WIAS Preprint No. 886, (2003)

An asymptotic maximum principle for essentially linear evolution models



Authors

  • Baake, Ellen
  • Baake, Michael
  • Bovier, Anton
  • Klein, Markus

2010 Mathematics Subject Classification

  • 15A18 95D15 60J80.

Keywords

  • asymptotics of leading eigenvalue, reversibility, mutation-selection models, ancestral distribution, lumping

Abstract

Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional variational principle in the limit $N to infty$ (where $N$ is the number of types). In order to generalize these results, we consider here a large family of reversible $N times N$ matrices and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order $1/N$. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types.

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