Open boundary condition and Glauber Dynamics

Warning: by background is in math, not physics. I've just recently started working with things that are close to theoretical physics. So please note that

1. I'm still very confused by the jargon.
2. Maybe I'm asking something trivial, in which case please just give me some reference and I'll be happy to read them.

I am reading some notes about Glauber Dynamics for discrete spin models (mainly stuff from Martinelli, Olivieri and Schonmann), and I'm surprised by how little space is given to models with "open boundary conditions" (which as far as I understood is the current terminology for no boundary conditions).

Googling the web didn't helped me understand if this is a choice of the authors, or there is some more stringent motivation beyond.

Let's suppose that we are working with a regular lattice (i.e. $\mathbb{Z}^d$), discrete spin space ($\{\pm 1\}$ is enough), and finite range interactions ($H(\sigma) = \sum_{A} J_A(\sigma_A)$, where the sum is taken over finite sets $A$ of diameter smaller than the interaction length).

1. Are there any problems with defining the Gibbs measure for finite volumes $V \subset \mathbb Z^d$ without boundary conditions, i.e. $$\mu_V(\sigma) = Z_V^{-1} \exp \left( \sum_{A \subset V} J_A(\sigma_A) \right)$$ where by "problem" I think of things like "is not a measure on the right space of configurations", or "does not verify some important compatibility rule" (DLR?), or "is not unique", etc.
2. Are known conditions for which Glauber Dynamics on this finite region $V$ will converge quickly to the above-defined distribution will be uniformly gapped?
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The choice of boundary conditions plays a crucial role in the phase coexistence regime. You should have a look at Martinelli's lecture notes, which are available at imap.ma.utexas.edu/mp_arc/c/97/97-578.ps.gz . There are, of course, more recent works, in particular concerning higher-dimensional systems. – Yvan Velenik Oct 6 '12 at 11:37
I knew the Martinelli's notes, but I overlooked at the chapter over phase coexistence. Thanks a lot for pointing it out! – Angelo Lucia Oct 8 '12 at 11:01

1. The Glauber dynamics preserves the Boltzmann distribution (this is easy to check by starting with the Boltzmann distribution, and showing it is obeys detailed balance, i.e. that $\rho_i K_{i\rightarrow j} = \rho_j K_{j\rightarrow i}$, in words: the rate of the i->j transition in equilibrium is balanced by the rate of the j->i transition. This shows that Glauber dynamics has a stationary distribution which is the Boltzmann distribution.