Transient random walks on the space of lattices

Given $d\geq2$, we construct a Zariski-dense random walk on the space of lattices SL$_d(\mathbb{R})/$SL$_d(\mathbb{Z})$ that exhibits escape of mass. This negates the suggestion of recurrence made by Benoist [Ben14] (ICM 2014) and by B\'enard-de Saxc\'e [BS22] (also asked in [BQ12]). For any $p \in (0,1)$, we also construct such a random walk with finite $L^p$-moment which shows that the moment assumption in [BS22] is sharp.