$p=p_{c1}$ in the infinite-size limit, whereas the probability to reach a finite *fraction* of the boundary points becomes finite at $p_{c2}$. This midpoint-percolation description offers a conceptually simple way of understanding the physics at the two thresholds. The hyperbolic lattices have, in comparison with the planar lattices, two complications: one is that there are few known exact results for the actual values of the thresholds and for the critical properties associated with them. The other is that, since the number of lattice points grows rapidly with the distance from the midpoint, it is hard to obtain very precise results numerically by using standard size scaling in conjunction with Monte Carlo techniques. The enhanced binary tree (EBT) model alleviates these difficulties to some extent.

![Hyperbolic lattice structures drawn on the Poincaré disk. (a) The solid lines show a heptagonal lattice, $\{7,3\}$, and the dotted lines show its dual, $\{3,7\}$. (b) A tree structure, represented as $\{\infty,3\}$, up to the maximum level $L=10$. (c) The enhanced tree obtained from (b) by connecting the tree branches.[]{data-label="fig:poin"}](fig1a.eps "fig:"){width="20.00000%"} ![Hyperbolic lattice structures drawn on the Poincaré disk. (a) The solid lines show a heptagonal lattice, $\{7,3\}$, and the dotted lines show its dual, $\{3,7\}$. (b) 