We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions d c. Behaviour at the critical point is non-universal in dimensions. Proliferation of the largest clusters, with fractal dimension 4, is associated with the breakdown of hyperscaling there when free boundary conditions are used. But when the boundary conditions are periodic, the maximal clusters have dimension D = 2d/3, and obey random-graph asymptotics. Universality is instead manifested at the pseudocritical point, where the failure of hyperscaling in its traditional form is universally associated with random-graph-type asymptotics for critical cluster sizes, independent of boundary conditions.
|Number of pages||27|
|Journal||Journal of Physics A: Mathematical and Theoretical|
|Publication status||Published - 11 May 2017|
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- critical phenomena
- finite-size scaling
- upper critical dimension
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Statistics and Probability
- Modelling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)