  For the exactly solvable model of exponential last passage percolation on
$\mathbb{Z}^2$, consider the geodesic $\Gamma_n$ joining $(0,0)$ and $(n,n)$
for large $n$. It is well known that the transversal fluctuation of $\Gamma_n$
around the line $x=y$ is $n^{2/3+o(1)}$ with high probability. We obtain the
exponent governing the decay of the small ball probability for $\Gamma_{n}$ and
establish that for small $\delta$, the probability that $\Gamma_{n}$ is
contained in a strip of width $\delta n^{2/3}$ around the diagonal is $\exp
(-\Theta(\delta^{-3/2}))$ uniformly in high $n$. We also obtain optimal small
deviation estimates for the one point distribution of the geodesic showing that
for $\frac{t}{2n}$ bounded away from $0$ and $1$, we have
$\mathbb{P}(|x(t)-y(t)|\leq \delta n^{2/3})=\Theta(\delta)$ uniformly in high
$n$, where $(x(t),y(t))$ is the unique point where $\Gamma_{n}$ intersects the
line $x+y=t$. Our methods are expected to go through for other exactly solvable
models of planar last passage percolation and, upon taking the $n\to \infty$
limit, provide analogous estimates for geodesics in the directed landscape.
