Sheffield's finite-volume continuum random tree prediction for critical FK planar maps with q > 4

Theorem 1.1: for every fixed $q>4$ there is $c_q>0$ such that the $n$-edge critical FK map, with graph distance scaled by $c_qn^{-1/2}$ and the normalised degree measure, converges in distribution to the Brownian continuum random tree in the GHP topology, through all positive integers $n$. It is proved directly under the finite law (block decomposition into a simply generated tree with finite offspring variance), not deduced from Feng's infinite-volume local limit. Not shown: the value of $c_q$, rates, uniformity in $q$, or anything at $q=4$ (treated in a companion).

Theorem 1.1: for every fixed $q>4$ there is $c_q>0$ such that the $n$-edge critical FK map, with graph distance scaled by $c_qn^{-1/2}$ and the normalised degree measure, converges in distribution to the Brownian continuum random tree in the GHP topology, through all positive integers $n$. It is proved directly under the finite law (block decomposition into a simply generated tree with finite offspring variance), not deduced from Feng's infinite-volume local limit. Not shown: the value of $c_q$, rates, uniformity in $q$, or anything at $q=4$ (treated in a companion).