- bi o . PE ] 1 3 N ov 2 00 3 DRAFT March 3 , 2008

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 lowdimensional variational principle in the limit N → ∞ (where N is the number of types). In order to generalize these results, we consider here a large family of reversible N × 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.