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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 @

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Asynchronously Parallelised Percolation on Distributed Machines

Gunnar Pruessner1 Nicholas R. Moloney2

1Mathematics Institute, University of Warwick, UK,

g.pruessner@warwick.ac.uk

2Loránd Eötvös University Budapest, Hungary

Warwick CSC @ Lunchtime Seminar, May 2007

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 1 / 24

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SLIDE 2

Outline

1

Introduction

2

Hoshen-Kopelman Algorithm

3

The Parallel Algorithm

4

Results

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 2 / 24

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SLIDE 3

Introduction

Illustration of the model

Sites occupied with probability ps Bonds active with probability pb

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

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SLIDE 4

Introduction

Illustration of the model

Sites occupied with probability ps Bonds active with probability pb

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

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SLIDE 5

Introduction

Illustration of the model

Sites occupied with probability ps Bonds active with probability pb

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

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SLIDE 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

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SLIDE 7

Introduction

Key features II

Cluster size distribution (density of s-clusters per site): P(s) = as−τG(b(p − pc)sσ) where τ = 187/91 and σ = 36/91.

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 5 / 24

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SLIDE 8

Introduction

Key features III

Crossing probability for different system sizes.

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 6 / 24

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 12

Introduction

A brief history I

Three dimensional polymers: Flory 1941 Mathematics: Hammersley and Broadbent 1954 pc = 1/2 in 2D bond percolation conjectured in 1955 θ(1/2) = 0 by Harris, 1960 pc = 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

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SLIDE 13

Introduction

A brief history II

Back on stage: Russo, and Seymour and Welsh, 1978 Kesten: pc = 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

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SLIDE 14

Hoshen-Kopelman Algorithm

Outline

1

Introduction

2

Hoshen-Kopelman Algorithm

3

The Parallel Algorithm

4

Results

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 10 / 24

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SLIDE 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

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SLIDE 16

Hoshen-Kopelman Algorithm

The Algorithm: Hoshen-Kopelman

Step by step

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

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SLIDE 17

Hoshen-Kopelman Algorithm

The Algorithm: Hoshen-Kopelman

Step by step

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

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SLIDE 18

Hoshen-Kopelman Algorithm

The Algorithm: Hoshen-Kopelman

Step by step

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

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SLIDE 19

Hoshen-Kopelman Algorithm

The Algorithm: Hoshen-Kopelman

Step by step

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

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SLIDE 20

Hoshen-Kopelman Algorithm

The Algorithm: Hoshen-Kopelman

Step by step

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

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SLIDE 21

Hoshen-Kopelman Algorithm

The Algorithm: Hoshen-Kopelman

Step by step

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 12 / 24

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SLIDE 22

The Parallel Algorithm

Outline

1

Introduction

2

Hoshen-Kopelman Algorithm

3

The Parallel Algorithm

4

Results

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 13 / 24

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 26

The Parallel Algorithm

The Algorithm

Features

Huge lattices: 106 realisations of 30 000 × 30 000 or a single lattice 22.2 · 106 × 22.2 · 106 (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

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SLIDE 27

Results

Outline

1

Introduction

2

Hoshen-Kopelman Algorithm

3

The Parallel Algorithm

4

Results

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 18 / 24

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SLIDE 28

Results

Results

Cluster Size Distribution

ps = 0.59274621, free boundaries, 30 000 × 30 000 sites

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 19 / 24

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SLIDE 29

Results

Results

Cluster Size Distribution

Site percolation, histogram normalised, shifted and binned ps = 0.59274621, free boundaries, L = 1 000 and 30 000

g.pruessner@warwick.ac.uk (WMI) Parallelised Percolation Warwick, 05/2007 20 / 24

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SLIDE 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

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SLIDE 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

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SLIDE 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

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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