GUE Fluctuations Near the Axis in One-Sided Ballistic Deposition

We introduce a variation of the classic ballistic deposition model in which vertically falling blocks can only stick to the top or the upper right corner of growing columns. We establish that fluctuations of the height function at points near the $t$-axis are given by the GUE ensemble and its corresponding Tracy-Widom limiting distribution. The proof is based on a graphical construction of the process in terms of a directed Last Passage Percolation model. Using this graphical construction, we define the notion of geodesics for the height function and show that the wandering exponent governing the transversal fluctuations of these geodesics is $2/3$.