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Cut-off for lamplighter chains on tori: dimension interpolation and Phase transition

Abstract

Given a finite, connected graph \(\mathsf {G}\), the lamplighter chain on \(\mathsf {G}\) is the lazy random walk \(X^\diamond \) on the associated lamplighter graph \(\mathsf {G}^\diamond =\mathbb {Z}_2 \wr \mathsf {G}\). The mixing time of the lamplighter chain on the torus \(\mathbb {Z}_n^d\) is known to have a cutoff at a time asymptotic to the cover time of \(\mathbb {Z}_n^d\) if \(d=2\), and to half the cover time if \(d \ge 3\). We show that the mixing time of the lamplighter chain on \(\mathsf {G}_n(a)=\mathbb {Z}_n^2 \times \mathbb {Z}_{a \log n}\) has a cutoff at \(\psi (a)\) times the cover time of \(\mathsf {G}_n(a)\) as \(n \rightarrow \infty \), where \(\psi \) is an explicit weakly decreasing map from \((0,\infty )\) onto [1 / 2, 1). In particular, as \(a > 0\) varies, the threshold continuously interpolates between the known thresholds for \(\mathbb {Z}_n^2\) and \(\mathbb {Z}_n^3\). Perhaps surprisingly, we find a phase transition (non-smoothness of \(\psi \)) at the point \(a_*=\pi r_3 (1+\sqrt{2})\), where high dimensional behavior (\(\psi (a)=1/2\) for all \(a \ge a_*\)) commences. Here \(r_3\) is the effective resistance from 0 to \(\infty \) in \(\mathbb {Z}^3\).

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Correspondence to Jian Ding.

Additional information

Amir Dembo: Research partially supported by NSF Grants DMS-1106627 and DMS-1613091. Jian Ding: Research partially supported by NSF Grant DMS-1313596, DMS-1757479 and an Alfred Sloan fellowship. Jason Miller: Research partially supported by NSF Grant DMS-1204894. Part of the work was done when A.D. and J.D. participated in the MSRI program on Random Spatial Processes.

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Dembo, A., Ding, J., Miller, J. et al. Cut-off for lamplighter chains on tori: dimension interpolation and Phase transition. Probab. Theory Relat. Fields 173, 605–650 (2019). https://doi.org/10.1007/s00440-018-0883-4

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  • DOI: https://doi.org/10.1007/s00440-018-0883-4

Keywords

  • Wreath product
  • Lamplighter walk
  • Mixing time
  • Cutoff
  • Uncovered set

Mathematics Subject Classification

  • 60J10
  • 60D05
  • 37A25