The height gap of planar Brownian motion is 5/π

We show that the occupation measure of planar Brownian motion exhibits a constant height gap of 5/π 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 SLE₄/CLE₄ curves. Heuristically, our result can also be thought of as the θ → 0⁺ limit of the height gap property of a field built out of a Brownian loop soup with subcritical intensity θ>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.