Biased randomly trapped random walks and applications to random walks on Galton-Watson trees

In this thesis we study biased randomly trapped random walks. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive. This application was initially considered model in its own right. We prove conditions under which the biased randomly trapped random walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. We also study the regime in which the walk is sub-ballistic; in this case we prove convergence to a stable subordinator. Furthermore, we study the fluctuations of the walk in the ballistic but sub-diffusive regime. In this setting we show that the walk can be properly centred and rescaled so that it converges to a stable process. The biased random walk on the subcritical GW-tree conditioned to survive fits suitably into the randomly trapped random walk model; however, due to a lattice effect, we cannot obtain such strong limiting results. We prove conditions under which the walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. In these cases the trapping is weak enough that the lattice effect does not have an influence; however, in the sub-ballistic regime it is only possible to obtain converge along specific subsequences. We also study biased random walks on infinite supercritical GW-trees with leaves. In this setting we determine critical upper and lower bounds on the bias such that the walk satisfies a quenched invariance principle

Ability Tracking and Social Capital in China's Rural Secondary School System

The goal of this paper is describe and analyze the relationship between ability tracking and student social capital, in the context of poor students in developing countries. Drawing on the results from a longitudinal study among 1,436 poor students across 132 schools in rural China, we find a significant lack of interpersonal trust and confidence in public institutions among poor rural young adults. We also find that there is a strong correlation between ability tracking during junior high school and levels of social capital. The disparities might serve to further widen the gap between the relatively privileged students who are staying in school and the less privileged students who are dropping out of school. This result suggests that making high school accessible to more students would improve social capital in the general population

Central limit theorems for biased randomly trapped random walks on Z

We prove CLTs for biased randomly trapped random walks in one dimension. By considering a sequence of regeneration times, we will establish an annealed invariance principle under a second moment condition on the trapping times. In the quenched setting, an environment dependent centring is necessary to achieve a central limit theorem. We determine a suitable expression for this centring. As our main motivation, we apply these results to biased walks on subcritical Galton–Watson trees conditioned to survive for a range of bias values

A quenched central limit theorem for biased random walks on supercritical Galton-Watson trees

In this note, we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton-Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves is considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp