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The height gap of planar Brownian motion is 5/π
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DOI：10.1007/s00440-026-01475-z.png)
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Abstract
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We show that the occupation measure of planar Brownian motion exhibits a constant height gap of across its outer boundary. This property bears similarities with the celebrated results of Schramm-Sheffield (Schramm and Sheffield in Probab Theory Related Fields 157(1-2):47-80, 2013) and Miller-Sheffield (Miller J, Sheffield S in CLE(4) and the Gaussian free field. In preparation) concerning the height gap of 5/pi the Gaussian free field across SLE4/CLE4 curves. Heuristically, our result can also be thought of as the theta -> 0(+ )limit of the height gap property of a field built out of a Brownian loop soup with subcritical intensity theta>0, proved in our recent paper (Jego et al. in Conformally invariant fields out of Brownian loop soups. arXiv:2307.10740, 2023). To obtain the explicit value of the height gap, we rely on the computation by Garban and Trujillo Ferreras (Garban and Trujillo Ferreras in The expected area of the filled planar Brownian loop is pi/5. Commun Math Phys 264(3):797-810, 2006) [3] of the expected area of the domain delimited by the outer boundary of a Brownian bridge.
Keywords:
INTERSECTION EXPONENTS
LOOP
POINTS
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