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The genotype-fitness map (that is, the fitness landscape) is a key determinant of evolution, yet it has mostly been used as a superficial metaphor because we know little about its structure. This is now changing, as real fitness landscapes are being analysed by constructing genotypes with all possible combinations of small sets of mutations observed in phylogenies or in evolution experiments. In turn, these first glimpses of empirical fitness landscapes inspire theoretical analyses of the predictability of evolution. Here, we review these recent empirical and theoretical developments, identify methodological issues and organizing principles, and discuss possibilities to develop more realistic fitness landscape models.

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Numerical and analytic results for the exponent θ describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear Langevin equations. The models are parametrized by the dynamic roughness exponent β, with 0 < β < 1; for β = 1/2 the time evolution is Markovian. Using simulations of solid-on-solid models, of the discretized continuum equations as well as of the associated zero-dimensional stationary Gaussian process, we address two problems: The return of an initially flat interface, and the return to an initial state with fully developed steady state roughness. The two problems are shown to be governed by different exponents. For the steady state case we point out the equivalence to fractional Brownian motion, which has a return exponent θS = 1 − β. The exponent θ0 for the flat initial condition appears to be nontrivial. We prove that θ0 → ∞ for β → 0, θ0 ≥ θS for β < 1/2 and θ0 ≤ θS for β > 1/2, and calculate θ0,S perturbatively to first order in an expansion around the Markovian case β = 1/2. Using the exact result θS = 1 − β, accurate upper and lower bounds on θ0 can be derived which show, in particular, that θ0 ≥ (1 − β) 2 /β for small β.