The exploration process of critical Boltzmann planar maps decorated by a triangular $O(n)$ loop model
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910430579916800 |
|---|---|
| author | Korzhenkova, Aleksandra |
| author_facet | Korzhenkova, Aleksandra |
| contents | In this paper we investigate pointed $(\mathbf{q}, g, n)$-Boltzmann loop-decorated maps with loops traversing only inner triangular faces. Using the peeling exploration of arXiv:1809.02012 modified to this setting we show that its law in the non-generic critical phase can be coded in terms of a random walk confined to the positive integers by a new specific boundary condition. Under a technical assumption that we believe to be true, combining this observation with explicit quantities for the peeling law we derive the large deviations property for the distribution of the so-called nesting statistic and show that the exploration process possesses exactly the same scaling limit as in the rigid loop model on bipartite maps that is a specific self-similar Markov process introduced in arXiv:1809.02012. Besides, we conclude the equivalence of the admissible weight sequences related by the so-called fixed point equation by proving the missing direction in the argument of arXiv:1202.5521. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_11576 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The exploration process of critical Boltzmann planar maps decorated by a triangular $O(n)$ loop model Korzhenkova, Aleksandra Probability In this paper we investigate pointed $(\mathbf{q}, g, n)$-Boltzmann loop-decorated maps with loops traversing only inner triangular faces. Using the peeling exploration of arXiv:1809.02012 modified to this setting we show that its law in the non-generic critical phase can be coded in terms of a random walk confined to the positive integers by a new specific boundary condition. Under a technical assumption that we believe to be true, combining this observation with explicit quantities for the peeling law we derive the large deviations property for the distribution of the so-called nesting statistic and show that the exploration process possesses exactly the same scaling limit as in the rigid loop model on bipartite maps that is a specific self-similar Markov process introduced in arXiv:1809.02012. Besides, we conclude the equivalence of the admissible weight sequences related by the so-called fixed point equation by proving the missing direction in the argument of arXiv:1202.5521. |
| title | The exploration process of critical Boltzmann planar maps decorated by a triangular $O(n)$ loop model |
| topic | Probability |
| url | https://arxiv.org/abs/2112.11576 |