On random quadratic forms: supports of potential local maxima

In the late eighties John Kingman studied the problem of maxima of a quadratic form, with independent, uniformly distributed, coefficients, on a simplex of growing dimension $n$. In particular, he proved that the largest support size (cardinality) $L_n$ of a potential local maximum is, in probability, $2.49n^{1\mathrm{/}2}$ at most, and for a non-biological case of independent exponentials on $[0,\mathrm{\infty })$ he reduced the constant to $2.14$. In this paper we show that the constant $2.14$ serves a broad class of the densities on $[0,1]$, which includes a linear non-decreasing (whence uniform) density and the exponential density conditioned on $[0,1]$. We also prove a qualitatively matching lower bound: in probability, $L_n\ge 2n^{1\mathrm{/}3}$ at least. Our argument shows also that the random counts of potential maxima supports, whose sizes range from $2$ to $\lceil 2n^{1\mathrm{/}3}\rceil$, are asymptotic to their expected values. Finally we show that a support of a local maximum, that does not contain a support of a local equilibrium, is very unlikely to have size exceeding $2{\log }_{2}n$.