# Asymptotics of 10j Symbols

#### August 5, 2002

**Abstract**: The Riemannian 10j symbols are spin networks that assign an
amplitude to each 4-simplex in the Barrett-Crane model of Riemannian
quantum gravity. This amplitude is a function of the areas of the 10
faces of the 4-simplex, and Barrett and Williams have shown that one
contribution to its asymptotics comes from the Regge action for all
non-degenerate 4-simplices with the specified face areas. However, we
show numerically that the dominant contribution comes from degenerate
4-simplices. As a consequence, one can compute the asymptotics of
the Riemannian 10j symbols by evaluating a `degenerate spin network',
where the rotation group SO(4) is replaced by the Euclidean group of
isometries of R^{3}. We conjecture formulas for the asymptotics of
a large class of Riemannian and Lorentzian spin networks, including
the Lorentzian 10j symbols, in terms of these degenerate spin networks.

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© 1992 - 2001 John Baez