Critical phenomena on $k$-booklets

We define a “$k$-booklet” to be a set of $k$ semi-infinite planes with $-\infty <x<\infty$ and $y\ge 0$, glued together at the edges (the “spine”) $y=0$. On such booklets we study three critical phenomena: self-avoiding random walks, the Ising model, and percolation. For $k=2$, a booklet is equivalent to a single infinite lattice, and for $k=1$ to a semi-infinite lattice. In both these cases the systems show standard critical phenomena. This is not so for $k\ge 3$. Self-avoiding walks starting at $y=0$ show a first-order transition at a shifted critical point, with no power-behaved scaling laws. The Ising model and percolation show hybrid transitions, i.e., the scaling laws of the standard models coexist with discontinuities of the order parameter at $y\approx 0$, and the critical points are not shifted. In the case of the Ising model, ergodicity is already broken at $T=T_c$, and not only for $T<T_c$ as in the standard geometry. In all three models, correlations (as measured by walk and cluster shapes) are highly anisotropic for small $y$.

Figure 1

Log-log plot of average numbers of self-avoiding walks of length $N$ on $k$ joined semi-infinite planes, divided by ${\mu }_k^N$. Here, ${\mu }_1={\mu }_2=\mu$ is the known connective constant of SAWs on ordinary square lattices, while ${\mu }_k$ for $k>2$ are connective constants fitted from these data. For $k\le 2$, the $\gamma$ exponents are in agreement with ${\gamma }_1=61/64$ [9] and ${\gamma }_2=\gamma =43/32$ [7], while they are equal to 1 for $k>2$.

Figure 2

Log-log plot of rms end-to-end distances of self-avoiding walks in the directions parallel and perpendicular to the spine, for $k=3$, 4, and 6. We do not show the data for $k=1$ and 2, since they scale according to the well-known Flory exponent, ${\left[\langle x^2\rangle \right]}^{1/2}\sim {\left[\langle y^2\rangle \right]}^{1/2}\sim N^{3/4}$. The asymptotic scaling is reached fastest for $k=6$, and slowest for $k=3$.

Figure 3

Log-log plot of the average squared magnetization density at fixed $y$, for $k$ Ising models joined at $y=0$. The temperature is the critical one for single lattices.

Figure 4

Non-normalized distributions of the Ising magnetization densities at $y=0$, (a) for $k=2$ and (b) for $k=3$.

Figure 5

Probabilities $P\left(t\right)$ that a percolation cluster at $p=p_c$ continues to grow for time $\ge t$. Clusters start growing at the line $y=0$.

Figure 6

Non-normalized distributions of the density ${\rho }_0$ of sites on the axis $y=0$ “wetted” by clusters that start to grow at $y=0$. In these simulations $L_x$ was finite, but $L_y$ was so large as to be effectively infinite. The left-hand peak corresponds to clusters that died early, while the right-hand peak corresponds to clusters that grow to values $y\gg L_x$.

Figure 7

Log-log plots of the (squared) cluster growth in the $x$ direction, for $k=2,3,...,8$. For $k=2$, we have of course the usual growth of two-dimensional (2D) percolation clusters, but for $k>2$, the growth follows power laws with larger exponents.

Figure 8

Log-log plots of average cluster sizes, for subcritical clusters grown from seeds at $y=0$ on infinitely large lattices. For $k\le 2$, we recover the known scaling laws, and for $k\ge 3$, we find again scaling laws with new exponents.