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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| Accès en ligne: | https://arxiv.org/abs/2101.01682 |
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| _version_ | 1866929214840635392 |
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| author | Kortchemski, Igor Marzouk, Cyril |
| author_facet | Kortchemski, Igor Marzouk, Cyril |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_01682 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps Kortchemski, Igor Marzouk, Cyril Probability 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. |
| title | Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps |
| topic | Probability |
| url | https://arxiv.org/abs/2101.01682 |