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The Directed Edge Reinforced Random Walk: The Ant Mill Phenomenon | | Dirk Erhard
; Tertuliano Franco
; Guilherme Reis
; | | Date: |
17 Nov 2019 | | Abstract: | We define here a extit{directed edge reinforced random walk} on a connected
locally finite graph. As the name suggests, this walk keeps track of its past,
and gives an exponential bias, proportional to the number of crossings, to
directed edges already crossed before. The model is inspired by the so called
extit{Ant Mill phenomenon}, in which a group of army ants forms a
continuously rotating circle until they die of exhaustion. For that reason we
refer to the walk defined in this work as the extit{Ant RW}. Our main result
justifies this name. Namely, we will show that on any finite graph which is not
a tree, and on $mathbb Z^d$ with $dgeq 2$, the Ant RW almost surely gets
eventually trapped into some directed cycle which will be followed forever. In
the case of $mathbb Z$ we show that the Ant RW eventually escapes to infinity
and satisfies a law of large number with a random limit which we explicitly
identify. | | Source: | arXiv, 1911.7295 | | Services: | Forum | Review | PDF | Favorites | |
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