This article presents the long-term behaviour analysis of Stochastic Diffusion Search (SDS), a distributed agent-based system for best-fit pattern matching. SDS operates by allocating simple agents into different regions of the search space. Agents independently pose hypotheses about the presence of the pattern in the search space and its potential distortion. Assuming a compositional structure of hypotheses about pattern matching agents perform an inference on the basis of partial evidence from the hypothesised solution. Agents posing mutually consistent hypotheses about the pattern support each other and inhibit agents with inconsistent hypotheses. This results in the emergence of a stable agent population identifying the desired solution. Positive feedback via diffusion of information between the agents significantly contributes to the speed with which the solution population is formed. The formulation of the SDS model in terms of interacting Markov Chains enables its characterisation in terms of the allocation of agents, or computational resources. The analysis characterises the stationary probability distribution of the activity of agents, which leads to the characterisation of the solution population in terms of its similarity to the target pattern.