On random quadratic forms: supports of potential local maxima

TL;DR

This paper extends Kingman's analysis of the support sizes of potential local maxima in quadratic forms, establishing new bounds and asymptotic behaviors for various density classes on [0,1], including exponential and uniform distributions.

Contribution

It broadens the class of densities for which the support size bounds are known, providing both upper and lower bounds and analyzing the asymptotic distribution of support sizes.

Findings

The constant 2.14 applies to a broad class of densities on [0,1].

Support sizes of potential maxima are asymptotic to their expected values.

Large supports exceeding 2 log_2 n are very unlikely.

Abstract

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, $2.49 n^{1/2}$ at most, and for a non-biological case of independent exponentials on $[0,\infty)$ 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, $L_n\ge 2n^{1/3}$ at least. Our argument shows also that the random counts of potential maxima supports, whose sizes range from $2$ to $\lceil 2n^{1/3}\rceil$, 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 $2\log_2 n$.