The fifth PhD students’ conference is jointly hosted by the IMPRS MIS Leipzig, the Research Academy Leipzig and the DMV Fachgruppe Stochastik. PhD students are expected to participate and/or present a 25- minute talk on an area within Probability.

In my minicourse I explain growth models in the Kardar-Parisi-Zhang universality class in one spatial dimension. In particular I cover various growth models, their stationary measures, scaling behavior, the link to tiling problems, directed last passage percolation, and to random matrices.

In the random conductance model, each edge of the hypercubic lattice is assigned a positive and finite random variable called the conductance. The resulting (random) resistor network is directly linked to a Markov chain --- referred to, invariably, as a random walk among random conductances --- where at each step the walk chooses a neighbor at random with probability proportional to the conductance of the corresponding edge. The conductances are distributed according to a translation-invariant, or even iid, law; the properties of the random walk are studied against a typical sample from this law (these are the so called quenched problems).
In my three lectures I will discuss the following facts/situations:(1) The proof of recurrence and transience for this random walk and the connection with effective resistivity.(2) The proof of an invariance principle for the path of the random walk in the case when the conductances are bounded away from zero and infinity (so called elliptic case).(3) Extensions to non-elliptic situations (random walk on the supercritical percolation cluster, regular and anomalous heat-kernel decay).
I will finish with an outlook of problems awaiting solutions and possible directions of future research. My contributions to this field are based on joint works with N. Berger, T. Prescott, G. Kozma and C. Hoffman.

Many calculations on the SK-Model use its quite special properties. In order to understand the problem more thoroughly we treat a generalized version where spins are not constrained to Ising type. We calculate its free energy by use of the ideas appearing in Talagrand (2009).

In this talk, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble $GCy$, whose eigenvalues PDF is given by $$\textrm{const}\cdot\prod_{1\leq j-1/2$ and where $N$ is the size of the matrix ensemble. We will see that for this ensemble, the appropriately rescaled largest eigenvalue converges in law. We also express the limiting probability distribution in terms of some non-linear second order differential equation. Eventually, we show that the convergence of the probability distribution function of the re-scaled largest eigenvalue to the limiting one is at least of order $(1/N)$.

We consider a particular Branching Random Walk in Random Environment (BRWRE) on $N_0$ started with one particle at the origin. Particles reproduce according to an offspring distribution (which depends on the location) and move either one step to the right (with a probability in $(0,1]$ which may also depend on the location) or stay in the same place. We give criteria for local and global survival and show that global survival is equivalent to exponential growth of the moments.

Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions $d\ge 2$. In 2002, Sznitman introduced for each $\gamma\in (0,1)$ the ballisticity conditions $(T)_\gamma$ and $(T'),$ the latter being defined as the fulfilment of $(T)_\gamma$ for all $\gamma\in (0,1)$. He proved that $(T')$ implies ballisticity and that for each $\gamma\in (0.5,1)$, $(T)_\gamma$ is equivalent to $(T')$. It is conjectured that this equivalence holds for all $\gamma\in (0,1)$. Here we prove that for $\gamma\in (\gamma_d,1)$, where $\gamma_d$ is a dimension dependent constant taking values in the interval $(0.366,0.388)$, $(T)_\gamma$ is equivalent to $(T')$.