Diffusivity and Ballistic Behavior of Random Walk in Random Environment
Xiaoqin Guo
TL;DR
This thesis investigates the diffusive and ballistic behaviors of random walks in random environments, establishing new laws, invariance principles, and the Einstein relation in various complex settings.
Contribution
It introduces a conditional law of large numbers, regeneration structures, quenched invariance principles, and proves the Einstein relation for specific classes of environments.
Findings
Conditional law of large numbers for RWRE in Gibbsian environments
Quenched invariance principles for balanced elliptic environments
Proof of Einstein relation for balanced iid uniformly elliptic environments
Abstract
In this thesis, we study the diffusive and ballistic behaviors of random walk in random environment (RWRE) in an integer lattice with dimension at least 2. Our contributions are in three directions: a conditional law of large numbers and regeneration structures for RWRE in Gibbsian environments, quenched invariance principles for balanced elliptic (but non uniformly elliptic) environments, and a proof of the Einstein relation for balanced iid uniformly elliptic environments.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Theoretical and Computational Physics · Markov Chains and Monte Carlo Methods