Doubly Free-Boundary Rail-Yard Dimers and Annular Gaussian Fluctuations

We study rail-yard dimer measures with free boundary conditions at both the left and the right boundary. The double free-boundary geometry produces an infinite family of reflected Cauchy factors in the partition function and in the exact contour formulas for height Laplace observables. These factors are absent from the empty-boundary model and survive in both the deterministic and second-order asymptotics.
For admissible piecewise periodic weights, we prove a Laplace-transform law of large numbers. In the natural moment variable $x=e^{-n\beta\kappa}$, the transform convergence gives a limit shape of the rescaled height function. The associated frozen-boundary satisfies the following system of equations \[
S_\chi(w)^\beta=e^{-n\beta\kappa},
\qquad
\frac{d}{dw}\log S_\chi(w)=0,
\qquad
S_\chi(w):=\mathcal G_\chi(w)\prod_{r\ge1}\mathcal F_{u,v,r}(w). \]
The main second-order result is a Gaussian fluctuation theorem for centered height Laplace observables. The covariance is the annular reflected-image kernel \[
\mathsf K_{LL}(z,w)
=
\partial_z\partial_w
\log\frac{\Theta_{\mathfrak q}(z/w)}
{\Theta_{\mathfrak q}(u^2zw)},
\qquad \mathfrak q=(uv)^2, \] with specified contour interpretation. Thus the two free boundaries do not merely alter the deterministic limit shape: they replace the usual Gaussian free field half-plane image structure by an annular prime-function covariance on the Laplace-test class.
As a final exact-solvability consequence, we construct an exact growth-diagram sampler for the finite reflected truncations of the doubly free-boundary model and prove that the resulting \(K\)-truncated laws converge, as \(K\to\infty\), in total variation to the full doubly free-boundary Gibbs measure on every fixed rail-yard graph with finitely many columns.