1. Percolation on a 2D Square Lattice and Cluster Distributions Kalin Arsov Second Year Undergraduate Student University of Sofia, Faculty of Physics Adviser: Prof. Dr. Ana Proykova University of Sofia, Department of Atomic Physics

2. CONTENTS What is Percolation? Applications Types of Percolation Size and dimensional effects Clusters and their distributions Hoshen-Kopelman labeling algorithm Results Acknowledgements

3. What is Percolation? Passage of a substance through a medium Every day examples: Coffee making with a coffee percolator Infiltration of gas through gas masks Mathematical theory

4. Some Applications of Percolation Theory Physical Applications: Flow of liquid in a porous medium Conductor/insulator transition in composite materials Polymer gelation, vulcanization Non-Physical Applications: Social models Forest fires Biological evolution Spread of diseases in a population

5. Types of Percolation Depending on the relevant entities –site percolation –bond percolation Depending on the lattice type we consider percolation on –a square lattice –a triangular lattice –a honeycomb lattice –a bow-tie lattice

6. Types of Percolation Site percolation –The connectivity is defined for squares sharing sites (the substance passes through squares sharing sites) Bond percolation –the substance passes through adjacent bonds

7. Main Lattice Types square latticetriangular lattice honeycomb lattice bow-tie lattice

8. Size and Dimensional Effects Influence of the linear size of the system – increase of the spanning probability with the system size Influence of the dimensionality –smaller spanning probabilities for higher dimensions

9. Some Percolation Thresholds Lattice p c (site percolation) p c (bond percolation) cubic (body-centered)0.2460.1803 cubic (face-centered)0.1980.119 cubic (simple)0.31160.2488 diamond0.430.388 honeycomb0.69620.65271 4-hypercubic0.1970.1601 5-hypercubic0.1410.1182 6-hypercubic0.1070.0942 7-hypercubic0.0890.0787 square0.5927460.50000 triangular0.500000.34729

10. Clusters and Their Distributions A cluster is a group of two or more neighboring sites (bonds) sharing a side (vertex) Cluster-size distribution n s (n s : total number of s -clusters divided by L 2, L – the linear size of the system ) Near the critical point: n s  s ,  = 187/91 ≈ 2.05(5) If then

11. Clusters and Their Distributions Mean cluster size S(p) is the mean size of the cluster (without the percolating cluster, if it exists) to which a randomly chosen occupied site belongs or

12. Hoshen-Kopelman Labeling Algorithm Developed in 1976 by Hoshen and Kopelman Advantages –simple –fast –no need of huge data files (the lattice is created on the fly) –uses less memory than other algorithms –gives us the clusters’ sizes as a secondary effect!!!

13. Hoshen-Kopelman Algorithm Clever ideas: one line, instead of a matrix, is kept; cluster labels are divided into good – a positive number, denoting the size of the cluster and bad – negative, denoting the opposite to the good cluster label they are connected to. N(1) = 2; N(2) = 3 N(3) = 1 N(1) = 11; N(2) = -1 N(3) = -2

14. Results: Cluster-Size Distribution n s  ≈ 2.065: Excellent agreement with the theoretically predicted  ≈ 2.055

15. Results: Spanning Probability W(p) Spanning probability W(p) is the ratio: No_of_percolated_systems all_systems

16. Results: Mean Cluster Size S(p)

17. Acknowledgements Ministry of Education and Science: Grant for Stimulation of Research at the Universities, 2003 The members of the “Monte Carlo” group My parents

18. “Monte Carlo” Group Members Prof. Dr. Ana Proykova, Group leader M.Sc. Stoyan Pisov, Ass. Prof. M.Sc. Evgenia P. Daykova, Ph.D. Student B.Sc. Histo Iliev, Ph.D. Student Mr. Kalin Arsov, Undergraduate Student M.Sc. Ivan P. Daykov, Ph.D. Student (Cornell USA/UoS)

19. More Information http://cluster.phys.uni-sofia.bg:8080/kalin/