Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps

Abstract

We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time $n$ at an arbitrary value, encompassing in particular large deviation regimes. This enables us to derive scaling limits of such random walks conditioned by their terminal value at time $n$ in various regimes. We believe both to be of independent interest. We then apply these results to obtain invariance principles for the Lukasiewicz path of Bienaymé-Galton-Watson trees conditioned on having a fixed number of leaves and of vertices at the same time, which constitutes a first step towards understanding their large scale geometry. We finally deduce from this scaling limit theorems for random bipartite planar maps under a new conditioning by fixing their number of vertices, edges, and faces at the same time. In the particular case of the uniform distribution, our results confirm a prediction of Fusy & Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian map.

Figures (3)

• Figure 1: A plane tree, with the depth-first search order indicated next to the nodes and its Łukasiewicz path.
• Figure 2: The big picture for scaling limits of biconditioned bipartite Boltzmann planar maps with weight sequence $\boldsymbol{q}$, conditioned to have $n-1$ edges and $K_{n}+1$ vertices (and $n-K_{n}$ faces). The generating function $F$, with radius of convergence $\rho$, as well as the function $A$ are defined in \ref{['eq:serie_gen_poids']}. For $\alpha \in (1,2)$, the sequence $r_{n}$ is of order $n^{1/\alpha}$.
• Figure 3: A pointed map (right) associated with a labelled tree (left): Labels on the map indicate, up a to a shift, the graph distance to the distinguished vertex, which is the one carrying the smallest label (here $-3$). The figure in the middle indicates the construction of the map from the tree, first by applying a Schaeffer-like rule, except visiting vertices in depth-first search order, and then merging each internal vertex of the tree with its right-most offspring (represented by dashed lines).

Theorems & Definitions (35)

• Remark 1.3
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm:LTT_large_endpoint']}
• proof : Proof of Theorem \ref{['thm:lls1']}
• Lemma 3.1
• proof
• Proposition 3.4: Subcritical conditioning
• proof
• proof : Proof of Theorems \ref{['thm:CVmarches']} and \ref{['thm:stable_marches_gaussien']}