The Ten Martini problem asked to show that the spectrum
of the Almost-Mathieu operator, that is, a Schroedinger operator with potential V (n) = 2 cos(2 pi alpha n), is a Cantor set. In particular, this means that the spectrum does not contain any intervals. The Ten Martini problem was solved in 2009 by Avila and Jitomirskaya. I will show that such a claim is false for the generalization to the potential V(n) =2 cos(2 pi alpha n^2), which is known as skew-shift Schroedinger operator. The proof relies on localization properties of this operator and that the phase space of the skew-shift is two dimensional, whereas it is one dimensional for the rotations underlying the Almost-Mathieu operator.

In statistical physics, systems like percolation and Ising models are of particular interest at their critical points. Critical systems have long-range correlations that typically decay like inverse powers. Their continuum scaling limits, in which the lattice spacing shrinks to zero, are believed to have universal dimension-dependent properties. In recent years critical two-dimensional scaling limits have been studied by Schramm, Lawler, Werner, Smirnov and others with a focus on the boundaries of large clusters. In the scaling limit these can be described by Schramm-Loewner Evolution (SLE) curves.

In this talk, I'll discuss a different but related approach, which focuses on cluster area measures. In the case of the two-dimensional Ising model, this leads to a representation of the continuum Ising magnetization field in terms of sums of certain measure ensembles with random signs. This is based on joint work with F. Camia (PNAS 106 (2009) 5457-5463) and on work in progress with F. Camia and C. Garban.