Percolation in the Miller-Abrahams random resistor network

The Miller-Abrahams random resistor network is used to study electron transport in amorphous solids. This resistor network is given by the complete random graph built on a marked homogeneous Poisson point process on R^d and each edge {x,y} is associated to a filament with conductance depending on the temperature, the distance between the points x,y and their associated marks. In this thesis we consider the subgraph containing only edges with lower bounded conductances and, using the method of randomized algorithms developed by Duminil-Copin et al. and the renormalization argument proposed by Grimmett and Marstrand, we analyze the connection probabilities and the left-right crossings in appropriate regimes. These percolation properties are key ingredients for understanding the asymptotic behavior at low temperature of the effective conductivity of the Miller-Abrahams random resistor network. Joint work with Alessandra Faggionato (Sapienza University, Rome).