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Random Walks in a Sparse Random Environment
Probability · Submitted 20 Feb 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Interestingly, in the critical (recurrent) regime, our model generalizes Sinai's scaling of $(\log n)^2$ for the location of the random walk after $n$ steps to $(\log n)^\alpha,$ where $\alpha>0$ is a parameter determined by the distribution of the distance between two successive impurities.
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The abstract, sentence by sentence
We introduce random walks in a sparse random environment on $\mathbb Z$ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a "locally strong" perturbation of a simple random walk by a random potential induced by "rare impurities," which are randomly distributed over the integer lattice. Interestingly, in the critical (recurrent) regime, our model generalizes Sinai's scaling of $(\log n)^2$ for the location of the random walk after $n$ steps to $(\log n)^\alpha,$ where $\alpha>0$ is a parameter determined by the distribution of the distance between two successive impurities. Similar scaling factors have appeared in the literature in different contexts and have been discussed in [28] and [30].
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We introduce random walks in a sparse random environment on $\mathbb Z$ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. | contribution 1.00 | 0.92 | 0.23 | asserted 0.96 |
| s1 | The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. | contribution 0.60 | 0.42 | 0.06 | asserted 1.00 |
| s2 | More specifically, a random walk in a sparse random environment can be characterized as a "locally strong" perturbation of a simple random walk by a random potential induced by "rare impurities," which are randomly distributed over the integer lattice. | method 0.68 | 0.40 | 0.07 | asserted 0.99 |
| s3 | Interestingly, in the critical (recurrent) regime, our model generalizes Sinai's scaling of $(\log n)^2$ for the location of the random walk after $n$ steps to $(\log n)^\alpha,$ where $\alpha>0$ is a parameter determined by the distribution of the distance between two successive impurities. | result 0.99 | 0.91 | 0.43 | asserted 1.00 |
| s4 | Similar scaling factors have appeared in the literature in different contexts and have been discussed in [28] and [30]. | background 0.99 | 0.04 | 0.03 | asserted 1.00 |
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The paper
Original abstract
We introduce random walks in a sparse random environment on $\mathbb Z$ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a "locally strong" perturbation of a simple random walk by a random potential induced by "rare impurities," which are randomly distributed over the integer lattice. Interestingly, in the critical (recurrent) regime, our model generalizes Sinai's scaling of $(\log n)^2$ for the location of the random walk after $n$ steps to $(\log n)^\alpha,$ where $\alpha>0$ is a parameter determined by the distribution of the distance between two successive impurities. Similar scaling factors have appeared in the literature in different contexts and have been discussed in [28] and [30].
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