### *New title* Hamiltonian cycles on Ammann Beenker tilings

Ergodic Theory and Dynamical Systems Seminar

9th December 2021, 3:15 pm – 4:15 pm

Fry Building, 2.04

Ammann-Beenker tilings are infinite aperiodic tilings with long-range order; the slightly lesser-known cousins of the Penrose tilings. In this talk I will outline a proof by myself and J. Lloyd (Geneva) of the following statement: for any set of vertices V on the Ammann-Beenker tiling, there exists a set of vertices U with V as a subset, where U admits a Hamiltonian cycle. I will demonstrate how to construct such cycles for the infinite tilings.

Proving the existence of a Hamiltonian cycle on an arbitrary graph lies in the complexity class NP-Complete. As a result we might expect a range of previously intractable problems on the (uncountably infinite) set of Ammann-Beenker tilings to now become tractable. Our result immediately implies the existence of a number of models of interest to physicists, including fully packed loops and perfect dimer and trimer matchings. I will outline some intriguing properties of the dimer matchings, including a statistical repulsion between excitations.

References:

[1] J. Lloyd, S. Biswas, S. H. Simon, S. A. Parameswaran, and F. Flicker, Statistical mechanics of dimers on quasiperiodic tilings, arXiv: 2103.01235

[2] F. Flicker, S. H. Simon, and S. A. Parameswaran, Classical dimers on Penrose tilings, Physical Review X 10, 011005 (2020).

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