Random-walk in Beta-distributed random environment

Understand

We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE.

• This is a random walk in $\mathbb{Z}$ which performs nearest neighbour jumps with transition probabilities drawn according to the Beta distribution.
• We also describe a related directed polymer model, which is a limit of the $q$-Hahn interacting particle system.
• Using a Fredholm determinant representation for the quenched probability distribution function of the walker's position, we are able to prove second order cube-root scale corrections to the large deviation principle satisfied by the walker's position, with convergence to the Tracy-Widom distribution.