  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 maximum principle in the limit N to infinity
(where N is the number of types). In order to extend this variational principle
to a larger class of models, we consider here a family of reversible N by 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.
