The first passage sets of the 2D Gaussian free field: convergence and isomorphisms

Abstract

In a previous article, we introduced 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 along which the GFF is greater than or equal to $-a$. This description can be taken as a definition of the FPS for the metric graph GFF, and it justifies the analogy with the first hitting time of $-a$ by a one-dimensional Brownian motion. In the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff metric. This allows us to show that the FPS of the continuum GFF can be represented as a union of clusters of Brownian excursions and Brownian loops, and to prove that Brownian loop soup clusters admit a non-trivial Minkowski content in the gauge $r\mapsto |\log r{|}^{1/2}{r}^2$. We also show that certain natural interfaces of the metric graph GFF converge to SLE${}_4$ processes.

arXiv commentOriginally contained in arXiv:1706.07737. Because of new results and to improve readability, it is now a paper on its own. 37 pages, 5 hand-drawn images

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Cited by in corpus (12)

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• Percolation for the Gaussian free field on the cable system: counterexamples
• Crossing exponent in the Brownian loop soup
• Percolation for two-dimensional excursion clouds and the discrete Gaussian free field
• The distance exponent for Liouville first passage percolation is positive
• On the chemical distance exponent for the two-sided level-set of the 2D Gaussian free field
• A level line of the Gaussian free field with measure-valued boundary conditions
• A Ray-Knight theorem for $\nabla \varphi$ interface models and scaling limits