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<center><math> \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} </math></center> | <center><math> \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} </math></center> | ||
where <math>Q_{M-1}(E) = [1-P(E)]^{M-1}</math>. | |||
'''Step 2: the Gumbel limit ''' So far, no approximations have been made. To proceed, we use <math> Q_{M-1}(E)\approx Q_M(E)</math> and its asymptotics Gumbel form: | '''Step 2: the Gumbel limit ''' So far, no approximations have been made. To proceed, we use <math> Q_{M-1}(E)\approx Q_M(E)</math> and its asymptotics Gumbel form: |
Revision as of 14:10, 31 August 2025
Nei seguente esercizio useremo le notazioni della statistica dei valori estremi usate nel corso.
exercise 1: La distribuzione di Gumbel
esercizio 2: The weakest link
Exercise 3: number of states above the minimum
Definition of :Given a realization of the random energies , define
that is, the number of random variables lying above the minimum but less than . This is itself a random variable. We are interested in its mean value:
The Final goal is to show that for large 'M', when the extremes are described by the Gumbel distribution :
Step 1: Exact manipulations: You start from the exact expression states in the interval:
To compute , you must sum over . Use the identity
to arrive at the form:
where . Step 2: the Gumbel limit So far, no approximations have been made. To proceed, we use and its asymptotics Gumbel form: