A phase transition in the directional growth of lamplighter groups

Let $G$ be a finitely generated group with a surjective homomorphism $\pi\colon G\to\Z$. The directional growth spectrum $I(\beta)$ is the exponential rate of the number of elements of length at most $n$ lying over $\lfloor\beta n\rfloor$; its maximum is the growth rate. Wherever $I$ has been computed---free, hyperbolic and relatively hyperbolic groups, free abelian groups---it is strictly concave and real-analytic, as Perron--Frobenius theory dictates. We compute $I$ in closed form for the lamplighter groups $F\wr\Z$, $|F|=r+1$, with the standard generators, and obtain a different picture: $I$ is affine on $[-\beta^{*},\beta^{*}]$ and strictly concave beyond it, with a second-order transition at $\beta^{*}$. Elements conditioned to the affine phase backtrack macroscopically, with lamp density independent of $\beta$. The series $\sum_x s^{|x|}y^{\pi(x)}$ is rational, and the transition is an exchange of dominant singularities, the inner one independent of $y$. The peak of $I$ is $\log\omega$ with $\omega^{2}=\omega+r$, the central value $\log\rho$ with $\rho^{3}=\rho+r$; the base group is therefore co-amenable yet grows strictly slower, by an amount unbounded in $r$. For $r=1$ these constants are the golden ratio and the plastic number.