# Research

I research at the intersection between combinatorics, probability theory and statistical mechanics. The principal stochastic objects are Bernoulli random fields and point processes in percolation and spatial Markov models. The combinatorial side enters mainly through graph theory and discrete objects appearing in (cluster) expansion techniques. I keep a passive interest in combinatorial optimization, data structures and discrete stochastic algorithms. The remainder of this page gives a bird's eyes view of my research. For details look at my publications.

## One-(in)dependence

A random process on a metric space is one-(in)dependent, if projections on subsets at distance at least one are independent. The term one-dependent is more common, although I still think that one-independent would be more appropriate. General R-dependence reduces to one-dependence by simple scaling of the metric. The definition of one-dependence is straightforward, but the classification and properties of one-dependent processes is scarcely known. One-dependent point processes and Bernoulli random fields are recurring objects in my research.

## Shearer's point process

We generalize the Lovász Local Lemma and Shearer's construction from the Bernoulli random field setting to the simple point process setting on a complete, separable metric space. For small intensities, there exists a unique point process minimizing the (conditional) avoidance functions. The set of small intensities, for which this uniform minimality holds, is non-trivial and there are sufficient conditions for an intensity to belong to this set. Another characterization of Shearer's point process is one-dependence together with a one-hardcore (for a fixed intensity measure).

## K-(in)dependent percolation

An application of the detailed knowledge about Shearer's Bernoulli random field on the k-fuzz of the integers to compare k-dependent percolation on a tree to the independent version.

## Hard-spheres

The hard-sphere model with radius R is a statistical mechanical model of non-penetrating spheres of radius R. In the discrete case, it is a hard-core lattice gas. Because of the binary nature of interaction (total exclusion in short range, none for longer ranges), it does not only appear on its own, but also pops up in calculations of other models (e.g., as abstract polymer system). The partition function of the hard-sphere model at negative real fugacity equals the avoidance probability of Shearer's point process at an intensity equal to the absolute value of the fugacity (as long as all the first quantities are all non-negative). This only works for small fugacities and the existence regions of Shearer's point process coincides with the region where a classic cluster expansion can be shown to work.

## Cluster expansion

The high temperature (or Mayer) expansion is a series expansion of the logarithm of the partition function of Markov point process. If one controls the resulting series, then one gets complete analyticity of the free energy and control over the reduced correlations at high temperature. Controlling the series is done by various approaches; I work on better bounds of the Ursell coefficients of this series, to show larger radii of convergence. Besides, there are results about the theoretical limit of this technique in the classic d-dimensional hard-sphere model and the large d limit behavior.

## Structure of one-dependence

The surprising structural relation between the partition function of the hard-sphere model and the avoidance probability of Shearer's point process leads to the following question: Is there a systematic link between R-dependent models and Markov models with range R interaction? Can we derive from every partition function an R-dependent model? Which are the models, for which the converse is possible (there are necessary natural constraints on the higher moment measures)?

## Stochastic domination

Two point processes are stochastically ordered (or, the first one dominates the second), if there exists a coupling of their laws such that the first one contains the second one almost-surely. Simpler said: the larger point process has always all the points of the smaller one plus some extra. Stochastic domination is a nice tool to compare expectations of monotone functions of the two point processes. Often, one of the two point processes to be a Poisson point process, which allows a comparison a well-known model.

Stochastic domination appears in disagreement percolation. Another result is that one-dependent Bernoulli random fields are uniformly stochastically dominated by a Bernoulli product field, if and only if Shearer's Bernoulli random field exists. A conjectured extension is: every one-independent simple point process of low enough intensity and smooth higher moment measures is dominated by a Poisson point process of low intensity.

## Disagreement percolation

Disagreement is a technique to show uniqueness of Gibbs state at high temperature. It does so by stochastically comparing the disagreement between two finite volume specifications with differing boundary conditions with a Boolean percolation model. If one is in the subcritical percolation phase, then the influence of the boundary conditions disappears as the volume tends to infinity.

## Lumpings and entropy

A lumping of a Markov chain is also known as a hidden Markov model. We investigate the structure of lumpings of finite, ergodic Markov chains preserving the entropy rate. These lumpings can be identified by only looking at the transition graph of the underlying Markov chain together with the lumping function.

## Infinite chordal graphs

Investigating the structure and properties of clique trees of infinite chordal graphs. The aim is to characterize the clique trees, i.e., tree representations, of infinite chordal graphs by various local properties. We use this knowledge to show that Shearer's Bernoulli random field on a chordal graph is always a block factor and give an easy description of its region of existence together with a perfect Lovász Local Lemma