Linear-time generation of inhomogeneous random directed walks

Directed random walks in dimension two describe the diffusion dynamics of particles in a line. Through a well-known bijection, excursions, i.e. walks in the half-plane, describe families of "simply-generated" Galton--Watson trees. These random objects can be generated in linear time, through an algorithm due to Devroye, and crucially based on the fact that the steps of the walk form an exchangeable sequence of random variables. We consider here the random generation of a more general family of structures, in which the transition rates, instead of being fixed once and for all, evolve in time (but not in space). Thus, the steps are not exchangeable anymore. On one side, this generalises diffusion into time-dependent diffusion. On the other side, among other things, this allows to consider effects of excluded volume, for Galton--Watson trees arising from exploration processes on finite random graphs, both directed and undirected. In the directed version, a special case concerns partitions of N objects into M blocks (counted by Stirling numbers of the second kind), and rooted K-maps which are accessible from the root, which in turn are related to the uniform generation of random accessible deterministic complete automata. We present an algorithm, based on the block-decomposition of the problem, and a crucial procedure consisting of a generalised Devroye algorithm, for transition rates which are well-approximated by piecewise exponential functions. The achieved (bit-)complexity remains linear.