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We study discontinuous percolation transitions (PT) in the diffusion-limitedcluster aggregation model of the sol-gel transition as an example of realphysical systems, in which the number of aggregation events is regarded as thenumber of bonds occupied in the system. When particles are Brownian, in whichcluster velocity depends on cluster size as $v_s \sim s^{\eta}$ with$\eta=-0.5$, a larger cluster has less probability to collide with otherclusters because of its smaller mobility. Thus, the cluster is effectively moresuppressed in growth of its size. Then the giant cluster size increasesdrastically by merging those suppressed clusters near the percolationthreshold, exhibiting a discontinuous PT. We also study the tricriticalbehavior by controlling the parameter $\eta$, and the tricritical point isdetermined by introducing an asymmetric Smoluchowski equation.

Discontinuous percolation transitions in real physical systems