A number of stochastic models have their meaning, interpretation and sense only if they are embedded in a spatial context. We mainly think of spatially distributed random structures such as ensembles of point clouds, paths (e.g., loops), geometric graphs, branching trees, etc., which interact with one another. Many of the models also have a time component, i.e., they are stochastic processes of such random objects. The goal is then always to develop mathematical methods for the macroscopic description of the system. Of particular interest are systems in which phase transitions are hidden, which are brought to the surface with such methods and whose existence is rigorously proven.
One of the main objects of investigation at WIAS are models of randomly interacting loops in a large box at the thermodynamic limit, where the total length of all loops is of the order of magnitude of the volume of the box. The most prominent representative of such models is the interacting Bose gas, in which the famous Bose-Einstein condensation phase transition is suspected: the occurrence of very long loops as soon as the temperature falls below a critical limit. Such models are important prototypes of spin models, i.e. Gibbs' models of particles whose spin space is unlimited and gives rise to new effects. Two different strategies are pursued at WIAS (see also the mathematical topic "Interacting stochastic many-particle systems" and "Large deviations"), namely the analysis of the free energy of the system in the thermodynamic limit in terms of a variational description and with the help of infinitely long Brownian movements as well as the application of manipulations such as reflections and the derivation of correlation inequalities.
Another direction in which the WIAS works are spatial models for large particle clouds with a coagulation mechanism (see the application area "Coagulation"), in which the accidental formation of particularly large (macroscopic) particles for certain coagulation nuclei is investigated after a sufficiently late period in the limit of large particle systems; this is called gelation. This phase transition can be seen as a kind of explosion transition, because all other particles continue to grow normally, and every now and then one particle size jumps over this transition limit. The novelty of the work of the WIAS is to consider spatial models. Simplified models are currently being considered, in which the coagulation is not expressed by a change in the location of the two particles involved, but by the insertion of an edge; in this way a unique geometrical growing graph is created, the connected components of which are studied. The main means here is a combinatorial expansion, as well as an approach using the theory of large deviations, see the mathematical topic of the same name.
There are also decisive spatial influences in the asymptotic analysis of the parabolic Anderson model (see also the mathematical topic "Spectra of random operators"), the spatial randomness of which is given as Gaussian white noise. A meaningful definition of this model was a task in itself and is only possible in dimensions up to three; we are interested in temporally asymptotic behavior, especially with regard to the phenomenon of intermittency. This phenomenon is now well understood for spatially discrete models, but in the continuous case with white noise this is still a challenge that the WIAS faces in dimension two. Since the solution of this equation is not a function but a distribution, a formulation of the effect (namely that the main mass of the solution is concentrated on small islands) is a priori unclear and the proof is difficult.
B. Jahnel, A. Tóbiás, Absence of percolation in graphs based on stationary point processes with degrees bounded by two, Preprint no. 2774, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2774 .
Abstract, PDF (548 kByte)
We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2.