Random mappings with one attracting center

A random mapping $T$ of the set $\{a_0,a_1,\dots,a_n\}$ into itself is determined by the following requirements: 1) images of the points $a_i$, $0\le i\le n$, are chosen at random and independently; 2) for any $i$
$$ \mathbf P(Ta_i=a_0)=\lambda/(n+\lambda),\quad\lambda\ge1;\quad\mathbf P(Ta_i=a_j)=1/(n+\lambda),\quad1\le j\le n. $$
Vertex $a_0$ is called an attracting center of weight $\lambda$. The graph component of mapping $T$ containing the center, the cycle belonging to it and all its vertices are called principal, and all the rest components, cycles and vertices are called free.
Limit distributions of various characteristics of random mappings with one attracting center of weight $\lambda$ are studied in this paper. For example, it is shown that if $\lambda$ varies an $n\to\infty$ so that $\lambda/\sqrt n\to\infty$ but $\lambda/n\to0$ the distribution of the random variable $\lambda^2\xi_n(\lambda)/n^2$ where $\xi_n(\lambda)$ is the number of free vertices converges to the $\chi^2$-distribution with one degree of freedom.