Geometric and spectral properties of causal maps
| Autor: | Asaf Nachmias, Nicolas Curien, Tom Hutchcroft |
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| Přispěvatelé: | Hutchcroft, Thomas [0000-0003-0061-593X], Apollo - University of Cambridge Repository, Université Paris-Saclay |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Causal maps Applied Mathematics General Mathematics Random trees 010102 general mathematics Spectral properties Probability (math.PR) FOS: Physical sciences random walks Mathematical Physics (math-ph) Random walk 01 natural sciences [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] Spectral dimension FOS: Mathematics [MATH]Mathematics [math] spectral dimension 0101 mathematics ComputingMilieux_MISCELLANEOUS Mathematics - Probability Mathematical Physics Mathematics |
| Zdroj: | Journal of the European Mathematical Society Journal of the European Mathematical Society, European Mathematical Society, 2020, 22 (12), pp.3997-4024. ⟨10.4171/JEMS/1001⟩ |
| ISSN: | 1435-9855 1435-9863 |
| DOI: | 10.4171/jems/1001 |
| Popis: | We study the random planar map obtained from a critical, finite variance, Galton-Watson plane tree by adding the horizontal connections between successive vertices at each level. This random graph is closely related to the well-known causal dynamical triangulation that was introduced by Ambj{\o}rn and Loll and has been studied extensively by physicists. We prove that the horizontal distances in the graph are smaller than the vertical distances, but only by a subpolynomial factor: The diameter of the set of vertices at level $n$ is both $o(n)$ and $n^{1-o(1)}$. This enables us to prove that the spectral dimension of the infinite version of the graph is almost surely equal to 2, and consequently that the random walk is diffusive almost surely. We also initiate an investigation of the case in which the offspring distribution is critical and belongs to the domain of attraction of an $\alpha$-stable law for $\alpha \in (1,2)$, for which our understanding is much less complete. Comment: 27 pages, 11 figures. Accepted version, to appear in JEMS |
| Databáze: | OpenAIRE |
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