Abstract
We study zero-temperature Ising Glauber Dynamics, on $2D$ slabs of thickness $k \geq 2$. In this model, $\pm 1$-valued spins at integer sites update according to majority vote dynamics with two opinions. We show that all spins reaches a final state (that is, the system fixates) for $k=2$ under free boundary conditions and for $k=2$ or $3$ under periodic boundary conditions. For thicker slabs there are sites that fixate and sites that do not.
Highlights
In this paper, we study some natural questions concerning coarsening on two-dimensional slabs and how the answers to those questions depend on the width k of the slab
For d = 1 this is a result about the standard one-dimensional voter model and holds for all p ∈ (0, 1) [1], while for d = 2, the
Motivated by non-fixation for d = 2 and the open d = 3 problem, in this paper we study graphs that interpolate between Z2 and Z3 by considering width-k slabs, Sk = Z2 × {0, 1, . . . , k − 1}, with free or periodic boundary conditions in the third coordinate
Summary
We study some natural questions concerning coarsening on two-dimensional slabs and how the answers to those questions depend on the width k of the slab. Motivated by non-fixation for d = 2 and the open d = 3 problem, in this paper we study graphs that interpolate between Z2 and Z3 by considering width-k slabs, Sk = Z2 × {0, 1, . The fact that the results for slabs do not depend on p ∈ (0, 1) is true on Z1 where there is never fixation This is not so in general, as it has been proved in [4] that on Zd with d ≥ 2, all sites fixate at +1 (resp., at −1) when p is close enough to 1 (resp., to 0). In the free boundary condition setting, one could consider the probability that the site at (0, 0, [(k − 1)/2]) fixates If it vanishes in this limit, that might supply a mechanism for proving that no fixation occurs in Z3
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