14,698 research outputs found
On the speed of once-reinforced biased random walk on trees
We study the asymptotic behaviour of once-reinforced biased random walk
(ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk
has a bias towards or away from the root. We prove that in the setting of
multiplicative once-reinforcement the ORbRW can be recurrent even when the
underlying biased random walk is ballistic. We also prove that, on
Galton-Watson trees without leaves, the speed is positive in the transient
regime. Finally, we prove that, on regular trees, the speed of the ORbRW is
monotone decreasing in the reinforcement parameter when the underlying random
walk has high speed, and the reinforcement parameter is small
Decoherence can be useful in quantum walks
We present a study of the effects of decoherence in the operation of a
discrete quantum walk on a line, cycle and hypercube. We find high sensitivity
to decoherence, increasing with the number of steps in the walk, as the
particle is becoming more delocalised with each step. However, the effect of a
small amount of decoherence is to enhance the properties of the quantum walk
that are desirable for the development of quantum algorithms. Specifically, we
observe a highly uniform distribution on the line, a very fast mixing time on
the cycle, and more reliable hitting times across the hypercube.Comment: (Imperial College London) 6 (+epsilon) pages, 6 embedded eps figures,
RevTex4. v2 minor changes to correct typos and refs, submitted version. v3
expanded into article format, extra figure, updated refs, Note on "glued
trees" adde
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