Abstract
AbstractWe consider the biased random walk on a critical Galton–Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.
Highlights
Biased random walks in inhomogeneous environments are a natural setting to witness trapping phenomena
In the case of supercritical Galton–Watson trees with leaves or the supercritical percolation cluster on Zd, for example, it has been observed that dead-ends found in the environment can, for suitably strong biases, create a sub-ballistic regime that is characteristic of trapping
To define m more precisely, suppose that (ξ(t))t≥0 is the symmetric Cauchy process, i.e., the Lévy process with Lévy measure given by μ((x, ∞)) = x−1/2 for x > 0, and set m(t) = max ξ(s), 0
Summary
Biased random walks in inhomogeneous environments are a natural setting to witness trapping phenomena. In the case of supercritical Galton–Watson trees with leaves (see [6,22,28]) or the supercritical percolation cluster on Zd (see [17]), for example, it has been observed that dead-ends found in the environment can, for suitably strong biases, create a sub-ballistic regime that is characteristic of trapping For both of these models, the distribution of the time spent in individual traps has polynomial tail decay, and this places them in the same universality class as the. Our main model—the biased random walk on critical Galton–Watson trees conditioned to survive—is presented, along with a summary of the results we are able to prove for it The arguments we apply for the one-dimensional model provide a useful template for the more complicated tree framework
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