  We consider planar stationary exponential Last Passage Percolation in the
positive quadrant with boundary weights. For $\rho\in (0,1)$ and points
$v_N=((1-\rho)^2 N,\rho^2 N)$ going to infinity along the characteristic
direction, we establish right tail estimates with the optimal exponent for the
exit time of the geodesic, along with optimal exponent estimates for the upper
tail moderate deviations for the passage time. For the case $\rho=\frac{1}{2}$
in the stationary model, we establish the lower bound estimate with the optimal
exponent for the lower tail of the passage time. Our arguments are based on
moderate deviation estimates for point-to-point and point-to-line exponential
Last Passage Percolation which are obtained via random matrix estimates.
