graph) in the vicinity of the phase transitions with respect to the magnetic field $h$ for $p \in \{4,5, \dots, 10\}$. The inset shows the detailed zoomed-in behavior for higher values of $p$.[]{data-label="Fig4"}](Fig4.pdf){width="70.00000%"}

![The phase transition field $h_t^{(p)}$ of the TFIM with respect to the lattice parameter $p$. The horizontal dot-dashed line represents the estimated asymptotic value $h_t^{(\infty)}=3.29332$.[]{data-label="Fig5"}](Fig5.pdf){width="70.00000%"}

The phase transition fields $h_t^{(p)}$, calculated according to the method described in [@DaniskaGendiar], are summarized in table \[Tab3\] together with their errors $\Delta^{(p)}$. Notice that $\Delta^{(p)}$ represents only the error of the method providing that the calculated magnetization $\langle S_p^{z}\rangle$ is considered accurate. The data are graphically plotted in figure \[Fig5\], whereas the error bars are too small to be displayed. Using an analogous exponential fitting function applied on the critical magnetic fields $h_t^{(p)}$ for $p \in \{6, \dots, 10\}$, (cf. ), we calculated the asymptotic phase transition field of the TFIM on the Bethe lattice $h_t^{(\infty)}=3.29332$ as listed in Tab. \[Tab3\].

        $p$               $4$                $5$                $6$                $7$
  ---------------- ------------------ ------------------ ------------------ ------------------
    $h_t^{(p)}$        $3.158034$         $3.263825$         $3.285405$         $3.291055$
   $\Delta^{(p)}$   $1\times10^{-6}$   $1\times10^{-6}$   $1\times10^{-6}$   $1\times10^{-6}$
        $p$               $8$                $9$                $10$             $\infty$
    $h_t^{(p)}$        $3.292647$         $3.293113$         $3.293263$         $3.29332$
   $\Delta^{(p)}$   $2\times10^{-6}$   $2\times10^{-6}$   $5\times10^{-6}$   $1\times10^{-5}$

  : The phase transition fields 