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On random quadratic forms: supports of potential local maxima

Part of: Asymptotic theory Combinatorial probability

Published online by Cambridge University Press:  16 January 2019

Boris Pittel
Boris Pittel*
Affiliation:
The Ohio State University
*
* Postal address: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA. Email address: bgp@math.ohio-state.edu
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Abstract

The selection model in population genetics is a dynamic system on the set of the probability distributions 𝒑=(p1,…,pn) of the alleles A1…,An, with pi(t+1) proportional to pi(t) multiplied by ∑jfi,jpj(t), and fi,j=fj,i interpreted as a fitness of the gene pair (Ai,Aj). It is known that 𝒑̂ is a locally stable equilibrium if and only if 𝒑̂ is a strict local maximum of the quadratic form 𝒑T𝒇𝒑. Usually, there are multiple local maxima and lim𝒑(t) depends on 𝒑(0). To address the question of a typical behavior of {𝒑(t)}, John Kingman considered the case when the fi,j are independent and [0,1]-uniform. He proved that with high probability (w.h.p.) no local maximum may have more than 2.49n1∕2 positive components, and reduced 2.49 to 2.14 for a nonbiological case of exponentials on [0,∞). We show that the constant 2.14 serves a broad class of smooth densities on [0,1] with the increasing hazard rate. As for a lower bound, we prove that w.h.p. for all k≤2n1∕3, there are many k-element subsets of [n] that pass a partial test to be a support of a local maximum. Still, it may well be that w.h.p. the actual supports are much smaller. In that direction, we prove that w.h.p. a support of a local maximum, which does not contain a support of a local equilibrium, is very unlikely to have size exceeding ⅔log2n and, for the uniform fitnesses, there are super-polynomially many potential supports free of local equilibriums of size close to ½log2n.

Keywords

Stable polymorphismrandom fitnessasymptotics

MSC classification

Primary: 34E05: Asymptotic expansions
Secondary: 60C05: Combinatorial probability
Type
Applied Probability Trust Lecture
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1113 - 1130
Copyright
Copyright © Applied Probability Trust 2018 

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