Profiles

Profiles for “Hofer-Temmel” (later publications):

Profiles for “Temmel” (early publications):

Publications

Articles

Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics
With Viktor Beneš and Günter Last and Jakub Večeřa.
Published in Journal of Applied Probability
See: arXiv,
doi.
Abstract
We study a stationary Gibbs particle process with deterministically bounded particles on Euclidean space defined in terms of an activity parameter and non-negative interaction potentials of finite range. Using disagreement percolation, we prove exponential decay of the correlation functions, provided a dominating Boolean model is subcritical. We also prove this property for the weighted moments of a U-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for such U-statistics of the Gibbs particle process. A by-product of our approach is a new uniqueness result for Gibbs particle processes.
Disagreement percolation for Gibbs ball models
With Pierre Houdebert.
Published in Stochastic Processes and Applications.
See: arXiv, doi.
Abstract
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.
Disagreement percolation for the hard-sphere model
Published in Electronic Journal of Probability.
See: arXiv, doi.
Abstract
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.
Shearer's point process and the hard-sphere model in one dimension
Preprint.
See: arXiv.
Clique trees of infinite, locally finite chordal graphs
With Florian Lehner.
Published in Electronic Journal of Combinatorics.
See: arXiv, journal, zbMATH, MathSciNet.
Abstract
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 hard-sphere gas and a continuum Lovász Local Lemma
Published in Advances in Applied Probability.
See: arXiv, doi, MathSciNet.
Abstract
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.
Graph-Based Lossless Markov Lumpings
Published in 2016 Proc. IEEE Int. Sym. on Information Theory (ISIT).
With Bernhard C. Geiger.
See: arXiv, doi.
Abstract
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.
Sufficient conditions for uniform bounds in abstract polymer systems and explorative partition schemes
Published in Journal of Statistical Physics.
See: arXiv, doi, zbMATH, MathSciNet.
Abstract
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.
Lumpings of Markov chains, entropy rate preservation, and higher-order lumpability
With Bernhard C. Geiger.
Published in Journal of Applied Probability
See: arXiv, doi, zbMATH, MathSciNet.
Abstract
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.
Shearer's measure and stochastic domination of product measures
Published in Journal of Theoretical Probability.
See: arXiv, doi, zbMATH, MathSciNet.
Abstract
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.
Information-preserving Markov aggregation
With Bernhard C. Geiger.
Published in 2013 IEEE ITW (Information Theory Workshop).
See: arXiv, doi.
Abstract
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.
K-independent percolation on trees
With Pierre Mathieu.
Published in Stochastic Processes and Applications.
See: arXiv, doi, zbMATH, MathSciNet.
Abstract
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.

Theses

Properties and applications of Bernoulli random fields with strong dependency graph
Supervised by Pierre Mathieu and Wolfgang Woess.
Doctoral thesis at Aix-Marseille Université and Graz University of Technology.
See: pdf.
Abstract
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.
K-independent percolation on infinite trees after Bollobàs and Balister
Supervised by Pierre Mathieu and Wolfgang Woess.
Master thesis at Graz University of Technology.
See: pdf.
Abstract
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.
La composante géante d'un graphe aléatoire
Supervised by Pierre Mathieu.
TER (travail d’étude et de recherche) at Aix-Marseille Université.
See: pdf.
Abstract
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.

Other

Entropy in Information Theory
With Bernhard C. Geiger and Gernot Kubin and Wolfgang Woess.
Published in TU Graz Research.
See: pdf.
Abstract
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).