= Participation ratio
{title2=$P=(\sum_n|\psi_n|^4)^{-1}$}
For an <eigenvector> normalized by $\sum_{n=1}^N|\psi_n|^2=1$, the <participation ratio> in its site basis is
$$
P=\frac{1}{\sum_{n=1}^N|\psi_n|^4},\qquad 1\leq P\leq N.
$$
It estimates how many sites carry appreciable probability. Equal weight on $K$ sites and zero elsewhere gives $P=K$, irrespective of their <wave phases>. A single occupied site gives $P=1$; uniform probability on all sites gives $P=N$. These are exact constructed examples, not claims about particular <Hamiltonian operators>.
To prove the bounds, write $p_n=|\psi_n|^2$. Since $0\leq p_n\leq1$, $\sum_np_n^2\leq\sum_np_n=1$, giving $P\geq1$. The <Cauchy-Schwarz inequality> gives $1=(\sum_np_n)^2\leq N\sum_np_n^2$, giving $P\leq N$. Thus the measure has useful limits without assuming an exponential shape. Comparing $P$ as $N$ increases helps distinguish a state filling the sample from one occupying a fixed region, provided <energy> selection and disorder statistics are controlled.
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