Two-dimensional Ising Gibbs coexistence at low temperature

Write $K=\beta J$ and $r=3e^{-2K}$. A cycle with the origin in its parity-inside has an odd number of crossings of the positive horizontal ray. Along the full horizontal line, each crossing changes the sign of a dual vertex's $y$-coordinate; after one closed traversal that sign returns to its starting value, so the total number of crossings is even. Therefore the negative ray also has a crossing. Dual vertices have half-integer coordinates, so positive-ray crossings occur on vertical dual edges at $x=j+\frac{1}{2}$ for integers $j\ge 0$. If such a cycle has length $\mathrm{\ell }$, its horizontal coordinate range includes a negative coordinate and this $x>0$; a closed lattice walk traverses that range in both directions, so it uses at least $2x$ horizontal edges and $x\le \mathrm{\ell }\mathrm{/}2$. Thus there are at most $\mathrm{\ell }+1$ possible positive-ray crossing edges. Choose the lower endpoint of each such edge; from it there are at most $4\cdot 3^{\mathrm{\ell }-1}$ nonbacktracking walks of length $\mathrm{\ell }$, which overcounts the simple cycles using that crossing. Hence there are at most $4(\mathrm{\ell }+1)3^{\mathrm{\ell }-1}$ candidate simple cycles of length $\mathrm{\ell }$.

By [F2] and the union bound, ${\gamma }_{B_n}^{+}(s_0=-1)\le \frac{4}{3}{\sum }_{\mathrm{\ell }\ge 4}(\mathrm{\ell }+1)r^{\mathrm{\ell }}=\frac{4}{3}{r}^4(5-4r)\mathrm{/}(1-r{)}^2=\delta (K)$, using [F5] for $0<r<1$.

The first seven nonnegative terms of [F4] give $\exp (4)>{\sum }_{n=0}^6{4}^n\mathrm{/}n!=437\mathrm{/}9>48$. Since $K\ge 2$, monotonicity and reciprocal positivity in [F6] imply $r=3\exp (-2K)\le 3\exp (-4)<1\mathrm{/}16$. Thus $\delta (K)<\frac{4}{3}(1\mathrm{/}16{)}^4\cdot 5\mathrm{/}(15\mathrm{/}16{)}^2<1\mathrm{/}2$. The same translated contour count applies to every site in the box, so the plus-boundary magnetization there is at least $m_{\ast }=1-2\delta (K)>0$.

The onsite and nearest-neighbor list in [F7] realizes the stated Hamiltonian, and its specification is proper and consistent. Take a subsequential limit of plus-boundary laws; by [F3] it is DLR and retains $\mu (s_i)\ge m_{\ast }$ for each fixed site. The zero-field nearest-neighbor Hamiltonian is invariant under simultaneous spin reversal, so the spin-reversed law is also DLR and has one-site expectations at most $-m_{\ast }$. Translation averaging as in [F3] preserves each bound and yields stationary ${\mu }_{+},{\mu }_{-}$. They are distinct because their one-site expectations have opposite signs.