Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation

For exponential last passage percolation on the plane we analyse the probability that the point-to-line geodesic exhibits an atypically large transversal fluctuation at the endpoint as well as the probability that the point-to-point geodesic exhibits an atypically large transversal fluctuation at the halfway point. In particular, we show that p^*ₙ(t), the probability that the point-to-line geodesic from the origin to the line $x+y=2n$ ends at (n-t(2n)2/3, n+t(2n)2/3) satisfies that n2/3p^*ₙ(t)=exp(-(4/3+o(1))t³) for t large and pn,1/2(t), the probability that the geodesic from the origin to the point $(n,n)$ passes through the point (1/2n-tn2/3, 1/2 n+tn2/3), satisfies n2/3pn,1/2(t)=exp(-(8/3+o(1))t³) for t large. The latter result solves a special case of a conjecture from Liu (PTRF, 2022).