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We analyze the existence and the size of the giant component in thestationary state of a Markovian model for bipartite multigraphs, in which themovement of the edge ends on one set of vertices of the bipartite graph is azero-range process, the degrees being static on the other set. The analysis isbased on approximations by independent variables and on the results of Molloyand Reed for graphs with prescribed degree sequences. The possible types ofphase diagrams are identified by studying the behavior below the zero-rangecondensation point. As a specific example, we consider the so-called Evansinteraction. In particular, we examine the values of a critical exponent,describing the growth of the giant component as the value of the dilutionparameter controlling the connectivity is increased above the criticalthreshold. Rigorous analysis spans a large portion of the parameter space ofthe model exactly at the point of zero-range condensation. These results,supplemented with conjectures supported by Monte Carlo simulations, suggestthat the phenomenological Landau theory for percolation on graphs is not brokenby the fluctuations.Comment: RevTeX

Phase Transitions on Markovian Bipartite Graphs—an Application of the Zero-range Process