  In last passage percolation models, the energy of a path is maximized over
all directed paths with given endpoints in a random environment, and the
maximizing paths are called geodesics. The geodesics and their energy can be
scaled so that transformed geodesics cross unit distance and have fluctuations
and scaled energy of unit order. Here we consider Poissonian last passage
percolation, a model lying in the KPZ universality class, and refer to scaled
geodesics as polymers and their scaled energies as weights. Polymers may be
viewed as random functions of the vertical coordinate and, when they are, we
show that they have modulus of continuity whose order is at most
$t^{2/3}\big(\log t^{-1}\big)^{1/3}$. The power of one-third in the logarithm
may be expected to be sharp and in a related problem we show that it is: among
polymers in the unit box whose endpoints have vertical separation $t$ (and a
horizontal separation of the same order), the maximum transversal fluctuation
has order $t^{2/3}\big(\log t^{-1}\big)^{1/3}$. Regarding the orthogonal
direction, in which growth occurs, we show that, when one endpoint of the
polymer is fixed at $(0,0)$ and the other is varied vertically over $(0,z)$,
$z\in [1,2]$, the resulting random weight profile has sharp modulus of
continuity of order $t^{1/3}\big(\log t^{-1}\big)^{2/3}$. In this way, we
identify exponent pairs of $(2/3,1/3)$ and $(1/3,2/3)$ in power law and
polylogarithmic correction, respectively for polymer fluctuation, and polymer
weight under vertical endpoint perturbation. The two exponent pairs describe
[10,11,9] the fluctuation of the boundary separating two phases in subcritical
planar random cluster models.
