Publication : t24/011

Abstract:
We consider maps with tight boundaries, i.e. maps whose boundaries have minimal length in their homotopy class, and discuss the properties of their generating functions T^{(g)}_{l_1,...,l_n} for fixed genus g and prescribed boundary lengths l_1,...,l_n, with a control on the degrees of inner faces. We find that these series appear as coefficients in the expansion of omega^{(g)}_n(z_1,...,z_n), a fundamental quantity in the Eynard-Orantin theory of topological recursion, thereby providing a combinatorial interpretation of the Zhukovsky transformation used in this context. This interpretation results from the so-called trumpet decomposition of maps with arbitrary boundaries. In the planar bipartite case, we obtain a fully explicit formula for T^{(0)}_{l_1,...,l_n} from the Collet-Fusy formula. We also find recursion relations satisfied by T^{(g)}_{l_1,...,l_n}, which consist in adding an extra tight boundary, keeping the genus g fixed. Building on a result of Norbury and Scott, we show that T^{(g)}_{l_1,...,l_n} is equal to a parity-dependent quasi-polynomial in l_1^2,...,l_n^2 times a simple power of the basic generating function R. In passing, we provide a bijective derivation in the case (g,n)=(0,3), generalizing a recent construction of ours to the non bipartite case.