Gwynne and Miller's graph-metric conjecture for critical FK planar maps (0 < q < 4) and spanning-tree-weighted maps, finite spherical case

Principal theorem: for fixed $q\in(0,4)$ (critical FK maps weighted by $q^{\ell/2}$, $\ell$ the number of interfaces) and for uniform map-spanning-tree pairs, $n$-edge spherical maps with degree measure and rescaled graph distance converge in GHP law to the unit-area $\gamma$-quantum sphere with its LQG metric and area. Companions: joint conformal (flag-triangle uniformization), metric, area and CLE-interface limits for $0<q<4$; the critical sphere and CLE$_4$-decorated limits at $q=4$; stationary random walk converging to Liouville Brownian motion with time acceleration exactly the number of edges for FK-Ising and spanning-tree maps; eigenvalue and heat-trace convergence for FK-Ising. Not shown: the other ensembles in Conjecture 1.7, infinite-volume or boundary versions, rates, or uniformity in $q$.

Principal theorem: for fixed $q\in(0,4)$ (critical FK maps weighted by $q^{\ell/2}$, $\ell$ the number of interfaces) and for uniform map-spanning-tree pairs, $n$-edge spherical maps with degree measure and rescaled graph distance converge in GHP law to the unit-area $\gamma$-quantum sphere with its LQG metric and area. Companions: joint conformal (flag-triangle uniformization), metric, area and CLE-interface limits for $0<q<4$; the critical sphere and CLE$_4$-decorated limits at $q=4$; stationary random walk converging to Liouville Brownian motion with time acceleration exactly the number of edges for FK-Ising and spanning-tree maps; eigenvalue and heat-trace convergence for FK-Ising. Not shown: the other ensembles in Conjecture 1.7, infinite-volume or boundary versions, rates, or uniformity in $q$.