| We are interested in the effects of various types of correlation on the structure of random fitness landscapes. In most models, we consider the space of individuals that have one of two alleles at each of n genetic loci; and we allow individuals to be either viable or inviable, thus considering binary fitness assignments. The random fitness landscape consists of the viable individuals, so it is a random subgraph of the discrete n-cube. We discuss the structure of the random fitness landscapes chosen according to these models, focusing on the connectivity of viable individuals. For most of the landscapes that we consider, we find that the landscape asymptotically almost surely undergoes a phase transition as n becomes large.; The first class of models that we consider is closely related to the theory of percolation in the n-cube. Traditionally, either edges or vertices are removed independently with a given probability. We generalize the vertex model to allow for phenotypic neutrality by first forming clusters via the edge model. We describe the mean field behavior for this model and conjecture a critical surface for percolation and find bounds for the subcritical and supercritical phases.; The second class of models that we study takes into account the possibility that certain combinations of traits are incompatible, and assumes that the viable individuals are those with no incompatible traits. These models are equivalent to random satisfiability models and the phase transition is characterized by the emergence of exponentially many individuals that satisfy the randomly generated problem.; We also obtain results for two other models: the discrete NK model and a model where random subsets of the continuous n-cube are chosen as a random fitness landscape on a phenotype space. |
