Through the consideration of a stochastic matching model dependent on a probability function built from the Heron's formula, we analyze the emergence and the dynamics of short agrifood sale circuits in form of time-evolving random hypergraphs. These marketing circuits, which are used as a supportive policy for promoting local food consumption, typify short distribution channels. Although bipartite matching can be easily derived from the framework, we move our focus on the matching of triplets of players representing buyers, retailers and sellers. Their conditional pairings with respect to standard and social preferences are both taken into consideration. Our results show that the emergence of short food supply chains is triggered by a three-dimensional stochastic matching mechanism. Their time evolution is found to be governed by both stable and unstable dynamics, the latter being subject to bounded antiperiodic oscillations. Via the use of a Poisson process, we then redirect our interest toward spatial randomness and the number of circuits attainable in a defined territory. The model unveils a restricted spread of short circuits over the entire territory, which, in expectation, can only be a partial substitute for long supply chains. The outcomes show consistency with the agribusiness patterns currently observable in France.