Phase diagram and correlation length bounds for Mandelbrot aerogels

Abstract

The authors study a variant of the Mandelbrot percolation process which is of current use as a model of aerogels. The model has two parameters: One of them, Q, is the usual multiscale parameter of the Mandelbrot percolation process and the other, p, is a Bernoulli percolation parameter that is reserved for 'the last step of the construction'. They investigate the phase diagram of this model in the (Q,p) plane. There are two phases, a sol phase and a gel phase, classified according to whether a limit of crossing probabilities vanishes or is nonzero. In the sol phase, they define a correlation length via the rate of decay of a rescaled connectivity function. They show that this length scale diverges at the phase boundary. Furthermore, they demonstrate that if the phase boundary is approached with Q fixed and p tending up to its critical value, PG(Q), then, up to logarithmic corrections, the divergence is at least as fast as mod PG-p mod -2dH/, where dH=2- mod log Q/log N mod can be identified as the Hausdorff dimension of the background medium.