Accessibility percolation on <i>N</i>-ary trees

Abstract

Consider a rooted <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M1.png"/></alternatives></inline-formula>-ary tree. To each of its vertices, we assign an independent and identically distributed continuous random variable. A vertex is called accessible if the assigned random variables along the path from the root to it are increasing. We study the number <inline-formula><tex-math id="M2">\begin{document}$C_{N,\,k}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M2.png"/></alternatives></inline-formula> of accessible vertices of the first <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M3.png"/></alternatives></inline-formula> levels and the number <inline-formula><tex-math id="M4">\begin{document}$ C_N $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M4.png"/></alternatives></inline-formula> of accessible vertices in the <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M5.png"/></alternatives></inline-formula>-ary tree. As <inline-formula><tex-math id="M6">\begin{document}$ N\rightarrow \infty $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M6.png"/></alternatives></inline-formula>, we obtain the limit distribution of <inline-formula><tex-math id="M7">\begin{document}$C_{N,\, \beta N}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M7.png"/></alternatives></inline-formula> as <inline-formula><tex-math id="M8">\begin{document}$ \beta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M8.png"/></alternatives></inline-formula> varies from <inline-formula><tex-math id="M9">\begin{document}$ 0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M9.png"/></alternatives></inline-formula> to <inline-formula><tex-math id="M10">\begin{document}$ +\infty $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M10.png"/></alternatives></inline-formula> and the joint limiting distribution of <inline-formula><tex-math id="M11">\begin{document}$(C_{N}, C_{N,\,\alpha N+t \sqrt{\alpha N}})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M11.png"/></alternatives></inline-formula> for <inline-formula><tex-math id="M12">\begin{document}$0 < \alpha\leqslant 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M12.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M13.png"/></alternatives></inline-formula>. In this work, we also obtain a weak law of large numbers for the longest increasing path in the first <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M14.png"/></alternatives></inline-formula> levels of the <inline-formula><tex-math id="M15">\begin{document}$ N $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M15.png"/></alternatives></inline-formula>-ary tree for fixed <inline-formula><tex-math id="M16">\begin{document}$ N $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0059_M16.png"/></alternatives></inline-formula>.