twice, and then we glue the $I$-vertex $i_l$ and $i_l+s$ in its pillar $M(Q)$, therefore, the degree of each $I$-vertex is multiple of four, which implies that the length of each subsequence in is multiple of four.

\[re2\] Note that in a characteristic sequence, subsequences in cannot intersect each other; for example, if we arrange two subsequences 1 0 0 -1 and 1 0 0 0 0 0 0 -1 in the characteristic sequence of length $4k=16$, then the following two structures are allowed: $$\begin{aligned}
\begin{array}{ll}
\text{0~{{\color{black}1}}~0~0~0~0~0~0~{{\color{black}-1}}~{{\color{blue}1}}~0~0~{{\color{blue}-1}}~0~0~0} & \text{(two subsequences are parallel)}~,\\
\text{0~{{\color{black}1}}~0~{{\color{blue}1}}~0~0~{{\color{blue}-1}}~0~{{\color{black}-1}}~0~0~0~0~0~0~0} & \text{(one is completely contained in another)}~;
\end{array}\end{aligned}$$ while $$\begin{aligned}
\begin{array}{ll}
\text{0~{{\color{black}1}}~0~0~0~0~0~{{\color{blue}1}}~{{\color{black}-1}}~0~{{\color{blue}-1}}~0~0~0~0~0} & \text{(two subsequences intersect each other)}
\end{array}\end{aligned}$$ is not.

Convergence of the largest eigenvalue of $A_T$ {#largest}
==============================================

Recall that due to in the previous section, we have the following: $$\begin{aligned}
{\mathbb{E}}(m_k(A_T))&=\sum_{i=0}^{k-1}\frac 1k \begin{pmatrix}
                               2k \\
                               i \\
                             \end{pmatrix}\begin{pmatrix}
                                            k \\
                                            i+1 \\
                                          \end{pmatrix}y_T^{2k-1-i}+o(1)\\
&\leq b(y_T)^k+o(1)\end{aligned}$$ for bounded $k$, where $b(y_T)$ is the value of $b$ in while substituting $y_T$ for $y$. The main point in this section is to improve this estimation in order to allow a [*growing*]{} $k$ such that: $$\begin{aligned}
\label{ae}
{\mathbb{E}}(m_k(A_T))&=\sum_{i=0}^{k-1}\frac 1k \begin{pmatrix}
                               2k \\
                               i \\
                             \end{pmatrix}\begin{pmatrix}
                                            k \\
                                            i+1 