Renormalization Group Approach to Percolation in Hierarchical Lattices

Percolation refers to an interesting class of problems related to the properties of disordered systems, usually formulated in terms of objects randomly placed on an underlying lattice or continuum. Despite the simplicity of the setup, most percolative systems undergo a phase transition from a disconnected state with many disjoint clusters to a state where a finite fraction of the lattice sites are connected to a single cluster. As in the case of thermodynamic phase transitions, power law dependencies generically near the critical percolation threshold. The origin of these dependencies can be understood through the lens of scaling and renormalization, and indeed many quantitative results can be acquired using these tools. In this paper we study the percolation problem on a hierarchical lattice, where exact results for the critical exponents can be obtained from a decimation procedure. We calculate analytic results for the full set of geometric critical exponents and confirm their consistency with simulation. Finally, we set up an interesting renormalization group for the conductivity of the system and use it to computationally extract the conductivity exponent t.