Asymptotic Distribution of Parameters in Trivalent Maps and Linear Lambda Terms

TL;DR

This paper investigates the asymptotic distributions of parameters in large random trivalent maps and linear lambda-terms, revealing their combinatorial relationships through bijections and advanced analytical tools.

Contribution

It introduces new combinatorial specifications and analytical methods to study the limit distributions of parameters in maps and lambda-terms, building on recent bijections between these domains.

Findings

Distribution of bridges in trivalent maps analyzed

Distribution of subterms in linear lambda-terms characterized

Asymptotic behavior of parameters established

Abstract

Structural properties of large random maps and lambda-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 lambda-calculus, building on recent bijective connections between these two domains. In such cases, parameters in maps naturally correspond to parameters in lambda-terms and vice versa. By an interplay between lambda-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 lambda-terms, the number of vertices of degree 1 in (1,3)-valent maps and of free variables in open linear lambda-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.