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

Abstract

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 SLE$_4$/CLE$_4$ curves. Heuristically, our result can also be thought of as the $θ\to 0^+$ 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.

Figures (5)

• Figure 3.1: On the left, we depict the decomposition of a Brownian bubble $\mathcal{P}$, where the event $E_\varepsilon$ occurs. The excursions in (1) (2) (3) are respectively drawn in blue, red and green. The picture on the left is mapped by $f$ to the picture on the right.
• Figure 4.1: Illustration of the two scenarios forbidden by the good event $G_\delta(z)$. On the left, a blue loop with one end point outside of $(z_-,z_+)$ hits the ball $B(z,\delta^{100})$ depicted in yellow. On the right, a blue loop with one endpoint in $(z_-,z_+)$ has a diameter exceeding $\delta^{1/100}$.
• Figure 4.2: Illustration of notations and an event appearing in the proof of Lemma \ref{['L:first_good']}. On the left, an excursion with an endpoint in $(e^{-i\delta} x/|x|,e^{+i\delta} x/|x|)$ has a diameter at least $\delta^{1/10}/2$ and thus intersects $C_\delta$. The right picture is the image of the left picture under the conformal map sending $x/|x|$ to infinity, $w_1$ to $1$ and $-x/|x|$ to $0$. The points $e^{\pm i\delta} x/|x|$ are then mapped to $\mp R$, $w_2$ to $-1$ and $x$ to $z$.
• Figure 4.3: Illustration of the setup of the proof of Lemma \ref{['L:second_good']}. The outer boundary $\gamma$ is the doted red curve and $e_{\max}$ is the blue excursion. On the event $E_1(\mathcal{E}_{[-1,1]})$, $e_{\max}$ is the only excursion reaching $\mathbb{H}_{\eta_2}$. On the event $E_2(\mathcal{E}_{[-1,1]})$, $e_{\max}$ does not visit $\varphi_\gamma(B(z,\delta^{100}))$ which is the region depicted in yellow.
• Figure 4.4: Schematic representation of all the excursions in $\varphi_\gamma^{-1}(\mathcal{E}_{[-1,1]})$ with at least one endpoint in $(z_-,z_+)$. $z_{\max,+}$ (resp. $z_{\max,-}$) is the endpoint of such an excursion that lies on the arc $(z,w)$ (resp. $(w,z)$) and whose distance to $z$ is maximal. By definition, there is no excursion in $\varphi_\gamma^{-1}(\mathcal{E}_{[-1,1]})$ with one endpoint on $(z_-,z_+)$ and one endpoint on $(z_{\max,+},z_{\max,-})$.

Theorems & Definitions (45)

• Theorem 1.1
• Definition 2.1: Space of loops $\mathcal{L}$
• Definition 2.2: Space of self-avoiding loops $\Gamma$
• Theorem 2.3
• Lemma 2.4: Proposition 3.38, MR2129588
• Lemma 2.5
• proof
• Corollary 2.6
• proof
• Proposition 3.1: Proposition 3.6, MR3901648