Self-similar but not conformally invariant traces obtained by modified Loewner forces.

The two-dimensional Loewner exploration process is generalized to the case where the random force is self-similar with positively correlated increments. We model this random force by a fractional Brownian motion with Hurst exponent H≥1/2≡H_{BM}, where H_{BM} stands for the one-dimensional Brownian motion. By manipulating the deterministic force, we design a scale-invariant equation describing self-similar traces which lack conformal invariance.

Invasion percolation in short-range and long-range disorder background.

In the original invasion percolation model, a random number quantifies the role of necks, or generally the quality of pores, ignoring the structure of pores and impermeable regions (to which the invader cannot enter). In this paper, we investigate invasion percolation (IP), taking into account the impermeable regions, the configuration of which is modeled by ordinary and Ising-correlated site percolation (with short-range interactions, SRI), on top of which the IP dynamics is defined. We model the long-ranged correlations of pores by a random Coulomb potential (RCP).