Replica symmetry breaking in finite connectivity systems: a large connectivity expansion at finite and zero temperature

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Abstract
Parisi replica symmetry breaking is extended to random bond systems with finite, fixed connectivity (M+1). The free energy, f is explicitly calculated for large M within the first stage of symmetry breaking. At finite temperature T, the 1/M expansion is found to diverge as T to O. Indeed, direct evaluation at zero temperature shows that the large M expansion is in powers of 1/M. Introduction of symmetry breaking brings the value of f close ( approximately 1% for 10<M<20) to numerical estimates of Banavar et al. The same techniques apply to systems with average finite connectivity.