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Random surface growth with a wall and Plancherel measures for O(infinity)

We consider a Markov evolution of lozenge tilings of a quarter-plane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall. We observe frozen and liquid regions, prove convergence of the local correlations to translation-invariant Gibbs measures in the liquid region, and obtain new discrete Jacobi and symmetric Pearcey determinantal point processes near the wall. The model can be viewed as the one-parameter family of Plancherel measures for the infinite-dimensional orthogonal group, and we use this interpretation to derive the determinantal formula for the correlation functions at any finite time moment.