  We study a class of Markovian systems of $N$ elements taking values in
$[0,1]$ that evolve in discrete time $t$ via randomized replacement rules based
on the ranks of the elements. These rank-driven processes are inspired by
variants of the Bak--Sneppen model of evolution, in which the system represents
an evolutionary 'fitness landscape' and which is famous as a simple model
displaying self-organized criticality. Our main results are concerned with
long-time large-$N$ asymptotics for the general model in which, at each time
step, $K$ randomly chosen elements are discarded and replaced by independent
$U[0,1]$ variables, where the ranks of the elements to be replaced are chosen,
independently at each time step, according to a distribution $\kappa_N$ on
$\{1,2,...,N\}^K$. Our main results are that, under appropriate conditions on
$\kappa_N$, the system exhibits threshold behaviour at $s^* \in [0,1]$, where
$s^*$ is a function of $\kappa_N$, and the marginal distribution of a randomly
selected element converges to $U[s^*, 1]$ as $t \to \infty$ and $N \to \infty$.
Of this class of models, results in the literature have previously been given
for special cases only, namely the 'mean-field' or 'random neighbour'
Bak--Sneppen model. Our proofs avoid the heuristic arguments of some of the
previous work and use Foster--Lyapunov ideas. Our results extend existing
results and establish their natural, more general context. We derive some more
specialized results for the particular case where K=2. One of our technical
tools is a result on convergence of stationary distributions for families of
uniformly ergodic Markov chains on increasing state-spaces, which may be of
independent interest.
