The critical phase of bond percolation on the random growing tree is examined. It is shown that the root cluster grows with the system size $N$ as $N^\psi$ and the mean number of clusters with size $s$ per node follows a power function $n_s \propto s^{ -\tau}$ in the whole range of open bond probability $p$. The exponent $\tau$ and the fractal exponent $\psi$ are also derived as a function of $p$ and the degree exponent $\gamma$, and are found to satisfy the scaling relation $\tau=1+\psi^{ -1}$. Numerical results with several network sizes are quite well fitted by a finite size scaling for a wide range of $p$ and $\gamma$, which gives a clear evidence for the existence of a critical phase.
Categories: cond-mat.dis-nn physics.soc-ph