We consider a model of the one dimensional discrete random Schrodinger operator on Z given by H = L + V, where L is the discrete Laplacian and V is a potential on Z. When V=0 (non-random) the spectrum is absolutely continuous and there are no L^2 eigenfunctions. For certain classes of random potentials it is known that the spectrum is pure point and there is complete L^2 basis of eigenfunctions which decay exponentially (Goldshield, Kunz and Souillard, Carmona et al), a phenomenon known as Anderson localization. In this talk I will discuss the shape of a typical eigenvector - it is an exponential of Brownian motion with drift. This result relies on the transfer matrix framework developed by Kritchevski, Valko, and Virag (2011) and is joint work with Balint Virag.

We study the competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following power-law distribution with exponent tau in (2,3). In this model, two colors spread with fixed but not necessarily equal speeds on the unweighted random graph. We show that almost all vertices ultimately get the `faster' color, while only a random subpolynomial fraction of the vertices get the `slower' color.
We also show that even if the speeds are equal, there is no coexistence with high probability, and further the `loser' color paints a polynomial fraction of the vertices with a random exponent.
This work is the counterpart of Deijfen and van der Hofstad, where there are exponential edge weights on the same graph model. Changing the edge weights significantly changes the picture: in the exponential case, even when the speeds are unequal, the `winner' can be either of the two colors, and the `loser' can only paint a set of vertices of constant order size.
Joint work with Enrico Baroni and Remco van der Hofstad.

We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on Z for q in the [0,1) interval where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of q-TASEP at time t is of order t^{1/3} and asymptotically distributed as the GUE Tracy-Widom distribution. The talk is based on joint work with Patrik Ferrari.