Note on the lamp lighting problem

Simulations and the Lamplighter group

We introduce a notion of "simulation" for labelled graphs, in which edges of the simulated graph are realized by regular expressions in the simulating graph, and prove that the tiling problem (aka "domino problem") for the simulating graph is at least as difficult as that for the simulated graph. We apply this to the Cayley graph of the "lamplighter group" $L=\mathbb Z/2\wr\mathbb Z$, and more generally to "Diestel-Leader graphs". We prove that these graphs simulate the plane, and thus deduce that the seeded tiling problem is unsolvable on the group $L$. We note that $L$ does not contain any p…

任意维度下三角plaquette模型的超Arrhenius弛豫

Super-Arrhenius relaxation of the triangular plaquette model in any dimension

研究三角plaquette模型的Glauber动力学弛豫时间，证明在任意维度d≥2下弛豫时间介于e^{β^2/C}和Ce^{e^{Cβ}}之间，验证了Newman和Moore的猜想，并揭示了脆弱玻璃现象。