Christoph Hofer-Temmel
(né Temmel)

Welcome to my homepage. I am a mathematician researching topics in discrete and spatial probability theory and related fields.



My email is


A short version of my curriculum vitae.


2015 - 2019
Assistant professor in mathematics at the Faculty of Military Sciences of the Dutch Defense Academy.
2013 - 2015
Postdoc at the Department of Mathematics of the VU University Amsterdam within the VIDI grant of Federico Camia.
2008 - 2013
Scientific assistant at the Department of Mathematical Structure Theory of Graz University of Technology.
2005 - 2013
Teaching assistant at the Department of Mathematical Structure Theory of Graz University of Technology


2008 - 2012
Doctoral studies within a cotutelle at Graz University of Technology and Aix-Marseille Université. My advisors have been Wolfgang Woess and Pierre Mathieu respectively. I have been member of the DK Discrete Mathematics. My dissertation was about "Properties and applications of Bernoulli random fields with strong dependency graphs".
2003 - 2008
Master studies in technical mathematics with a specialization in combinatorial optimization at the Graz University of Technology. I spent three semesters at the Université de Provence. My master thesis was about "K-independent percolation on infinite trees after Bollobàs and Balister".


My BibTeX file.

My profiles on arXiv, MathSciNet, zbMATH, dblp, Google Scholar.

Old profiles for "Temmel": zbMATH, Google Scholar.


  1. Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics With Viktor Beneš and Günter Last and Jakub Večeřa Submitted arXiv
  2. Disagreement percolation for Gibbs ball models With Pierre Houdebert Published in Stochastic Processes and Applications abstract arXiv doi

    We generalise disagreement percolation to Gibbs point processes of balls with varying radii. This allows to establish the uniqueness of the Gibbs measure and exponential decay of pair correlations in the low activity regime by comparison with a sub-critical Boolean model. Applications to the Continuum Random Cluster model and the Quermass-interaction model are presented. At the core of our proof lies an explicit dependent thinning from a Poisson point process to a dominated Gibbs point process.

  3. Disagreement percolation for the hard-sphere model Published in Electronic Journal of Probability abstract arXiv doi

    Disagreement percolation connects a Gibbs lattice gas and i.i.d. site percolation on the same lattice such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds improve upon known bounds obtained by cluster expansion techniques. The proof uses a novel dependent thinning from a Poisson point process to the hard-sphere model, with the thinning probability related to a derivative of the free energy.

  4. Shearer's point process and the hard-sphere model in one dimension Preprint arXiv
  5. Clique trees of infinite, locally finite chordal graphs With Florian Lehner Published in Electronic Journal of Combinatorics abstract arXiv journal zbMATH MathSciNet

    We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting.

  6. Shearer's point process, the hard-sphere gas and a continuum Lovász Local Lemma Published in Advances in Applied Probability abstract arXiv doi MathSciNet

    A point process is R-independent, if it behaves independently beyond the minimum distance R. We investigate uniform positive lower bounds on the avoidance functions of R-independent point processes of the same intensity. We characterise those intensities by the existence of Shearer's point process, the unique R-independent point process with an R-hard-core. Shearer's point process is intimately related to the hard-sphere gas with radius R, the unique Gibbsian point process with an R-hard-core. The continuum Lovász Local Lemma is a sufficient condition on the intensity and R to guarantee uniform exponential lower bounds on the avoidance function for all R-independent point processes of this intensity. Hence, the continuum Lovász Local Lemma guarantees the existence of Shearer's point process. Because it is also a lower bound on the radius of convergence of the hard-sphere gas, we recover classic bounds by Ruelle via an inductive approach à la Dobrushin.

  7. Graph-Based Lossless Markov Lumpings Published in 2016 Proc. IEEE Int. Sym. on Information Theory (ISIT) With Bernhard C. Geiger abstract arXiv doi

    We use results from zero-error information theory to project a Markov chain through a non-injective function without losing information. We show that this lossless lumping is made possible by exploiting the (sufficiently sparse) temporal structure of the Markov chain, and that our scheme is asymptotically optimal. Eliminating edges in the transition graph of the Markov chain trades the required output alphabet size versus information loss, for which we present bounds. Finally, we present a first zero-error source coding result for non-Markov processes with a deterministic dependence structure.

  8. Sufficient conditions for uniform bounds in abstract polymer systems and explorative partition schemes Published in Journal of Statistical Physics abstract arXiv doi zbMATH MathSciNet

    We present several new sufficient conditions for uniform boundedness of the reduced correlations and free energy of an abstract polymer system in a small complex disc around zero fugacity. In particular, we solve a discrepancy between several known conditions, which are not always comparable, and show how the arise from a common approach. The main tool is an extension of the tree-operator approach introduced by Fernández & Procacci combined with novel partition schemes of the spanning subgraph complex of a cluster.

  9. Lumpings of Markov chains, entropy rate preservation, and higher-order lumpability With Bernhard C. Geiger Published in Journal of Applied Probability abstract arXiv doi zbMATH MathSciNet

    A lumping of a Markov chain is a coordinate-wise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, iff the lumped process is a k-th order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.

  10. Shearer's measure and stochastic domination of product measures Published in Journal of Theoretical Probability abstract arXiv doi zbMATH MathSciNet

    Let G=(V,E) be a locally finite graph. Let \vec{p}\in[0,1]^V. We show that Shearer's measure, introduced in the context of the Lovàsz Local Lemma, with marginal distribution determined by \vec{p} exists on G iff every Bernoulli random field with the same marginals and dependency graph G dominates stochastically a non-trivial Bernoulli product field. Additionaly we derive a lower non-trivial uniform bound for the parameter vector of the dominated Bernoulli product field. This generalizes previous results by Liggett, Schonmann & Stacey in the homogeneous case, in particular on the k-fuzz of Z. Using the connection between Shearer's measure and lattice gases with hardcore interaction established by Scott & Sokal, we apply bounds derived from cluster expansions of lattice gas partition functions to the stochastic domination problem.

  11. Information-preserving Markov aggregation With Bernhard C. Geiger Published in 2013 IEEE ITW (Information Theory Workshop) abstract arXiv doi

    We present a sufficient condition for a non-injective function of a Markov chain to be a second-order Markov chain with the same entropy rate as the original chain. This permits an information-preserving state space reduction by merging states or, equivalently, lossless compression of a Markov source on a sample-by-sample basis. The cardinality of the reduced state space is bounded from below by the node degrees of the transition graph associated with the original Markov chain. We also present an algorithm listing all possible information-preserving state space reductions, for a given transition graph. We illustrate our results by applying the algorithm to a bi-gram letter model of an English text.

  12. K-independent percolation on trees With Pierre Mathieu Published in Stochastic Processes and Applications abstract arXiv doi zbMATH MathSciNet

    Consider the class of k-independent bond, respectively site, percolations with parameter p on an infinite tree T. We derive bounds on p in terms of k and the branching number br(T) of T for both a.s. percolation and a.s. nonpercolation. The bounds are tight for the whole class, coincide for bond and site percolation and are continuous functions of br(T). This extends previous results by Lyons for the independent (k=0) case and by Balister & Bollobàs for the 1-independent bond percolation case. Central to our argumentation are moment method and capacity estimates à la Lyons supplemented by explicit percolation models inspired by the work of Balister & Bollobàs. An indispensable tool is the minimality and explicit construction of Shearer's measure on the k-fuzz of Z.


  1. Properties and applications of Bernoulli random fields with strong dependency graphs Supervised by Wolfgang Woess and Pierre Mathieu PhD thesis at Graz University of Technology and Aix-Marseille Université abstract pdf

    A Bernoulli random field (short BRF) is a collection of {0,1}-valued random variables indexed by the vertices of a graph. We investigate BRFs with prescribed marginal parameters and a dependency structure encoded by a graph. A prominent example is Shearer's measure, derived as the extreme case of the Lovàsz Local Lemma (short LLL). In the case of a finite graph it is constructed from the weighted independent set polynomial of this graph, with weights derived from the prescribed marginal parameters. The LLL is a classic sufficient condition for the existence of Shearer's measure.

    The first part of this thesis recapitulates the properties of Shearer's measure, in particular its minimality for certain conditional probabilities of a large class of BRFs. This minimality, specialized to k-independent BRFs indexed by the integers, lets us determine critical probabilities for k-independent homogeneous percolation on trees. The critical probabilities are smooth functions of the branching number of the tree. Furthermore, the minimality allows to characterize the uniform domination of Bernoulli product fields by the above class of Bernoulli random fields through the existence of Shearer's measure alone. Thus the LLL also yields sufficient condition for this uniform domination problem.

    The second part of this thesis deals with a second BRF linked intimately to the weighted independent set polynomial of a graph. It is the Boltzmann measure of the model of a repulsive hardcore lattice gas. A classic question is to find estimates of the domain of absolute and uniform convergence and analyticity of the free energy of the above model. We extend cluster expansion techniques to derive improved estimates in the spirit of Dobrushin's condition. The link with the weighted independent set polynomial allows a straightforward interpretation of these estimates as improvements of the LLL. We conclude with a series of specializations and improvements aimed at improving estimates for regular, transitive, grid-like graphs. These graphs are of particular relevance in statistical mechanics.

  2. K-independent percolation on infinite trees after Bollobàs and Balister Supervised by Wolfgang Woess and Pierre Mathieu Master thesis at Graz University of Technology abstract pdf

    Retracing the proof by Lyons in the independent case using first and second moment methods as well as the proof of Balister & Bollobàs in the 1-independent case with the same methods, replacing their inductive proofs by more direct probabilistic arguments. Based on Pierre Mathieu's notes of Béla Bollobàs' talk at the IHP/Paris during the trimester on Phenomena in High Dimensions in 2006.

  3. La composante géante d'un graphe aléatoire Supervised by Pierre Mathieu TER (travail d’étude et de recherche) at Aix-Marseille Université abstract pdf

    Using a graph exploration process of the Erdös-Rèny G(n,p) random graph model to prove the sub- and supercritical regimes. Retraces the martingale methods employed in The critical random graph, with martingales by Nachmias and Peres.


  1. Entropy in Information Theory with Gernot Kubin and Bernhard C. Geiger and Wolfgang Woess Published in TU Graz Research abstract pdf

    Do you know how many bits of data memory your mp3 music files occupy on your smartphone? And do you still remember the crazy amounts paid by the mobile operators for the radio spectrum required for offering, first, voice telephony, and then later, a variety of digital communications services with ever increasing data rates? Then you have already encountered entropy as the measure of information content of data, processes, and signals. Mathematics and signal processing team up to extend the theory and application of this foundation of ICC (Information, Communication & Computing).

Research interests

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.


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.


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



I regularly play the game of Go. A wonderful resource is Sensei's Library, one of the oldest wikis on the web dedicated to it. For slow playing I recommend the venerable Dragon Go Server.

Programming languages

There are many forums about programming language theory, but some of the oldest and best are Lambda the Ultimate and C2 Wiki.