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We study random triangulations of the integer points [0,n]^2 ∩Z^2, where each triangulation σ has probability measure λ^|σ| with |σ| denoting the sum of the length of the edges in σ. Such triangulations are called lattice triangulations. We construct a height function on lattice triangulations and prove that, in the whole subcritical regime λ<1, the function behaves as a Lyapunov function with respect to Glauber dynamics; that is, the function is a supermartingale. We show the applicability ofarXiv:1504.07980v1 fatcat:auail6wjsff2zdo56pwuquf7rm