  Bounds are obtained for the efficiency or mean to peak ratio $E(\Omega)$ for
the first Dirichlet eigenfunction (positive) for open, connected sets $\Omega$
with finite measure in Euclidean space $\R^m$. It is shown that (i)
localisation implies vanishing efficiency, (ii) a vanishing upper bound for the
efficiency implies localisation, (iii) localisation occurs for the first
Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex
and planar sets, (iv) if $\Omega_n$ is any quadrilateral with perpendicular
diagonals of lengths $1$ and $n$ respectively, then the sequence of first
Dirichlet eigenfunctions localises, and $E(\Omega_n)=O\big(n^{-2/3}\log
n\big)$. This disproves some claims in the literature. A key technical tool is
the Feynman-Kac formula.
