  Random walk in a dynamic i.i.d. beta random environment, conditioned to
escape at an atypical velocity, converges to a Doob transform of the original
walk. The Doob-transformed environment is correlated in time, i.i.d. in space,
and its marginal density function is a product of a beta density and a
hypergeometric function. Under its averaged distribution the transformed walk
obeys the wandering exponent 2/3 that agrees with Kardar-Parisi-Zhang
universality. The harmonic function in the Doob transform comes from a
Busemann-type limit and appears as an extremal in a variational problem for the
quenched large deviation rate function.
