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arXiv · 1712.00210

An upper bound on the size of avoidance couplings

Erik Bates·Lisa Sauermann
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Abstract

We show that a coupling of non-colliding simple random walkers on the complete graph on $n$ vertices can include at most $n - \log n$ walkers. This improves the only previously known upper bound of $n-2$ due to Angel, Holroyd, Martin, Wilson, and Winkler ({\it Electron.~Commun.~Probab.~18}, 2013). The proof considers couplings of i.i.d.~sequences of Bernoulli random variables satisfying a similar avoidance property, for which there is separate interest. Our bound in this setting should be closer to optimal.

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Erik Bates, Lisa Sauermann. 2017-12-01. An upper bound on the size of avoidance couplings. https://doi.org/10.1017/s0963548318000500

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