Leaf-excluded percolation in two and three dimensions

We introduce the leaf-excluded percolation model, which corresponds to independent bond percolation conditioned on the absence of leaves (vertices of degree one). We study the leaf-excluded model on the square and simple-cubic lattices via Monte Carlo simulation, using a worm-like algorithm. By studying wrapping probabilities, we precisely estimate the critical thresholds to be $0.3552475\left(8\right)$ (square) and $0.185022\left(3\right)$ (simple-cubic). Our estimates for the thermal and magnetic exponents are consistent with those for percolation, implying that the phase transition of the leaf-excluded model belongs to the standard percolation universality class.

Figure 1

A typical configuration (denoted by bold edges) of the leaf-excluded model on a $6\times 6$ patch of the square lattice.

Figure 2

Plots of $R^{\left(x\right)}$ and $Q_1$ versus $v$ for the leaf-excluded model on the square lattice.

Figure 3

Plots of $R^{\left(x\right)}(v,L)$ versus $L$ for fixed values of $v$, for the two-dimensional leaf-excluded model. The curves correspond to our preferred fit of the Monte Carlo data. The shaded gray strips indicate an interval of one error bar above and below the estimate $R_c^{\left(x\right)}=0.5212\left(2\right)$.

Figure 4

Plots of $L^{-3/4}{g}_{bR}^{\left(x\right)}$ and $L^{-91/48}{C}_1$ versus $L^{-3/2}$. The straight lines are simply to guide the eye.

Figure 5

Plots of $R^{\left(x\right)}$ and $Q_1$ versus $v$ for the leaf-excluded model on the simple-cubic lattice.

Figure 6

Plots of $R^{\left(x\right)}(p,L)$ versus $L$ for fixed values of $p$, for the three-dimensional leaf-excluded model. The curves correspond to our preferred fit of the Monte Carlo data. The shaded gray strips indicate an interval of one error bar above and below the estimate $R_c^{\left(x\right)}=0.260\left(4\right)$.

Figure 7

Plot of $L^{-1.1415}{g}_{bR}^{\left(x\right)}$ and $L^{-2.52295}{C}_1$ versus $L^{-1.2}$. The values of the critical exponents used on the vertical axis correspond to the estimates $y_t=1.1415\left(15\right)$ and $y_h=2.52295\left(10\right)$ [4, 18]. The straight lines are simply to guide the eye.