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Betz V, Meiners M, Tomic I. Speed function for biased random walks with traps. Stat Probab Lett 2022. [DOI: 10.1016/j.spl.2022.109765] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 12/29/2022]
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Abstract
Conditioning independent and identically distributed bond percolation with retention parameter p on a one-dimensional periodic lattice on the event of having a bi-infinite path from -∞ to ∞ is shown to make sense, and the resulting model exhibits a Markovian structure that facilitates its analysis. Stochastic monotonicity in p turns out to fail in general for this model, but a weaker monotonicity property does hold: the average edge density is increasing in p.
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Croydon DA. Slow movement of a random walk on the range of a random walk in the presence of an external field. Probab Theory Relat Fields 2012. [DOI: 10.1007/s00440-012-0463-y] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
AbstractIn this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions ($$d\ge 5$$). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon as a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai’s regime. Via this approach, a corresponding aging result is also proved.
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