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| =Detour: Extreme Value Statistics= | | =Detour: Extreme Value Statistics= |
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| Consider the <math>M</math> energies <math>E_1, \dots, E_M</math> as independent and identically distributed (i.i.d.) random variables drawn from a distribution <math>p(E)</math>. It is useful to introduce the cumulative probability of finding an energy smaller than ''E''
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| <center> <math>P(E) = \int_{-\infty}^E dx \, p(x)</math> </center>
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| We define:
| | '''Definition of <math> n(x) </math>:''' |
| <center> <math>E_{\min} = \min(E_1, \dots, E_M)</math> </center>
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| Our goal is to compute the cumulative distribution:
| | Given a realization, <math> n(x) </math> is defined as the number of random variables above the minimum <math>E_{\min} </math> such that their value is smaller than <math>E_{\min} +x</math>. This quantity is a random variable, and we are interested in its average value: |
| <center> <math>Q_M(E) \equiv \text{Prob}(E_{\min} > E)</math> </center>
| | <center><math> \overline{n(x)} = \sum_k k \, \text{Prob}[n(x) = k] </math></center> |
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| for large <math>M</math>. To achieve this, we rely on three key relations:
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| * '''First relation''':
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| <center> <math>Q_M(E) = (1-P(E))^M</math> </center>
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| This relation is exact but depends on ''M'' and the precise form of <math>p(E)</math>. However, in the large ''M'' limit, a universal behavior emerges.
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| * '''Second relation''':
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| <center> <math>P(E_{\min}^{\text{typ}}) = 1/M</math> </center>
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| This is an estimation of the typical value of the minimum. It is a crucial relation that will be used frequently in this context. | |
| * '''Third relation'''
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| <center> <math>Q_M(E) = e^{M \log(1 - P(E))} \sim \exp\left(-M P(E)\right)</math> </center>
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| This is an approximation valid for large ''M'' and around the typical value of the minimum energy.
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Revision as of 20:53, 30 August 2025
![{\displaystyle {\overline {n(x)}}=\sum _{k}k\,{\text{Prob}}[n(x)=k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60b7ab18af629ce7a3211174fae3b391ee86a3f2)