Random walks in Dirichlet random environment in dimension $d+1$

The atypical behaviour of random walks in time-dependent random environment was recently related to Kardar-Parisi-Zhang (KPZ) growth.

While this is now well-understood in spatial dimension $d=1$, further efforts are necessary to better understand these connections in dimensions $d>1$.

In this paper, we study this problem numerically for $d=1, 2$ and $3$, focusing on a discrete model with Dirichlet distributed transition probabilities.

This model is a generalization of an integrable model in $d=1$, and it has the advantage of admitting an explicit, product-form, stationary measure.

We verify that the growth of the variance of the logarithm of point-to-point probabilities, namely from the origin to position $x$ in time $t$, is compatible with KPZ growth in dimension $d=1$ and $d=2$.

In spatial dimension $d=3$, we confirm the existence of a phase transition as the angle $\vert x\vert /t$ increases and we obtain a lower bound based on an exact second moment calculation.

We find that in the weak disorder phase the point-to-point probability acquires a heavy tailed distribution, and that in the strong disorder phase the cumulants of its logarithm grow with time.

Further, we show that for this model, we can compute exactly the sample to sample variance of the thermal average $\overline{ \langle x \rangle^2}$ and that it is related to the extreme diffusion coefficient introduced recently.