abstract:
In this talk we will present some results that we obtained with Frank den Hollander about a model of copolymer in an emulsion. This model was introduced by F. den Hollander and S. Whittington in their paper “Diffusion of a heteropolymer in a multi interface medium”. We will focus on the super-critical case (when one of the two types of droplets percolates), and more particularly on the phase transition between full-delocalization in the infinite cluster and partial-localization at the interface between the infinite cluster and the other solvent. We will see that the order of the transition is exactly 2.
(Joint work with Frank den Hollander.)
abstract:
We consider invasion percolation on a
rooted regular tree. The edges of the tree are assigned i.i.d. uniform (0,1)
random variables, and an infinite cluster is grown from the root by
successively adjoining the edge in the outer boundary of the cluster that
carries the smallest weight. It is known that, with probability 1,
\limsup_{m\to\infty} w_m = p_c, where w_m is the weight of the m-th edge
accepted in the invasion percolation cluster (IPC) and p_c is the
critical probability for ordinary percolation.
For the IPC, we identify the scaling behaviour of its r-point function for
any integer r >= 2, and of its volume both at and below a given height above
the root. In addition, we derive scaling estimates for simple random walk on
the IPC starting from the root. We find that, while the power laws of the
scaling are the same as for the incipient infinite cluster (IIC) for
ordinary percolation, the scaling functions differ. Thus, somewhat
surprisingly, IPC and IIC have different scaling limits.
(Joint work with Omer Angel, Jesse
Goodman and Gordon Slade.)