Abstract
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.
| Original language | English |
|---|---|
| Article number | 235001 |
| Number of pages | 27 |
| Journal | Journal of Physics A: Mathematical and Theoretical |
| Volume | 50 |
| Issue number | 23 |
| DOIs | |
| Publication status | Published - 11 May 2017 |
Bibliographical note
Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.Keywords
- clusters
- critical phenomena
- finite-size scaling
- hyperscaling
- percolation
- universality
- upper critical dimension
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Statistics and Probability
- Modelling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)