Extrema statistics in the dynamics of a non-Gaussian random field

When the equations that govern the dynamics of a random field are nonlinear, the field can develop with time non-Gaussian statistics even if its initial condition is Gaussian. Here, we provide a general framework for calculating the effect of the underlying nonlinear dynamics on the relative densities of maxima and minima of the field. Using this simple geometrical probe, we can identify the size of the non-Gaussian contributions in the random field, or alternatively the magnitude of the nonlinear terms in the underlying equations of motion. We demonstrate our approach by applying it to an initially Gaussian field that evolves according to the deterministic KPZ equation, which models surface growth and shock dynamics.Comment: 9 pages, 3 figure

Stochastic geometry and topology of non-Gaussian fields

Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher order correlation functions. In the present work, we take a different route. We show how geometric and topological properties of Gaussian fields, such as the statistics of extrema, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. In our treatment, we consider both local and nonlocal mechanisms that generate non-Gaussian fields, both statically and dynamically through nonlinear diffusion.Comment: 8 pages, 4 figure

Critical and umbilical points of a non-Gaussian random field

Random fields in nature often have, to a good approximation, Gaussian characteristics. For such fields, the relative densities of umbilical points -- topological defects which can be classified into three types -- have certain fixed values. Phenomena described by nonlinear laws can however give rise to a non-Gaussian contribution, causing a deviation from these universal values. We consider a Gaussian field with a perturbation added to it, given by a nonlinear function of that field, and calculate the change in the relative density of umbilical points. This allows us not only to detect a perturbation, but to determine its size as well. This geometric approach offers an independent way of detecting non-Gaussianity, which even works in cases where the field itself cannot be probed directly.Comment: 13 pages, 3 figure

Gaining new insights by going local: determinants of coalition formation in mixed democratic polities

We develop a simple spatial model suggesting that Members of Parliament (MPs) strive for the inclusion of the head of state’s party in coalitions formed in mixed democratic polities, and that parliamentary parties try to assemble coalitions that minimize the ideological distance to the head of state. We identify the German local level of government as functionally equivalent to a parliamentary setting, such that the directly elected mayor has competencies similar to a president in a mixed national polity. Our findings show that the party affiliation of the head of state is a key factor considered by party members in the legislature when forming coalitions: coalitions in the legislature are more likely to form if they include the party of the head of the executive branch. Furthermore, the policy preferences of the head of the executive branch matter for the legislators’ behavior in the coalition formation process: the smaller the ideological distance between the position of a coalition and the position of the head of state, the more likely a coalition is to be formed