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Proof: Limit measure \(\mu^x\) exists

We follow an argument given by Adams, \cite{A_12}. We note that the measure \(\mu^x\) can also be seen as the limit of a sequence of image measures \(\mu_{N}^{\nabla,\psi_{x}}=\mu_{N}^{\psi_{x}}\circ\nabla^{-1}\), where \(\mu_{N}^{\psi_{x}}\) measures height configurations on \(\Lambda_N\) with boundary condition \(\psi_{x}(y)=xy\), and \(\nabla:\Lambda_{N}\rightarrow \mathcal B ( \Lambda_N )\) maps height configurations to bond configurations, \cite{F_03} pp.152-153. Note that \(\nabla\) is not invertible, however \(\mu_{\Lambda_{N}}^{\psi_{u}}\circ\nabla^{-1}\) is still well defined because the preimage \(\nabla^{-1}(\eta)\) consists of height configurations that all have the same mass under \(\mu_{N}^{\psi_{x}}\), thus there is no ambiguity.

Denoting \(b_i = (i, i+1) \in \mathcal{B}(\Lambda)\), we define the height function of a configuration \(\eta \in \Omega_N\) to be

\[

\phi_\eta(y)\coloneqq\sum_{i=1}^{y}\eta(b_i), \qquad y \in \Lambda_N

\]

so that \(\nabla(\phi_\eta) = \eta\). Now if \((\eta(b_i) )_{i=0}^N\) are i.i.d.\@ with distribution \(\nu(\cdot) \propto \exp(- V( \cdot ) )\), we see that the measure obtained by conditioning on the final height satisfies \(\nu^{\otimes N}( \cdot \, | \phi(N) = xN ) = \mu_N^{\psi_x} \circ \nabla ^{-1}\), where equality is in law. Hence the limit \(\mu^x\) is the weak limit of this sequence of conditioned measures.

In Theorem 3.5 of \cite{GPV_88}, Guo et al.\@ show that in the limit, the conditioned probability measure above can be expressed as a product measure of the relevant Cram\'er transformed measures; in particular letting \(\lambda = \lambda(x)\), they show that this limit is an infinite product of \(\hat \nu_\lambda\). Now we are done since

\[

\mu^x(\cdot) = \lim_{N \rightarrow \infty} \mu_N^{\nabla, \psi_x}(\cdot) = \lim_{N\rightarrow \infty} \nu^{\otimes N}( \, \cdot | \phi(N) = xN ) = \prod_{i \in \mathbb Z} \hat \nu_\lambda(\cdot) \qedhere

\]