Abstract：We study the question of whether there exist double points on the boundaries of clusters in planar Brownian loop-soups - an object introduced by Lawler and Werner in 2004. This question is motivated by our earlier works (with Werner) on the decomposition of Brownian loop-soup clusters. More concretely, we introduce a notion of disconnection exponents which generalizes the Brownian disconnection exponents derived by Lawler, Schramm and Werner in 2001. By computing the generalized disconnection exponents, we can predict the dimension of multiple points on the cluster boundaries in Brownian loop-soups - their dimensions have not been conjectured before. We plan to rigorously confirm the prediction in a future work. According to our prediction, there exist double points on the cluster boundaries of Brownian loop-soups with any subcritical intensity. Interestingly, for the critical intensity, the dimension of double points on the cluster boundaries becomes zero, leaving an open question of whether such points exist for the critical loop-soup.
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