We consider the edge Hall conductance and show it is invariant under perturbations located in a strip along the edge. This enables us to prove for the edge conductances a general sum rule relating currents due to the presence of two different media located respectively on the left and on the right half plane. As a particular interesting case we put forward a general quantization formula for the difference of edge Hall conductances in semi-infinite samples with and without a confining wall. Applications to disordered Hall systems with and without a confining potential are discussed.

We study the following model for the spread of a rumor or infection: There is a ``gas'' of so-called $A$-particles, each of which performs a continuous time simple random walk on $\Bbb Z^d$, with jumprate $D_A$. We assume that ``just before the start'' the number of $A$-particles at $x$, $N_A(x,0-)$, has a mean $\mu_A$ Poisson distribution and that the $N_A(x,0-), \, x \in \Bbb Z^d$, are independent.
In addition, there are $B$-particles which perform continuous time simple random walks with jumprate $D_B$. We start with a finite number of $B$-particles in the system at time 0. The positions of these initial $B$-particles are arbitrary, but they are non-random. The $B$-particles move independently of each other. The only interaction is that when a $B$-particle and an $A$-particle coincide, the latter instantaneously turns into a $B$-particle. \cite {KSb} gave some basic estimates for
the growth of the set $\wt B(t):= \{x \in \Bbb Z^d:$ a $B$-particle visits $x$ during $[0,t]$\}. In this article we show that if $D_A=D_B$, then $B(t) = \wt B(t) + [-\frac 12, \frac 12]^d$ grows linearly in time with an asymptotic shape, i.e., there
exists a non-random set $B_0$ such that $(1/t)B(t) \to B_0$, in a sensewhich will be made precise. Joint work with H. Kesten.

The integral QHE can be explained either as resulting from bulk or
edge currents (or, in reality, as a combination of both). The equality
of the two conductances at zero temperature was recently established
for the case that the Fermi energy falls in the spectral gap of the bulk
system. We define the edge conductance via a suitable time averaging
procedure in the more general case of a bulk system which exhibits
dynamical localization in the vicinity of the Fermi energy, and show
that the two conductances are equal.
This is a joint work with G.-M. Graf and J. Schenker.

Bond-percolation graphs are random subgraphs of the d-dimensional
integer lattice generated by a standard Bernoulli bond-percolation
process. The
associated graph Laplacians, subject to Dirichlet or Neumann conditions at
cluster boundaries, represent bounded, self-adjoint, ergodic random
operators. They possess almost surely the
non-random spectrum [0,4d] and a self-averaging integrated density
of states. This integrated density of states is shown to exhibit Lifshits
tails at both spectral edges in the non-percolating phase. Depending
on the boundary condition and on the spectral edge, the Lifshits tail
discriminates between different cluster geometries (linear clusters
versus cube-like
clusters) which contribute the dominating eigenvalues. Lifshits tails
arising
from cube-like clusters continue to show up above the percolation
threshold.
In contrast, the other type of Lifshits tails cannot be observed in the
percolating
phase any more because they are hidden by van Hove singularities from the
percolating cluster.