Math science technology

percolation threshold xc for a wire mesh is the ratio of non-blocked sites to the total number of sites when the electrical conductance vanishes. Watson and Leath (1974) found xc=0.59 using a mesh with 18 769 nodes (137 by 137). In class we looked at a mesh with 64 nodes (8 by 8) and found a similar result.

Please follow these steps to get your own guess for site percolation threshold on the square lattice with open boundary conditions on the 64 node mesh (8 by 8)! We can then combine our guesses to get a sense of the spread of the critical regime where conductance goes to zero. Please prepare and drop-off at the beginning of class on Thursday 29 November a report with your name and student identification number at the top documenting your estimate along the following lines:

1) Take four sheets of paper and draw on each sheet a square mesh with 64 squares. For each mesh, label the rows 1,2,3,4,5,6,7,8 and the columns a,b,c,d,e,f,g,h. Each square now has a unique label such as e1, f4, and g7.

We will use the first mesh to generate a random sequence of sites to block in the second, third and fourth meshes. In the squares of the first mesh, write the label of the square. Now slice up the first mesh into its squares, put the pieces in a hat, and draw squares in order without replacing them from the hat and record the sequence you obtain. Please include the sequence in your report.

2) Now using the random sequence of labels we got in (1), mark an x in the corresponding square of the second, third and fourth meshes. As you proceed, keep track of the number of sites that you have blocked and the size of the biggest cluster of non-blocked sites, where, we say two non-blocked sites are in the same cluster if they share the same row and differ by one column or share the same column and differ by one row (in other words they are nearest-neighbors on the mesh). Please include in your report the list of sizes of the biggest cluster of non-blocked sites and a plot showing the size of the biggest cluster as a function of the number of blocked sites.

3) As you draw site labels, please record the number of blocked sites Nh when there no longer exists a cluster of non-blocked sites connecting the left edge to the right edge. Please stop adding blocked sites to the second mesh after blocking the Nh-th site, highlight the last blocked site in some way, and include this mesh with the label “Critical Percolation Cluster (Horizontal)” in your report. Similarly make a note of the number of blocked sites Nv when there no longer exists a cluster of non-blocked sites connecting the top edge and bottom edge of the mesh. Please stop adding sites the third mesh after blocking the Nv-th site, highlight the last blocked site in some way, and include this mesh with the label “Critical Percolation Cluster (Vertical)” in your report.

Please include numbers Nh and Nv in your report.

4) Please make an estimate of xc using your data: compute and include in your report xc(h)=(N-Nh)/N and xc(v)=(N-Nv)/N, where, N=8×8=64 is the number of sites in the mesh. Please include also the estimate xc(1/2)=(N-N1/2)/N, where, N1/2 is the number of blocked sites when the size of the biggest cluster of non-blocked sites is closest to thirty-two (32) or one-half of the number of sites.

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