2024/06/19 by Bouttier, Jérémie, Guitter, Emmanuel, Miermont, Grégory · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2406.13528
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)ℓ1,…,ℓn for fixed genus g and prescribed boundary lengths ℓ1,…,ℓn, with a control on the degrees of inner faces. We find that these series appear as coefficients in the expansion of ω(g)n(z1,…,zn), 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)2ℓ1,…,2ℓn from the Collet-Fusy formula. We also find recursion relations satisfied by T(g)ℓ1,…,ℓn, 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)ℓ1,…,ℓn is equal to a parity-dependent quasi-polynomial in ℓ12,…,ℓn2 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.