The O(n) model on random lattices of all topologies

The one-cut property implies ∀x ∈ [a(t), b(t)] and → 0 : [3] W 1 (x + i ) + W (0) 1 (x− i ) + nW (0) 1 (−x) = V ′(x) [3] W 2 (x1 + i , x2) + W (0) 2 (x1 − i , x2) + nW (0) 2 (−x1, x2) = −1 (x1−x2) [4] W k (x1 + i , J) + W (g) k (x1 − i , J) + nW (g) k (−x1, J) = 0 else → Riemann-Hilbert problem: solution in terms of functions on a spectral curve S Def. of spectral curve data S = (Σ, x , y), where . Σ is a Riemann surface . x and y are two meromorphic functions on Σ 5 Topological recursion