Java Applet: RG

This applet demonstrates the renormalization group applied to site percolation in two dimensions. The 6 lattices pictured have dimensions 32x32, 16x16, 8x8, 4x4, 2x2, and 1x1. Occupied sites are colored red and empty sites are colored black. The 32x32 lattice represents the physical percolation system. The successive lattices 16x16 - 1x1 are derived from the 32x32 system by block transformations: the lattice is divided into 2x2 blocks of 4 sites; the super-site representing this block is occupied if the 2x2 block has a spanning cluster, and is empty if the 2x2 block does not have such a cluster. The "fixed point" p^* of the renormalization group is indicated on the 1x1 lattice: it is simply defined as the average occupancy of this single site.

The control panel allows you to adjust the following parameters:

• You can select the definition of a spanning cluster with the H and V toggle buttons:
  • The default is vertical spanning (V selected, H unselected)
  • Horizontal spanning if H is selected and V unselected
  • Horizontal and vertical spanning are both required if both H and V are selected
  • Either horizontal or vertical spanning if neither H nor V is selected
• The parameter p for the 32x32 lattice can be set using the horizontal slider
• The New Sample button generates a new sample by assigning a fresh set of random numbers to each site of the 32x32 lattice
• The Animate button is equivalent to pressing the New Sample button multiple times
• The Show Grids button colors occupied sites red and unoccupied sites white, and also draws gray grid lines on the lattice

Phenomena to observe:

• When p < p_c = 0.5927, the system should tend to the trivial fixed point p^* = 0
• When p > p_c = 0.5927, the system should tend to the trivial fixed point p^* = 1
• When p > p_c = 0.5927, the system is scale invariant and tends to the non-trivial fixed point p^* = p_c

Click for java source file: RG.java.
This applet uses the comphys package.

Maintained by: Dr. R.J. Gonsalves
Last updated: