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We present a controlled method for computing the exchange coupling instrongly correlated one-dimensional electron systems. It is based on theasymptotically exact relation between the exchange constant and thepair-correlation function of spinless electrons. Explicit results are obtainedfor thin quantum rings with realistic Coulomb interactions, by calculating thisfunction via a many-body instanton approach.Comment: 7 pages, 2 figures. Changes in the text and figures to improve  readability; added reference

Spin exchange in quantum rings and wires in the Wigner-crystal limit