Efficiency and localisation for the first Dirichlet eigenfunction

Bounds are obtained for the efficiency or mean to peak ratio E(Ω) for the first Dirichlet eigenfunction (positive) for open, connected sets Ω 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 Ωn is any quadrilateral with perpendicular diagonals of lengths 1 and n respectively, then the sequence of first Dirichlet eigenfunctions localises, and E(Ωn)=O(n−2/3logn). This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.