are the fitting parameters, which were determined in the following way. First we defined a function $f(E)$, which returns the residual sum of squares ($RSS$) of the linear regression $\ln\vert E-E_0^{(p)}\vert = \ln\vert a_1\vert +a_2 p $. Then, $E_0^{(\infty)}$ was chosen as the argument, which minimizes the function $f(E)$. The corresponding linear regression $\ln\vert E_0^{(\infty)}-E_0^{(p)}\vert = \ln\vert a_1\vert + a_2 p $ specifies the parameters $a_1$ and $a_2$. If considering another way, $E_0^{(\infty)}$ is such a value that the curve $\ln\vert E_0^{(\infty)}-E_0^{(p)}\vert$ is as close as possible to a line, where the closeness is measured by the $RSS$. Thus obtained parameters $E_0^{(\infty)}$, $a_1$, and $a_2$ are listed in the inset of figure \[Fig2\], where the dot-dashed line represents the estimate of the ground-state energy per bond of the quantum XY model on the Bethe lattice $E_0^{(\infty)}=\lim\limits_{p \rightarrow \infty} E_0^{(p)}=-1.08083446$.

Analogously, the ground-state energies $E_0^{(p)}$ of the Heisenberg model are plotted in figure \[Fig3\]. Again, rapid convergence of the energy to the asymptotic values is obvious from the data. Although we have not clarified the physical origin of the non-monotonic convergence (saw-like pattern) of $E_0^{(p)}$ yet, a detailed analysis indicates that the exponential fitting function in  can successfully describe the data, 