点灯人群定向增长中的相变
A phase transition in the directional growth of lamplighter groups
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中文总结 AI 辅助
研究点灯人群定向增长谱\(I(\beta)\)，通过计算得出其在\([-\beta^{*},\beta^{*}]\)上仿射、外严格凹且有二阶相变，元素有回溯现象，级数有理，还给出了相关常数，揭示了基础群增长特性与\(r\)的关系。
中文摘要 AI 辅助
设\(G\)为有限生成群，有满同态\(\pi:G\to\mathbb{Z}\)。定向增长谱\(I(\beta)\)是长度至多为\(n\)且位于\(\lfloor\beta n\rfloor\)之上元素数量的指数率，其最大值为增长率。在已计算\(I\)的自由群、双曲群、相对双曲群、自由阿贝尔群中，\(I\)严格凹且实解析。本文计算了具有标准生成元的点灯人群\(F\wr\mathbb{Z}\)（\(|F| = r + 1\)）的\(I\)，发现其在\([-\beta^{*},\beta^{*}]\)上仿射，在其外严格凹，在\(\beta^{*}\)处有二阶相变。处于仿射阶段的元素宏观上会回溯，灯密度与\(\beta\)无关。级数\(\sum_x s^{|x|}y^{\pi(x)}\)是有理的，相变是主导奇点的交换，内部奇点与\(y\)无关。\(I\)的峰值为\(\log\omega\)，其中\(\omega^{2}=\omega + r\)，中心值为\(\log\rho\)，其中\(\rho^{3}=\rho + r\)；基础群是余顺从的，但增长严格更慢，差值在\(r\)中无界。对于\(r = 1\)，这些常数是黄金比例和塑性数。
英文摘要
Let $G$ be a finitely generated group with a surjective homomorphism $π\colon G\to\Z$. The directional growth spectrum $I(β)$ is the exponential rate of the number of elements of length at most $n$ lying over $\lfloorβn\rfloor$; 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 $[-β^{*},β^{*}]$ and strictly concave beyond it, with a second-order transition at $β^{*}$. Elements conditioned to the affine phase backtrack macroscopically, with lamp density independent of $β$. The series $\sum_x s^{|x|}y^{π(x)}$ is rational, and the transition is an exchange of dominant singularities, the inner one independent of $y$. The peak of $I$ is $\logω$ with $ω^{2}=ω+r$, the central value $\logρ$ with $ρ^{3}=ρ+r$; 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.
