Bowditch, Adam (2017) Biased randomly trapped random walks and applications to random walks on Galton-Watson trees. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Random walks (Mathematics), Trees (Graph theory)
Official Date:	July 2017
Dates:	Date Event July 2017 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Croydon, David A.
Sponsors:	Engineering and Physical Sciences Research Council
Format of File:	pdf
Extent:	v, 194 leaves : illustrations, charts
Language:	eng

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