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

We show that the occupation measure of planar Brownian motion exhibits a constant height gap of $5\mathrm{/}\pi$ across its outer boundary. This property bears similarities with the celebrated results of Schramm–Sheffield [18] and Miller–Sheffield [12] concerning the height gap of the Gaussian free field across SLE${}_4$/CLE${}_4$ curves. Heuristically, our result can also be thought of as the $\theta \to 0^{+}$ 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 [3]. To obtain the explicit value of the height gap, we rely on the computation by Garban and Trujillo Ferreras [1] of the expected area of the domain delimited by the outer boundary of a Brownian bridge.