Christoph HoferTemmel
(né Temmel)
Welcome to my homepage. I am a mathematician researching topics in discrete and spatial probability theory and related fields.
About
Contact
My email is math@temmel.me.
CV
A short version of my curriculum vitae.
Professional
 2015  present
 Assistant professor in mathematics at the Faculty of Military Sciences of the Dutch Defense Academy.
 2013  2015
 Postdoc with Federico Camia at the Department of Mathematics of the VU University Amsterdam.
 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
Education
 2008  2012
 Doctoral studies within a cotutelle at Graz University of Technology and AixMarseille Université. My advisors have been Wolfgang Woess and Pierre Mathieu respectively. I have been member of the DK Discrete Mathematics. My thesis 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 thesis was about "Kindependent percolation on infinite trees after Bollobàs and Balister".
Publications
My BibTeX file.
My profiles on arXiv, MathSciNet, zbMATH, dblp, Google Scholar.
Old profiles for "Temmel": zbMATH, Google Scholar.
Articles
 Disagreement percolation for Gibbs ball models Submitted arXiv
 Disagreement percolation for the hardsphere model Submitted arXiv
 Shearer's point process and the hardsphere model in one dimension Preprint arXiv

Clique trees of infinite, locally finite chordal graphs
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.

Shearer's point process, the hardsphere gas and a continuum Lovász Local Lemma
Published in Advances in Applied Probability
abstract
arXiv
doi
MathSciNet
A point process is Rindependent, if it behaves independently beyond the minimum distance R. We investigate uniform positive lower bounds on the avoidance functions of Rindependent point processes of the same intensity. We characterise those intensities by the existence of Shearer's point process, the unique Rindependent point process with an Rhardcore. Shearer's point process is intimately related to the hardsphere gas with radius R, the unique Gibbsian point process with an Rhardcore. 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 Rindependent 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 hardsphere gas, we recover classic bounds by Ruelle via an inductive approach à la Dobrushin.

GraphBased Lossless Markov Lumpings
Published in 2016 Proc. IEEE Int. Sym. on Information Theory (ISIT)
abstract
arXiv
doi
We use results from zeroerror information theory to project a Markov chain through a noninjective 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 zeroerror source coding result for nonMarkov processes with a deterministic dependence structure.

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 treeoperator approach introduced by Fernández & Procacci combined with novel partition schemes of the spanning subgraph complex of a cluster.

Lumpings of Markov chains, entropy rate preservation, and higherorder lumpability
Published in Journal of Applied Probability
abstract
arXiv
doi
zbMATH
MathSciNet
A lumping of a Markov chain is a coordinatewise 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 finitelength 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 klumpable, iff the lumped process is a kth order Markov chain for each starting distribution of the original Markov chain. We characterise strong klumpability 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 klumpable.

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 nontrivial Bernoulli product field. Additionaly we derive a lower nontrivial 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 kfuzz 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.

Informationpreserving Markov aggregation
Published in 2013 IEEE ITW (Information Theory Workshop)
abstract
arXiv
doi
We present a sufficient condition for a noninjective function of a Markov chain to be a secondorder Markov chain with the same entropy rate as the original chain. This permits an informationpreserving state space reduction by merging states or, equivalently, lossless compression of a Markov source on a samplebysample 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 informationpreserving state space reductions, for a given transition graph. We illustrate our results by applying the algorithm to a bigram letter model of an English text.

Kindependent percolation on trees
Published in Stochastic Processes and Applications
abstract
arXiv
doi
zbMATH
MathSciNet
Consider the class of kindependent 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 1independent 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 kfuzz of Z.
Theses

Properties and applications of Bernoulli random fields with strong dependency graphs
PhD thesis at
Graz University of Technology
and
AixMarseille 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 kindependent BRFs indexed by the integers, lets us determine critical probabilities for kindependent 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, gridlike graphs. These graphs are of particular relevance in statistical mechanics.

Kindependent percolation on infinite trees after Bollobàs and Balister
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 1independent 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.

La composante géante d'un graphe aléatoire
TER (travail d’étude et de recherche) at AixMarseille Université
abstract
pdf
Using a graph exploration process of the ErdösRèny 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.
Other

Entropy in Information Theory
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.
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 onedependent is more common, although I still think that oneindependent would be more appropriate. General Rdependence reduces to onedependence by simple scaling of the metric. The definition of onedependence is straightforward, but the classification and properties of onedependent processes is scarcely known. Onedependent 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 nontrivial and there are sufficient conditions for an intensity to belong to this set. Another characterization of Shearer's point process is onedependence together with a onehardcore (for a fixed intensity measure).
K(in)dependent percolation
An application of the detailed knowledge about Shearer's Bernoulli random field on the kfuzz of the integers to compare kdependent percolation on a tree to the independent version.
Hardspheres
The hardsphere model with radius R is a statistical mechanical model of nonpenetrating spheres of radius R. In the discrete case, it is a hardcore 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 hardsphere 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 nonnegative). 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 ddimensional hardsphere model and the large d limit behavior.
Structure of onedependence
The surprising structural relation between the partition function of the hardsphere model and the avoidance probability of Shearer's point process leads to the following question: Is there a systematic link between Rdependent models and Markov models with range R interaction? Can we derive from every partition function an Rdependent 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 almostsurely. 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 wellknown model.
Stochastic domination appears in disagreement percolation. Another result is that onedependent 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 oneindependent 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