Massive CLE2 boundary conjecture for the canonical evaporated UST component

Condition on the event t(r)=n|{\bf t}(r)|=n, and let Canonical(t<n(r)){\sf Canonical}({\bf t}^{<n}(r)) be the canonical embedding of the root component obtained by edge evaporation. The massive CLE2CLE_2 boundary conjecture. The rescaled vertex sets

Canonical(t<n(r))n\frac{{\sf Canonical}({\bf t}^{<n}(r))}{\sqrt n}

converge in distribution for the Hausdorff metric on compact subsets of R2\mathbb{R}^2 to a limiting compact set KK with Lebesgue measure 11 that is simply connected, and the boundary of KK is a massive version of CLE2CLE_2 conditioned to have area 11. This is a conjectural description of the continuum boundary of the conditioned evaporated-tree shape; the source gives no proof or resolution.

Sources & referencesView supporting material

Primary source

Luis Fredes and Jean-Francois Marckert, “Models of random subtrees of a graph”, arXiv:2102.12738 (2023).

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