Abstract
We introduce the first passage set (FPS) of constant level \(-a\) of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path on which the GFF does not go below \(-a\). It is, thus, the two-dimensional analogue of the first hitting time of \(-a\) by a one-dimensional Brownian motion. We provide an axiomatic characterization of the FPS, a continuum construction using level lines, and study its properties: it is a fractal set of zero Lebesgue measure and Minkowski dimension 2 that is coupled with the GFF \(\Phi \) as a local set A so that \(\Phi +a\) restricted to A is a positive measure. One of the highlights of this paper is identifying this measure as a Minkowski content measure in the non-integer gauge \(r \mapsto \vert \log (r)\vert ^{1/2}r^{2}\), by using Gaussian multiplicative chaos theory.
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Notes
Here and elsewhere this means piecewise constant that changes only finitely many times.
The continuation threshold is the first time \(\tau \) in which the level line hits a boundary point, \(x\in \partial D\), such that there is no level line of \(\Phi +u+h_{\eta _\tau }\) starting from x in O. Here O is the non bounded connected component of \(D\backslash \eta _\tau \). This condition can be described explicitly using boundary values. See, for example, Definition 2.14 in [42].
In that paper we only use a specific construction of the FPS, basically the one given by Proposition 4.4, and properties stemming from that construction.
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Acknowledgements
The authors wish to thank F. Viklund for helpful comments on an earlier draft, W. Werner for his vision, and B. Werness for his beautiful simulations and interesting discussions. Also, the authors are very thankful to the two anonymous referees for their careful reading and helpful remarks. This work was partially supported by the SNF grants SNF-155922 and SNF-175505. A. Sepúlveda was supported by the ERC grant LiKo 676999. The authors are thankful to the NCCR Swissmap. T. Lupu acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation. The work of this paper was finished during the memorable visit of J.Aru and A. Sepúlveda to Paris in May 2018, on the invitation by T. Lupu, funded by PEPS “Jeunes chercheuses et jeunes chercheurs” 2018 of INSMI.
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Aru, J., Lupu, T. & Sepúlveda, A. First passage sets of the 2D continuum Gaussian free field. Probab. Theory Relat. Fields 176, 1303–1355 (2020). https://doi.org/10.1007/s00440-019-00941-1
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DOI: https://doi.org/10.1007/s00440-019-00941-1
Keywords
- First passage sets
- Gaussian free field
- Gaussian multiplicative chaos
- Local set
- Schramm–Loewner evolution
- Two-valued local sets
Mathematics Subject Classification
- 60G15
- 60G60
- 60J65
- 60J67
- 81T40