A Monte Carlo simulation of a simple statistical physics model is decomposed onto a multi-processor (transputer) array in two essentially different ways: using ‘geometric’ and ‘algorithmic’ concurrency. The geometric decomposition (in which each processor handles a small sector of the physical system) is characterized by high efficiency in utilization of processors, and relative simplicity in programming. The algorithmic decomposition (in which each processor handles a small sub-task of the full algorithm, typically in a pipelined mode) is characterized by greater flexibility in the data-size (size of the physical system) and minimal memory requirements for a majority of the processors in the array. These assertions are made concrete in relation to our specific problem (a two-dimensional spin system simulation) which is, in many respects representative of a wide class of problems of interest to theoretical physicists.