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The Model
In this section, we describe the dilute Ising model underlying our Monte Carlo simulations. It is defined as a two-dimensional (d=2) square lattice of dimension L x L, with sites denoted by a pair of integers, r=(x,y). The boundary conditions are fully periodic in all directions. To model the two species of particles, black (``spin up'') and white (``spin down'', displayed as gray in the figures), and the vacancies, we introduce a spin variable sr at each site which can take three values: +1 (-1) if the site is occupied by a black (white) particle, and sr =0 if it is empty. Multiple occupancy is forbidden. The numbers of black (N+)and white (N- )particles are conserved and differ by at most 1. The number of vacancies, M, is much smaller: M < < N+. In fact, most of our simulations will be restricted to M=1, to model the minute vacancy concentrations in real systems.

The particles and vacancies interact with one another according to a dilute Ising model:

H[ s r] = -J S srsr'           (1)

with a ferromagnetic, nearest-neighbor coupling J>0. Since the dilution is so small, the behavior of the system is that of the ordinary (non-dilute) two-dimensional Ising model. Thus, it has a phase transition, from a disordered to a phase-segregated phase, at the Onsager critical temperature, Tc=2.267...J/kB[18] . The ground state is doubly degenerate. It consists of a strip of black and a strip of white particles, each filling half the system, separated by two planar interfaces running parallel to a lattice axis. Since the particle-particle interactions are ferromagnetic, only few vacancies will accumulate at the interfaces.

Next, we turn to the dynamics of the model. Only particle-hole exchange is allowed and performed with the usual Metropolis rate[19]: min {1,exp ( -bDH)}, where DH is the energy difference of the system before and after the jump, and b =1/(kBT) is the inverse temperature. The initial configuration of the system is perfectly phase-segregated, with two interfaces chosen to lie along the x-axis. The vacancies are located at random positions along one of the interfaces. Since this is technically a zero-temperature configuration, while the dynamics occurs at temperature T > 0, the vacancies will move around, disordering the interfaces and, if T > Tc, dissolving them eventually. The final steady state is an equilibrium state of the usual two-dimensional Ising model. Therefore, many of its properties are exactly known[20].

In our simulations, the system size is L = 60. The final temperature T , measured in units of the Onsager temperature Tc, varied between 1.1 Tcand infinity. Our data are averages over 102 to 104 realizations (or runs) depending on the desired quality of the data. The time unit is one Monte Carlo step (MCS) which corresponds to M attempted particle-hole exchange. All systems investigated equilibrate after about 108 MCS.

To monitor the evolution of the system, we measure a ``disorder parameter'', defined as the average number of black and white nearest-neighbor pairs, A(L,t), as a function of (Monte Carlo) time, t. This quantity is easily related to the Ising energy,

A(L,t)= (d/2)Ld + (1/2J)< H >+O(Lg)            (2)

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where < . > denotes the configurational average over runs. The correction is due to the vacancies that remains much smaller than the two leading terms.

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