A conditional limit theorem for tree-indexed random walk
Let \(\mathcal{T}\) be a Galton-Watson tree with critical aperiodic offspring distribution with analytic characteristic function and conditioned to have exactly \(n\) vertices. Let \(Y_e\) be i.i.d. r.v.'s with symmetric distribution \(\gamma\) such that as \(x\to \infty\), \(\gamma(x,\infty)= o(x^{-4})\). It is assumed that these \(Y_e\) are associated to different edges of the tree, and the spatial position \(U_v\) of a vertex \(v\) is obtained by summing the displacements \(Y_e\) along the ray from the root to the \(v\). The resulting object \((\mathcal{T}, \{U_v, v\in \mathcal{T}\})\) is called discrete spatial tree. The main result of the paper states that the scaling limit of the pair consisting of the contour function and the spatial contour function of the discrete spatial trees conditioned to have nonnegative spatial positions is a conditioned Brownian snake with initial point \(0\). Also the author applies his main result to give a short proof of some asymptotics for random quadrangulations which were obtained by \textit{P. Chassaing} and \textit{G. Schaeffer} [Probab. Theory Relat. Fields 128, No. 2, 161-212 (2004; Zbl 1041.60008)].
scientific article