Branching Interfaces with Infinitely Strong Couplings

Abstract

A hierarchical froth model of the interface of a random $q$-state Potts ferromagnet in $2D$ is studied by recursive methods. A fraction $p$ of the nearest neighbour bonds is made inaccessible to domain walls by infinitely strong ferromagnetic couplings. Energetic and geometric scaling properties of the interface are controlled by zero temperature fixed distributions. For $p<p_c$, the directed percolation threshold, the interface behaves as for $p=0$, and scaling supports random Ising ($q=2$) critical behavior for all $q$'s. At $ p=p_c$ three regimes are obtained for different ratios of ferro vs. antiferromagnetic couplings. With rates above a threshold value the interface is linear ( fractal dimension $d_f=1$) and its energy fluctuations, $\Delta E$ scale with length as $\Delta E\propto L^{\omega}$, with $\omega\simeq 0.48$. When the threshold is reached the interface branches at all scales and is fractal ($d_f\simeq 1.046$) with $\omega_c \simeq 0.51$. Thus, at $p_c$, dilution modifies both low temperature interfacial properties and critical scaling. Below threshold the interface becomes a probe of the backbone geometry ($\df\simeq{\bar d}\simeq 1.305$; $\bar d$ = backbone fractal dimension ), which even controls energy fluctuations ($\omega\simeq d_f\simeq\bar d$). Numerical determinations of directed percolation exponents on diamond hierarchical lattice are also presented.