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Asynchronously Parallelised Percolation on Distributed Machines Gunnar Pruessner 1 Nicholas R. Moloney 2 1 Mathematics Institute, University of Warwick, UK, g.pruessner@warwick.ac.uk 2 Lornd Etvs University Budapest, Hungary Warwick CSC @

  1. Asynchronously Parallelised Percolation on Distributed Machines Gunnar Pruessner 1 Nicholas R. Moloney 2 1 Mathematics Institute, University of Warwick, UK, g.pruessner@warwick.ac.uk 2 Loránd Eötvös University Budapest, Hungary Warwick CSC @ Lunchtime Seminar, May 2007 g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 1 / 24

  2. Outline Introduction 1 Hoshen-Kopelman Algorithm 2 The Parallel Algorithm 3 Results 4 g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 2 / 24

  3. Introduction Illustration of the model Sites occupied with Bonds active with probability p s probability p b Definition of a cluster sites connected through occupied sites and active bonds g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 3 / 24

  4. Introduction Illustration of the model Sites occupied with Bonds active with probability p s probability p b Definition of a cluster sites connected through occupied sites and active bonds g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 3 / 24

  5. Introduction Illustration of the model Sites occupied with Bonds active with probability p s probability p b Definition of a cluster sites connected through occupied sites and active bonds g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 3 / 24

  6. Introduction Key features I Order parameter θ : fraction in the “infinite” cluster In 2D: β = 5 / 36 g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 4 / 24

  7. Introduction Key features II Cluster size distribution (density of s -clusters per site): P ( s ) = as − τ G ( b ( p − p c ) s σ ) where τ = 187 / 91 and σ = 36 / 91. g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 5 / 24

  8. Introduction Key features III Crossing probability for different system sizes. g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 6 / 24

  9. Introduction Why percolation? Long history (Flory 1941) Renaissance because of Conformal Field Theory (Langlands et al. 1992, Cardy 1992) Numerics as a guide ◮ Study leading to Conformal Field Theory ◮ Multiple spanning clusters Open questions ◮ Higher dimensions ◮ universality ◮ relation lattice ← → Conformal Field Theory g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 7 / 24

  10. Introduction Why percolation? Long history (Flory 1941) Renaissance because of Conformal Field Theory (Langlands et al. 1992, Cardy 1992) Numerics as a guide ◮ Study leading to Conformal Field Theory ◮ Multiple spanning clusters Open questions ◮ Higher dimensions ◮ universality ◮ relation lattice ← → Conformal Field Theory g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 7 / 24

  11. Introduction Why percolation? Long history (Flory 1941) Renaissance because of Conformal Field Theory (Langlands et al. 1992, Cardy 1992) Numerics as a guide ◮ Study leading to Conformal Field Theory ◮ Multiple spanning clusters Open questions ◮ Higher dimensions ◮ universality ◮ relation lattice ← → Conformal Field Theory g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 7 / 24

  12. Introduction A brief history I Three dimensional polymers: Flory 1941 Mathematics: Hammersley and Broadbent 1954 p c = 1 / 2 in 2D bond percolation conjectured in 1955 θ ( 1 / 2 ) = 0 by Harris, 1960 p c = 1 / 2 tackled by Sykes and Essam, 1963 “Dormant state” Details: Grimmet, Percolation , 2000 g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 8 / 24

  13. Introduction A brief history II Back on stage: Russo, and Seymour and Welsh, 1978 Kesten: p c = 1 / 2, 1980 Uniqueness of infinite cluster: Newman and Schulman, 1981 Renaissance because of Conformal Field Theory for crossing probabilities: Langlands et al. 1992, Cardy 1992 Multiple spanning clusters: Hu and Lin 1996, Aizenman 1997, Cardy 1998 Percolation is SLE with κ = 6, Smirnov 2001 g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 9 / 24

  14. Hoshen-Kopelman Algorithm Outline Introduction 1 Hoshen-Kopelman Algorithm 2 The Parallel Algorithm 3 Results 4 g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 10 / 24

  15. Hoshen-Kopelman Algorithm The Algorithm: Hoshen-Kopelman Overview scan row by row label clusters using list of labels remember configuration of “active” sites (Hoshen and Kopelman, 1976) g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 11 / 24

  16. Hoshen-Kopelman Algorithm The Algorithm: Hoshen-Kopelman Step by step g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

  17. Hoshen-Kopelman Algorithm The Algorithm: Hoshen-Kopelman Step by step g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

  18. Hoshen-Kopelman Algorithm The Algorithm: Hoshen-Kopelman Step by step g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

  19. Hoshen-Kopelman Algorithm The Algorithm: Hoshen-Kopelman Step by step g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

  20. Hoshen-Kopelman Algorithm The Algorithm: Hoshen-Kopelman Step by step g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

  21. Hoshen-Kopelman Algorithm The Algorithm: Hoshen-Kopelman Step by step g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

  22. The Parallel Algorithm Outline Introduction 1 Hoshen-Kopelman Algorithm 2 The Parallel Algorithm 3 Results 4 g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 13 / 24

  23. The Parallel Algorithm The parallel algorithm Border Preparation (N. R. Moloney and G.P . 2003) scan around the boundary move roots into boundary g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 14 / 24

  24. The Parallel Algorithm The parallel algorithm Border Preparation – Comparison (N. R. Moloney and G.P . 2003) scan around the boundary move roots into boundary g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 15 / 24

  25. The Parallel Algorithm The Algorithm Gluing Small patches produced at slave nodes (asynchronous) Assembly at master nodes: ◮ Shift labels for uniqueness ◮ Redirect roots — larger cluster prevails related: (Rapaport, 1992) g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 16 / 24

  26. The Parallel Algorithm The Algorithm Features Huge lattices: 10 6 realisations of 30 000 × 30 000 or a single lattice 22 . 2 · 10 6 × 22 . 2 · 10 6 (hierarchical nodes). Flexibility (boundary conditions, aspect ratios) Reduced correlations by rotating, mirroring and permuting Asynchronous Minimal hardware (CPU, memory, network) g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 17 / 24

  27. Results Outline Introduction 1 Hoshen-Kopelman Algorithm 2 The Parallel Algorithm 3 Results 4 g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 18 / 24

  28. Results Results Cluster Size Distribution p s = 0 . 59274621, free boundaries, 30 000 × 30 000 sites g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 19 / 24

  29. Results Results Cluster Size Distribution Site percolation, histogram normalised, shifted and binned p s = 0 . 59274621, free boundaries, L = 1 000 and 30 000 g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 20 / 24

  30. Results Results Winding on a Torus Probability of winding clusters with particular winding numbers g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 21 / 24

  31. Results Results Winding on a Torus bond percolation, probability of ( 1 , 2 ) winding circles numerical, line analytical (Pinson, 1994) (G. P . and N. R. Moloney 2004) g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 22 / 24

  32. Results Summary Very large systems Very flexible (reduced correlations) Minimal hardware Asynchronous Numerical test of CFT results New open questions (exotic clusters, universality) g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 23 / 24

  33. Results Acknowledgements This work has been carried out at Imperial College London Supported partly by EPSRC (NRM + GP), the Beit fellowship (NRM), the NSF (GP) and the Humboldt Foundation (GP) Thanks to A. Thomas, K. Christensen, P . Anderson, M. Kaulke, O. Kilian, D. Moore, B. Maguire, and D. Erickson for their help g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 24 / 24

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