STAT 206A: Gibbs Measures

Recall that E denotes the number of red edges, and F denotes the total number of edges. Then E = ξF = ξNΛ′(1) with ξ defined appropriately. In this lecture we use the saddle point method to derive an approximation to coeff [∏kmax k=2 qk(z) MPk , zE ] , where qk(z) = 12 ((1 + z) n + (1− z)n) = ∑ r even ( k r ) zr. This coefficient counts the number of ways to choose which edges incident to the check nodes are colored red. The other coeff[. . .] can be dealt with similarly.