  The Magnetic Eden Model (MEM) with ferromagnetic interactions between
nearest-neighbor spins is studied in $(d+1)-$dimensional rectangular geometries
for $d = 1,2$. In the MEM, magnetic clusters are grown by adding spins at the
boundaries of the clusters. The orientation of the added spins depends on both
the energetic interaction with already deposited spins and the temperature,
through a Boltzmann factor. A numerical Monte Carlo investigation of the MEM
has been performed and the results of the simulations have been analyzed using
finite-size scaling arguments. As in the case of the Ising model, the MEM in $d
= 1 $ is non-critical (only exhibits an ordered phase at $T= 0$). In $d = 2$
the MEM exhibits an order-disorder transition of second-order at a finite
temperature. Such transition has been characterized in detail and the relevant
critical exponents have been determined. These exponents are in agreement
(within error bars) with those of the Ising model in 2 dimensions. Further
similarities between both models have been found by evaluating the probability
distribution of the order parameter, the magnetization and the susceptibility.
Results obtained by means of extensive computer simulations allow us to put
forward a conjecture which establishes a nontrivial correspondence between the
MEM for the irreversible growth of spins and the equilibrium Ising model. This
conjecture is certainly a theoretical challenge and its confirmation will
contribute to the development of a framework for the study of irreversible
growth processes.
