Directed percolation in the two-dimensional continuum.

Abstract

We report the first study of directed percolation in the continuum. The percolation threshold is found to be at ${B}_{c}$=5.0\ifmmode\pm\else\textpm\fi{}0.1, where ${B}_{c}$ is the average number of intersections per diode at the threshold. This is to be compared with ${B}_{c}$=3.2\ifmmode\pm\else\textpm\fi{}0.1 in the corresponding nondirected problem. It is found that the critical exponents of this system are ${\ensuremath{\nu}}_{\mathrm{para}}$=0.74\ifmmode\pm\else\textpm\fi{}0.05, ${\ensuremath{\nu}}_{\ensuremath{\perp}}$=0.46\ifmmode\pm\else\textpm\fi{}0.08, \ensuremath{\beta}=0.33\ifmmode\pm\else\textpm\fi{}0.07, and \ensuremath{\beta}'=(2.00 \ifmmode\pm\else\textpm\fi{}0.05)\ensuremath{\beta}. The good agreement with values found for directed lattices appears to be a confirmation of universality for these systems as well as a demonstration that geometrical and physical properties of directed systems in the continuum can be computed.