Enumeration of maps with tight boundaries and the Zhukovsky transformation

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 $T_{{\ell }_1,\dots ,{\ell }_n}^{(g)}$ for fixed genus $g$ and prescribed boundary lengths ${\ell }_1,\dots ,{\ell }_n$, with a control on the degrees of inner faces. We find that these series appear as coefficients in the expansion of ${\omega }_n^{(g)}(z_1,\dots ,z_n)$, 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 $T_{2{\ell }_1,\dots ,2{\ell }_n}^{(0)}$ from the Collet–Fusy formula. We also find recursion relations satisfied by $T_{{\ell }_1,\dots ,{\ell }_n}^{(g)}$, which consist in adding an extra tight boundary, keeping the genus $g$ fixed. Building on a result of Norbury and Scott, we show that $T_{{\ell }_1,\dots ,{\ell }_n}^{(g)}$ is equal to a parity-dependent quasi-polynomial in ${\ell }_1^2,\dots ,{\ell }_n^2$ 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.