Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices

We consider the stochastic dynamics of the pure and random ferromagnetic Ising model on a hierarchical diamond lattice of branching ratio K with fractal dimension d(f) = (ln(2K))/ln 2. We adapt the real-space renormalization procedure introduced in our previous work (Monthus and Garel, 2013 J. Stat. Mech. P02037) to study the equilibrium time t(eq)(L) as a function of the system size L near zero temperature. For the pure Ising model, we obtain the behavior t(eq)(L) similar to L(alpha)e(beta 2JLds), where d(s) = d(f) - 1 is the interface dimension, and we compute the prefactor exponent alpha. For the random ferromagnetic Ising model, we derive the renormalization rules for dynamical barriers B-eq(L) equivalent to (lnt(eq)/beta) near zero temperature. For the fractal dimension d(f) = 2, we obtain that the dynamical barrier scales as B-eq(L) = cL + L(1/2)u, where u is a Gaussian random variable of non-zero mean. While the non-random term scaling as L corresponds to the energy cost of the creation of a system-size domain-wall, the fluctuation part scaling as L-1/2 characterizes the barriers for the motion of the system-size domain-wall after its creation. This scaling corresponds to the dynamical exponent psi = 1/2, in agreement with the conjecture psi = d(s)/2 proposed by Monthus and Garel (2008 J. Phys. A: Math. Theor. 41 115002). In particular, it is clearly different from the droplet exponent theta similar or equal to 0.299 involved in the statics of the random ferromagnet on the same lattice.