<p>We show that the occupation measure of planar Brownian motion exhibits a constant height gap of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$5/\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>5</mn> <mo stretchy="false">/</mo> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> across its outer boundary. This property bears similarities with the celebrated results of Schramm–Sheffield (Schramm and Sheffield in Probab Theory Related Fields 157(1-2):47–80, 2013) and Miller–Sheffield (Miller J, Sheffield S in CLE(4) and the Gaussian free field. In preparation) concerning the height gap of the Gaussian free field across <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\hbox {SLE}_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>SLE</mtext> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>/<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\hbox {CLE}_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>CLE</mtext> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> curves. Heuristically, our result can also be thought of as the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\theta \rightarrow 0^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> limit of the height gap property of a field built out of a Brownian loop soup with subcritical intensity <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\theta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, proved in our recent paper (Jego et al. in Conformally invariant fields out of Brownian loop soups. <a href="http://arxiv.org/abs/2307.10740">arXiv:2307.10740</a>, 2023). To obtain the explicit value of the height gap, we rely on the computation by Garban and Trujillo Ferreras (Garban and Trujillo Ferreras in The expected area of the filled planar Brownian loop is <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\pi /5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo stretchy="false">/</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. Commun Math Phys 264(3):797–810, 2006) [<CitationRef CitationID="CR3">3</CitationRef>] of the expected area of the domain delimited by the outer boundary of a Brownian bridge.</p>

The height gap of planar Brownian motion is $\frac{5}{\pi }$

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2026年3月11日

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Antoine Jego,
Titus Lupu,
Wei Qian

摘要

We show that the occupation measure of planar Brownian motion exhibits a constant height gap of $5/\pi $ $5 / π$ 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 the Gaussian free field across $\hbox {SLE}_4$ ${SLE}_{4}$ / $\hbox {CLE}_4$ ${CLE}_{4}$ curves. Heuristically, our result can also be thought of as the $\theta \rightarrow 0^+$ $θ \to 0^{+}$ limit of the height gap property of a field built out of a Brownian loop soup with subcritical intensity $\theta >0$ $θ > 0$ , 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 $\pi /5$ $π / 5$ . 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.

The height gap of planar Brownian motion is \(\frac{5}{\pi }\)

摘要