We call network of automata a family of partially synchronised automata, i.e. a family of deterministic automata which are synchronised via shared letters, and evolve independently otherwise. We address the problem of uniform random sampling of words recognised by a network of automata. To that purpose, we define the reduced automaton of the model, which involves only the product of the synchronised part of the component automata. We provide uniform sampling algorithms which are polynomial with respect to the size of the reduced automaton, greatly improving on the best known algorithms. Our sampling algorithms rely on combinatorial and probabilistic methods and are of three different types: exact, Boltzmann and Parry sampling.

The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically “computes” the number 2/π ≐ 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithmic-series distributions. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities of success are expressible in terms of numbers such as . Generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to design Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function.

We present a general theorem that can be used to identify the limiting distribution for a class of combinatorial schemata. For example, many parameters in random mappings can be covered in this way. In particular, we can derive the limiting distribution of those points with a given number of total predecessors.

A direct generating function construction is given for cycles of combinatorial structures.