Cluster distribution in short- and long-range percolation

This dissertation addresses the cluster distribution in both short- and long-range percolation. The long-range version of the problem is also known as mean-field percolation. The form of the mean-field cluster distribution function that is derived is in excellent agreement with the Monte Carlo simulation data. A field-theoretical approach is then used to determine the explicit expression of the cluster distribution function, up to one-loop corrections. The analytical result is compared to the Monte Carlo data for both bond and site percolation in the vicinity of the percolation transition on two-dimensional square lattices, three-dimensional simple cubic lattices and higher-order hypercubic lattices up to seven dimensions with periodic boundary condition. Excellent agreement is achieved at the upper critical dimension; however, the Monte Carlo simulation data implies that one must include higher-order terms in the Landau theory in the case of low dimensional systems above the percolation transition.