# The many faces of KPZ

### Random walks in space-time random environment

Consider a discrete-time random walk on the integer lattice with nearest-neighbour jumps, starting from 0. We assume that
$\hspace{2cm} \mathsf{P}\big( X_{t+1}=X_t+1\vert X_t=x\big) = B_{t,x} \text{ and } \mathsf{P}\big( X_{t+1}=X_t+1\vert X_t=x\big) = 1- B_{t,x},$
where B_{t,x} are i.i.d. random variables with Beta distribution. In particular, they can be uniform random variables in (0,1).

The quenched large deviation behaviour of this random walk is related to the universal free energy fluctuations of directed polymers in the KPZ universality class. In particular, when time goes to infinity
$\hspace{2cm} \dfrac{\log\Big(\mathsf{P}\big( X_t>x t\big)\Big) - I(x)t}{\sigma(x)t^{1/3}} \Longrightarrow \mathcal{L}_{GUE},$

More details in Random walks in Beta-distributed random environment(Guillaume Barraquand and Ivan Corwin).