For random two-component grid diagrams, the odd moments of the linking number vanish, the even moments are polynomials in the grid size, and the normalized linking number converges weakly.
Cromwell, Embedding knots and links in an open book I: Basic properties , Mathematical Proceedings of the Cambridge Philosophical Society 119 , 2 (1996), 309
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Linking number of grid models
For random two-component grid diagrams, the odd moments of the linking number vanish, the even moments are polynomials in the grid size, and the normalized linking number converges weakly.