An upper bound on the size of avoidance couplings

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.

8 pages; minor formatting changes

References in corpus (4)

• Shy couplings, CAT(0) spaces, and the lion and man
• Rubber Bands, Pursuit Games and Shy Couplings
• Avoidance Coupling
• Monotonicity of Avoidance Coupling on $K_N$