Multiple points on the boundaries of Brownian loop-soup clusters

TL;DR

This work determines the fractal structure of boundary points on Brownian loop-soup clusters in the unit disk for $c\in(0,1]$. By introducing a separation lemma for loop-soup arm events and establishing up-to-constants estimates that link generalized non-intersection/disconnection exponents $\xi_c(k,\lambda)$ to $\xi_c(k)$, the authors prove that the Hausdorff dimensions of simple and double points on cluster boundaries equal $2-\xi_c(2)$ and $2-\xi_c(4)$, respectively, with no triple points. They also show that these points are dense on boundary arcs when the dimension is positive and provide zero-one laws via CLE-based explorations. The results bridge generalized restriction/interaction exponents with loop-soup geometry, confirming predictions from prior works and enriching the understanding of conformally invariant random fractals in two dimensions. The approach combines extremal-distance techniques, two-speed excursion decompositions, and detailed moment estimates, yielding sharp dimension results and a robust probabilistic framework for non-intersection and non-disconnection phenomena in loop soups.

Abstract

For a Brownian loop soup with intensity $c\in(0,1]$ in the unit disk, we show that almost surely, the set of simple (resp. double) points on any portion of boundary of any of its clusters has Hausdorff dimension $2-ξ_c(2)$ (resp. $2-ξ_c(4)$), where $ξ_c(k)$ is the generalized disconnection exponent computed in arxiv:1901.05436. As a consequence, when the dimension is positive, such points are a.s. dense on every boundary of every cluster. There are a.s. no triple points on the cluster boundaries. As an intermediate result, we establish a separation lemma for Brownian loop soups, which is a powerful tool for obtaining sharp estimates on non-intersection and non-disconnection probabilities in the setting of loop soups. In particular, it allows us to define a family of generalized intersection exponents $ξ_c(k, λ)$, and show that $ξ_c(k)$ is the limit as $λ\searrow 0$ of $ξ_c(k, λ)$.

Figures (12)

• Figure 1.1: We illustrate the event that $\Theta_r$ does not disconnect $\mathcal{C}_0$ from $\infty$ in the case $k=3$. We depict in grey the (filled) clusters of the Brownian loops in $\Gamma_r \setminus \Gamma_0$.
• Figure 2.1: Left: Illustration of Lemma \ref{['lem:obs']}. The black loops represent the outermost boundaries of the outermost clusters of a loop soup with intensity $c\in(0,1]$ in $\mathbb{U}$. The grey region is a sample $K$ with law $\mathbf{P}^{\alpha, \beta}_{8/3}$. The yellow line is the outer boundary of the union of $K$ with all the clusters that it intersects. The region encircled by the yellow line has law $\mathbf{P}^{\alpha, \beta}_{\kappa(c)}$. Right: A similar picture as the left one where $K$ is replaced by $\beta\in\mathbb{N}$ independent Brownian excursions in $\mathbb{U}$ between $0$ and $1$. The region encircled by the yellow line has law $\mathbf{P}^{0, \beta}_{\kappa(c)}$.
• Figure 3.1: Left: The concentric circles are $\mathcal{C}_s$ and $\mathcal{C}_r$ respectively. The black loops represent the outermost boundaries of the outermost clusters in $\Lambda_{s,r}$. The red and blue curves are excursions $Y^1_{s,r}$ and $Y^2_{s,r}$ respectively. Right: The yellow and green regions are two connected components $\widetilde{O}^1_{s,r}$ and $\widetilde{O}^2_{s,r}$ respectively.
• Figure 3.2: An illustration of the configuration that is very nice at the end (see Definition \ref{['def:nice_end']}). The three concentric circles are $\mathcal{C}_s$, $\mathcal{C}_{r-1/2}$ and $\mathcal{C}_r$, respectively. The black loops represent the outermost boundaries of the outermost clusters in $\Lambda_{s,r}$, and the red and blue curves are $Y_{s,r}^1$ and $Y_{s,r}^2$ respectively. The angles of the purple and green lines are indicated in the figure.
• Figure 3.3: Illustration for wedges defined above Lemma \ref{['lem:concatenation']} and a typical sample from the event $J$ defined in \ref{['eq:H-def']}. For convenience, we only sketch the configuration on the right hand side, corresponding to the excursion $Y^1_{s+r+2}$. First, $Y^1_{s+r+2}$ is decomposed into three parts according to its first hitting on $\mathcal{C}_s$ and its last hitting on $\mathcal{C}_{s+2}$, and they are in red, blue and red, respectively. The three wedges are $W_1$, $W_1'$ and $W_1^s$ in decreasing order. The clusters encountered by $Y^1_{s+r+2}$ and intersect with $\mathcal{A}(s,s+2)$ are sketched in green. The very nice event $G^+(s)\cap G^-(s+2,s+r+2)$ makes the red curves in $W_1$ stay in the cone with absolute angle $1/20$. The event $U$ requires the blue curve to stay in the smallest wedge $W_1^s$. The event $E$ makes sure the green clusters are sufficiently small.

Theorems & Definitions (86)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Lemma 2.1: The comparison principle, MR0357743
• Lemma 2.2: The composition laws, MR0357743
• Definition 2.3: Good domains
• Lemma 2.4: MR1901950
• Lemma 2.5
• proof
• Lemma 2.6