By Wei WU, NYU Shanghai

The minimal spanning tree model has been widely applied to combinatorial optimizations and to the study of disordered physical systems. For the infinite lattice $\mathbb{Z}^d$, rigorous results for the geometry of the minimal spanning forest were recently proved for $d= 2$ and still  remain open for $d \geq 3$. We made partial progress by proving that the minimal spanning forest measure is supported on a single tree for quasi-planar graphs, such as the two dimensional slabs. Our proof uses the connections between the minimal spanning forests and critical bond percolations, and certain generalizations of gluing lemmas for bond percolation.

Based on joint work with Charles Newman and Vincent Tassion.