Consider the spectrum of the adjacency matrix $A$ of the Cayley graph of the standard, 3x3x3 Rubik's cube generated with the usual quarter-turn and half-turn twists of each face (the Singmaster generators). The Perron-Frobenius theorem posits that the largest eigenvalue $\lambda_1$ of $A$ is one, and that the unique eigenvector corresponds to the uniform distribution over all scrambles or positions of the cube. But the spectral gap - the difference to the next largest eigenvalue $\lambda_2$ - is, as far as I know, presently unknown. This gap controls the mixing properties of the cube - i.e., how fast the cube can be scrambled. A larger gap may correspond to a faster mixing.
Let us now ponder how the spectral gap of the generalized $n\times n\times n$ cube (under similarly generalized Singmaster moves) grows with $n$. There may be a constant gap between $\lambda_2$ and $\lambda_1$ for all $n$, but if this gap vanishes as $n$ tends to infinity, we can say that the gap closes. This growth rate with $n$ may be an interesting object to consider.
Does the spectral gap of the adjacency matrix of the Cayley graph of the $n\times n \times n$ Rubik's cube tend to $0$ with $n$, and if so how quickly?
This is partly motivated because there are lovely videos of people scrambling and then solving very large cubes ($n=15$ or $17$...). I suspect that their initial scrambles may be far from sufficient to randomly permute their respective cubes, however, as I guess that larger cubes are in general much harder to scramble than smaller cubes. The number of Singmaster generators grows linearly with $n$, but the number of cubies/cells grows as $O(n^2)$. How fast does the mixing time grow? As $O(n^2)$, for example?
This is also partly motivated by curiosity about Motzkin and Fredkin spin chains, which are simple quantum many-body systems (QMBS's) with really nice combinatorial properties. In both the QMBS problems and the Rubik's cube, finding the gap as $n$ go off to infinity corresponds to finding the thermodynamic limit. Furthermore, the Motzkin and Fredkin chains violate the area law with respect to their entanglement entropy as their gap closes; does it make sense to ask about an area law for the Rubik's cube?
