on a lattice of size $L$ in $d$ dimensions. In this case, 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 = L^d} and the number of configurations 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 M = 2^N = e^{N \log 2}} .

In the presence of disorder, the energy associated with a given configuration becomes a random quantity. For instance, in the Edwards-Anderson model:

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 = - \sum_{\langle i, j \rangle} J_{ij} \sigma_i \sigma_j, }

where the sum runs over nearest neighbors 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 \langle i, j \rangle} , and the couplings 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 J_{ij}} are independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and unit variance.

The energy of a given configuration is a random quantity because each system corresponds to a different realization of the disorder. In an experiment, this means that each of us has a different physical sample; in a numerical simulation, it means that each of us has generated a different set of couplings 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 J_{ij}} .

To illustrate this, consider a single configuration, for example the one where all spins are up. The energy of this configuration is given by the sum of all the couplings between neighboring spins:

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[\sigma_1=1,\sigma_2=1,\ldots] = - \sum_{\langle i, j \rangle} J_{ij}. }

Since the the couplings are random, the energy associated with this particular configuration is itself a Gaussian random variable, with zero mean and a variance proportional to the number of terms in the sum — that is, of order 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} . The same reasoning applies to each of the 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 M = 2^N} configurations. So, in a disordered system, the entire energy landscape is random and sample-dependent.

Self-averaging observables

A crucial question is whether the macroscopic properties measured on a given sample are themselves random or not. Our everyday experience suggests that they are not: materials like glass, ceramics, or bronze have well-defined, reproducible physical properties that can be reliably controlled for industrial applications.

From a more mathematical point of view, it means that the free energy 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_N(\beta)=N f_N(\beta)} and its derivatives (magnetization, specific heat, susceptibility, etc.), in the limit 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 \to \infty } , these random quantities concentrates around a well defined value. These observables are called self-averaging. This means that,

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 \lim_{N \to \infty} f_N (\beta)= \lim_{N \to \infty} f_N^{\text{typ}}(\beta) =\lim_{N \to \infty} \overline{f_N(\beta)} =f_\infty(\beta) }

Hence 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_N(\beta) } becomes effectively deterministic and its sample-to sample fluctuations vanish in relative terms:

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 \lim_{N \to \infty} \frac{\overline{f_N^2(\beta)}}{\overline{f_N(\beta)}^2}=1. }

The partition function

The partition function

$Z_{N}=\exp(-\beta Nf_{N}(\beta ))$

is itself a random variable in disordered systems. Analytical methods can capture the statistical properties of this variable. We can define to average over the disorder realizations:

• The annealed average corresponds to the calculation of the moments of the partition function. The annealed free energy 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 f^{\text{ann.}} = -\frac{1}{\beta N} \ln \, \overline{Z_N} }

• the quenched average corresponds to the average of the logarithm of the partition function, which is self-averaging for sure.

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_\infty (\beta)\sim \overline{f_N(\beta)} = - \ln Z_N(\beta)/ (\beta N)}

Do these two averages coincide?

If the partition function is self-averaging in the thermodynamic limit, then

$\lim _{N\to \infty }Z_{N}(\beta )=\lim _{N\to \infty }Z_{N}^{\text{typ}}(\beta )=\lim _{N\to \infty }{\overline {Z_{N}(\beta )}}=e^{-\beta Nf_{\infty }(\beta )}$

As a consequence, the annealed and the quenched averages coincide.

If the partition function is not self-averaging, only typical partition function concentrates, but extremely rare configurations contribute disproportionately to its moments:

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 \lim_{N \to \infty} Z_N^{\text{typ}}(\beta) = e^{-\beta N f_\infty(\beta)} < \lim_{N \to \infty} \overline{Z_N(\beta)} = e^{-\beta N f^{\text{ann.}}(\beta)} }

There are then two main strategies to determine the deterministic value of the observable :

• Compute directly the quenched average using methods such as the replica trick and the Parisi solution.

• Determine 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 Z_N^{\text{typ}}(\beta) } and evaluate $f_{\infty }(\beta )=-{\frac {1}{\beta N}}\ln Z_{N}^{\text{typ}}(\beta )$

Part II

Random Energy Model

The Random energy model (REM) neglects the correlations between the 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 M=2^N} configurations. The energy associated to each configuration is an independent Gaussian variable with zero mean and variance 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} . The simplest solution of the model is with the microcanonical ensemble.

Microcanonical calculation

Step 1: Number of states .

Let ${\mathcal {N}}_{N}(E)dE$ the number of states of energy in the interval (E,E+dE). It is a random number and we use the representation

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 \mathcal{N}_N(E)dE= \sum_{\alpha=1}^{2^N} \chi_\alpha(E) dE \;}

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 \chi_\alpha(E)=1} if $E_{\alpha }\in [E,E+dE]$ 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 \chi_\alpha(E)=0} otherwise. We can cumpute its average

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{\mathcal{N}_N(E)}= \sum_{\alpha=1}^{2^N} \overline{\chi_\alpha(E)} = \frac{2^N}{\sqrt{2 \pi N}} \exp\left(-\frac{E^2}{2 N}\right) \sim \exp \left[N (\ln 2 -\epsilon^2/2)\right] }

Here 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 \epsilon =E/N } is the energy density and the annealed entropy density in the thermodynamic limit 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 s^{\text{ann.}}(\epsilon)=\ln 2 -\epsilon^2/2 }

Step 2: Self-averaging.

Let compute now the second moment

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{\mathcal{N}_N^2(E)} = \sum_{\alpha=1}^{2^N} \overline{\chi_\alpha} \left(\sum_{\beta\ne \alpha} \overline{\chi_\beta} \right) + \sum_{\alpha=1}^{2^N} \overline{\chi_\alpha^2} \sim \mathcal{N}_N(E) \left( \mathcal{N}_N(E) - \exp\left(-\frac{E^2}{2 N}\right) \right) + \mathcal{N}_N(E) }

We can then check the self averaging condition:

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{\overline{\mathcal{N}_N^2(E)}}{\overline{\mathcal{N}_N(E)}^2} \sim 1+ \frac{1}{\overline{\mathcal{N}_N(E)}}}

A critical energy density 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 \epsilon^* = \sqrt{2 \ln 2}} separates a self-averaging regime for 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 |\epsilon| < \epsilon^*} and a non self-averaging regime where for $|\epsilon |>\epsilon ^{*}$. In the first regime, 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{\mathcal{N}_N(E)}} is exponentially large and its value is determinstic (average, typical, median are the same). In the secon regime, 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{\mathcal{N}_N(E)}} is exponentially small but nonzero. The typical value instead is exactly zero, ${\mathcal {N}}_{N}^{\text{typ}}(E)=0$: for most disorder realizations, there are no configurations with energy below 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 - \epsilon^* N} and only a vanishingly small fraction of rare samples gives a positive contribution to the average. As a result, the quenched average on the entropy density 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 s_\infty(\epsilon) = \begin{cases} \ln 2 - \dfrac{\epsilon^2}{2}, & \text{for } |\epsilon| < \epsilon^* \\ -\infty, & \text{for } |\epsilon| > \epsilon^* \end{cases} }

Back to canonical ensemble: the freezing transition

The annealed partition function is the average of the partition function over the disorder:

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{Z_N(\beta)} = \int_{-\infty}^{\infty} d\epsilon \; \overline{\mathcal{N}_N(\epsilon)} \, e^{-\beta N \epsilon} = \int_{-\infty}^{\infty} d\epsilon \; \exp \left[N (\ln 2 -\epsilon^2/2 -\beta \epsilon)\right]. }

Using the saddle point for large N we find $\epsilon _{\min }=-\beta$ and thus

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^{\text{ann.}}(\beta) = \ln 2/\beta + \dfrac{\beta}{2}}

The quenched partition function is obtained replacing the mean with 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 Z_N^{\text{typ.}}(\beta) = \int_{-\infty}^{\infty} d\epsilon \; \mathcal{N}_N^{\text{typ.}}(\epsilon) \, e^{-\beta N \epsilon} = \int_{-\epsilon^*}^{\epsilon^*} d\epsilon \; \exp \left[N (\ln 2 -\epsilon^2/2 -\beta \epsilon)\right]. }

Using the saddle point for large N we find a critical inverse temperature 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 \beta_c = \sqrt{2 \ln 2}} separating two phases:

• For 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 \beta < \beta_c } , 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 \epsilon_\min =-\beta} and the annealed calculation works
• For 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 \beta > \beta_c } , 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 \epsilon_\min =-\beta_c} and the free energy freezes to a temperature independent value. As a result, the quenched average on the free energy density 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 f_\infty(\beta) = \begin{cases} \ln 2/\beta + \dfrac{\beta}{2}, & \text{for } \beta < \beta_c \\ \sqrt{2 \ln 2}, & \text{for } \beta> \beta_c \end{cases} }

Part III

Detour: Extreme Value Statistics

Consider the REM spectrum 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 M} 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, \dots, E_M} drawn from a distribution 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)} . It is useful to introduce the cumulative probability of finding an energy smaller than E

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) = \int_{-\infty}^E dx \, p(x)}

We also define:

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} = \min(E_1, \dots, E_M), \quad Q_M(E) \equiv \text{Prob}(E_{\min} > E) }

The statistical properties 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 E_{\min}} are derived using three key relations:

• First relation:

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(E) = (1-P(E))^M}

This relation is exact but depends on M and the precise form 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 p(E)} .

• Second relation:

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_{\min}^{\text{typ}}) = 1/M}

This is an estimation of the typical value of the minimum. It is a crucial relation that will be used frequently.

• Third relation

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(E) = e^{M \log(1 - P(E))} \sim \exp\left(-M P(E)\right)}

This is an approximation valid for large M and around the typical value of the minimum energy. It allows to extract the universal scaling.

Back to REM

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

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) = \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 :

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 \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 this case it is convenient to introduce the 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 A(E)} defined as 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) = \exp(A(E))} . hence we 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 A_{\text{gauss}}(E)= -\frac{E^2}{2 \sigma^2} - \log(\frac{\sqrt{2 \pi} |E|}{\sigma})+\ldots }

From the second relation we impose 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_{\text{gauss}}(E_{\min}^{\text{typ}})=- \ln M} . For large 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 M} we get:

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}^{\text{typ}} = -\sigma \sqrt{2 \log M} \left( 1- \frac{1}{4} \frac{\log (4 \pi \log M)}{\log M} + \ldots \right)}

We look for a scaling form of the random variable 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} = a_M + b_M z} such that 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_M} , 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_M} are deterministic and M dependent while 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} is random and M independent.

In the Gaussian case, we start from the third relation introduced earlier and expand 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(E)} around 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_M} :

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(E) \sim \exp\left(-M P(E)\right) \sim \exp\left(-M e^{A(a_M) +A'(a_M)\cdot (E-a_M)}\right) }

By setting

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_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}}}