### Abstract

We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e., the behavior of the “ant in the labyrinth.” It is natural to conjecture that the scaling limit for random walks on large critical random graphs exists in high dimensions and is universal. This scaling limit is simply the natural Brownian motion on the integrated super-Brownian excursion. We give here a set of four natural, sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, and spread-out percolation in high enough dimension. In a companion paper, we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above in dimensions larger than 14.

Original language | English (US) |
---|---|

Pages (from-to) | 669-763 |

Number of pages | 95 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 72 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*72*(4), 669-763. https://doi.org/10.1002/cpa.21813

**Scaling Limit for the Ant in High-Dimensional Labyrinths.** / Ben Arous, Gerard; Cabezas, Manuel; Fribergh, Alexander.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 72, no. 4, pp. 669-763. https://doi.org/10.1002/cpa.21813

}

TY - JOUR

T1 - Scaling Limit for the Ant in High-Dimensional Labyrinths

AU - Ben Arous, Gerard

AU - Cabezas, Manuel

AU - Fribergh, Alexander

PY - 2019/4/1

Y1 - 2019/4/1

N2 - We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e., the behavior of the “ant in the labyrinth.” It is natural to conjecture that the scaling limit for random walks on large critical random graphs exists in high dimensions and is universal. This scaling limit is simply the natural Brownian motion on the integrated super-Brownian excursion. We give here a set of four natural, sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, and spread-out percolation in high enough dimension. In a companion paper, we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above in dimensions larger than 14.

AB - We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e., the behavior of the “ant in the labyrinth.” It is natural to conjecture that the scaling limit for random walks on large critical random graphs exists in high dimensions and is universal. This scaling limit is simply the natural Brownian motion on the integrated super-Brownian excursion. We give here a set of four natural, sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, and spread-out percolation in high enough dimension. In a companion paper, we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above in dimensions larger than 14.

UR - http://www.scopus.com/inward/record.url?scp=85060658300&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060658300&partnerID=8YFLogxK

U2 - 10.1002/cpa.21813

DO - 10.1002/cpa.21813

M3 - Article

VL - 72

SP - 669

EP - 763

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 4

ER -