  We consider the Beta polymer, an exactly solvable model of directed polymer
on the square lattice, introduced by Barraquand and Corwin. We study the
statistical properties of its point to point partition sum. The problem is
equivalent to a model of a random walk in a time-dependent (and in general
biased) 1D random environment. In this formulation, we study the sample to
sample fluctuations of the transition probability distribution function (PDF)
of the random walk. Using the Bethe ansatz we obtain exact formulas for the
integer moments, and Fredholm determinant formulas for the Laplace transform of
the directed polymer partition sum/random walk transition probability. The
asymptotic analysis of these formulas at large time $t$ is performed both (i)
in a diffusive vicinity, $x \sim t^{1/2}$, of the optimal direction (in
space-time) chosen by the random walk, where the fluctuations of the PDF are
found to be Gamma distributed; (ii) in the large deviations regime, $x \sim t$,
of the random walk, where the fluctuations of the logarithm of the PDF are
found to grow with time as $t^{1/3}$ and to be distributed according to the
Tracy-Widom GUE distribution. Our exact results complement those of BC for the
cumulative distribution function of the random walk in regime (ii), and in
regime (i) they unveil a novel fluctuation behavior. We also discuss the
crossover regime between (i) and (ii), identified as $x \sim t^{3/4}$. Our
results are confronted to extensive numerical simulations of the model.
