Asymptotic Distribution of Parameters in Trivalent Maps and Linear Lambda Terms

Structural properties of large random maps and λ-terms may be gleaned by studying the limit distributions of various parameters of interest. In our work we focus on restricted classes of maps and their counterparts in the λ-calculus, building on recent bijective connections between these two domains. In such cases, parameters in maps naturally correspond to parameters in λ-terms and vice versa. By an interplay between λ-terms and maps, we obtain various combinatorial specifications which allow us to access the distributions of pairs of related parameters such as: the number of bridges in rooted trivalent maps and of subterms in closed linear λterms, the number of vertices of degree 1 in (1, 3)-valent maps and of free variables in open linear λ-terms etc. To analyse asymptotically these distributions, we introduce appropriate tools: a moment-pumping schema for differential equations and a composition schema inspired by Bender’s theorem.