Erratum to Components in Meandric Systems and the Infinite Noodle.

This document is an erratum to a paper titled "Components in Meandric Systems and the Infinite Noodle." The erratum acknowledges a missing argument in the proof of Proposition 5 in the original paper. It also refers to Proposition 1 and Proposition 2 from the original paper, which discuss the convergence of random meandric systems and well-parenthesized words, respectively. The document provides equations and a lemma related to these propositions. The erratum concludes by stating that the difference between two equations is the use of the same root in both well-parenthesized words. The document also introduces notation and provides a proof for a swapping lemma. The given text is a proof of Proposition 1, provided by Valentin Féray and Paul Thévenin. The authors use equations and arguments to demonstrate the convergence in expectation and convergence in probability for multiplicative functions. They acknowledge Svante Janson for pointing out an incomplete proof in their original paper. [Extracted from the article]