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Building it rigorously would be a two-dimensional model of the Yang-Mills construction problem.","posed_since":"","precise":"Show that lattice $O(N)$ correlations ($N \\ge 3$) at separations $x = y\\,\\xi(\\beta),\\ y\\ \\text{fixed}$, converge as $\\beta \\to \\infty$ to the Schwinger functions of a non-Gaussian Euclidean field theory on $R^2$ satisfying the Osterwalder-Schrader axioms with a mass gap, and identify its scattering with the factorized S-matrix of A. B. and Al. B. Zamolodchikov (1979). Answer: a construction with proof.","problem_ref":null,"references":"","settled_by":"A proof of convergence of the rescaled lattice correlations to an Osterwalder-Schrader theory with non-Gaussian four-point function.","status_note":"","title":"Construct the continuum $2\\mathrm{D}$ $O(N)$ sigma model as an interacting 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