The Philosophers' Process on ladder graphs

The Philosophers' Process is a Markovian version of the famous Dining Philosophers Problem introduced by Dijkstra as a model for resource sharing. It can be viewed as a reversible interacting particle system. This paper deals with the problem of computing its reversible measure in the case of a large but finite number of sites. This equilibrium measure is a Markov field on the underlying graph of interactions. When this graph has a linear structure (ladder graph), the reversible measure can be interpreted as the distribution of a stationary Markov chain. Its computation reduces to matrix products. This leads to explicit algorithms of computation in most cases of practical interest (grids, hypercubes, trees). Some examples of explicit formulae illustrate the results.