Random walks on Galton-Watson trees with random conductances

We consider random walks on supercritical Galton-Watson trees without leaves and with random conductances. That is, given a Galton-Watson tree, we assign to each edge a positive random weight (conductance) and the random walk traverses an edge with a probability proportional to its conductance. On these trees, the random walk is transient and the distance of the walker to the root satisfies a law of large numbers with limit the speed of the walk. The speed is given as an expectation of ratios of effective conductances, which means that it cannot be calculated explicitly. In this thesis, we investigate how the speed of the random walk depends on the distribution of the environment. We first study the regularity of the speed as a function of distribution of the conductances. In particular, we investigate how the speed changes when the conductances of a positive fraction of edges tend to zero. We show that in this case the limit of the speed is smaller than the speed of the random walk as usually defined on trees with positive extinction probability. Following this, we focus on the behavior of the speed as a function of the offspring distribution. We show that the speed is a continuous function of the offspring law. Given a law of large numbers, the natural question arises whether the distance of the random walk to the root satisfies a central limit theorem. We prove a functional central limit theorem when the edges of the tree are assigned randomly uniformly elliptic conductances. Moreover, we investigate the effect of small conductances on the fluctuations of the random walk. In order to do this, we assign to a positive fraction of edges a small conductance ε. When ε → 0, we show that the variance is bounded away from zero, provided that the tree formed by larger conductances is supercritical. This implies that the slowdown induced by the small edges is not too strong. The proof relies on the existence of a renewal structure with good moments to decouple the increments of the random walk. To control the volatility for small ε, we need the moment bounds to hold uniformly in ε.