三角形引导渗流基的拉伸指数渐近性
Stretched exponential asymptotics for bases of triangular bootstrap percolation
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中文总结 AI 辅助
研究有限三角形网格上引导渗流的大小为\(n\)的基，通过证明计数基的生成函数满足代数微分方程，分析得出其基数\(t_n\)有拉伸指数渐近行为\(t_n \sim c n!e^{\sqrt{12n}}n^{5/12}\)，这是模式避免排列研究中首个此类例子。
中文摘要 AI 辅助
本文研究边长为\(n\)的有限三角形网格\(\mathfrak{T}_n\)上的引导渗流过程。若从子集\(\eta\)开始的最终构型是整个网格\(\mathfrak{T}_n\)，则称\(\eta\)渗流。大小为\(n\)的基是\(\mathfrak{T}_n\)中渗流的最小基数点子集。首先证明计数基的生成函数满足代数微分方程，通过分析其修正方程，证明大小为\(n\)的基数\(t_n\)呈现拉伸指数渐近行为，即\(t_n \sim c n!e^{\sqrt{12n}}n^{5/12}\)，\(c>0\)。这些基与避免模式\((12, 12)\)和\((231, 312)\)的\(3 -\)排列双射，是模式避免排列研究中首个被证明的渐近拉伸指数例子。
英文摘要
In this paper, we study a bootstrap percolation process on the finite triangular grid $\mathfrak{T}_n$ of side length $n$. We say that a subset $η$ of points in $\mathfrak{T}_n$ percolates if the final configuration, starting from $η$, is the whole grid $\mathfrak{T}_n$. A basis of size $n$ is then a subset of points of $\mathfrak{T}_n$ of minimum cardinality which percolates. In this paper, we first prove that the generating function counting bases satisfies an algebraic differential equation. Then, by analysing a modified version of this equation, we prove that the number $t_n$ of bases of size $n$ exhibits a stretched exponential asymptotic behaviour. More precisely, we show that $t_n \sim c n!e^{\sqrt{12n}}n^{5/12}$, for some constant $c>0$. These bases were recently shown by the second author to be in bijection with $3$-permutations avoiding the patterns $(12, 12)$ and $(231, 312)$, so this represents to our knowledge the first proven example of an asymptotic stretched exponential appearing in the study of pattern avoiding permutations.
