Characterisation of planar Brownian multiplicative chaos
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866911290406993920 |
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| author | Jego, Antoine |
| author_facet | Jego, Antoine |
| contents | We characterise the multiplicative chaos measure $\mathcal{M}$ associated to planar Brownian motion introduced in [BBK94,AHS20,Jeg20a] by showing that it is the only random Borel measure satisfying a list of natural properties. These properties only serve to fix the average value of the measure and to express a spatial Markov property. As a consequence of our characterisation, we establish the scaling limit of the set of thick points of planar simple random walk, stopped at the first exit time of a domain, by showing the weak convergence towards $\mathcal{M}$ of the point measure associated to the thick points. In particular, we obtain the convergence of the appropriately normalised number of thick points of random walk to a nondegenerate random variable. The normalising constant is different from that of the Gaussian free field, as conjectured in [Jeg20b]. These results cover the entire subcritical regime.
A key new idea for this characterisation is to introduce measures describing the intersection between different Brownian trajectories and how they interact to create thick points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_05067 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Characterisation of planar Brownian multiplicative chaos Jego, Antoine Probability We characterise the multiplicative chaos measure $\mathcal{M}$ associated to planar Brownian motion introduced in [BBK94,AHS20,Jeg20a] by showing that it is the only random Borel measure satisfying a list of natural properties. These properties only serve to fix the average value of the measure and to express a spatial Markov property. As a consequence of our characterisation, we establish the scaling limit of the set of thick points of planar simple random walk, stopped at the first exit time of a domain, by showing the weak convergence towards $\mathcal{M}$ of the point measure associated to the thick points. In particular, we obtain the convergence of the appropriately normalised number of thick points of random walk to a nondegenerate random variable. The normalising constant is different from that of the Gaussian free field, as conjectured in [Jeg20b]. These results cover the entire subcritical regime. A key new idea for this characterisation is to introduce measures describing the intersection between different Brownian trajectories and how they interact to create thick points. |
| title | Characterisation of planar Brownian multiplicative chaos |
| topic | Probability |
| url | https://arxiv.org/abs/1909.05067 |