Numerical estimates for arm exponents and the acceptance profile in two-dimensional invasion percolation

Abstract

The main object of this thesis is to numerically estimate some conjectured arm exponents when there exist a number of open paths and closed dual paths that extend to the boundary of different sizes of boxes centering at the origin in bond invasion percolation on a plane square lattice by Monte-Carlo simulations. The results turn out to be supportive of the conjectured value in some case. The numerical estimate for the acceptance profile of invasion percolation at the critical point is also obtained, which suggests a neighborhood in which the liminf and limsup of the acceptance profile might fall. An efficient algorithm to simulate invasion percolation and to find disjoint paths on most regular 2-dimensional lattices are discussed.