Publications
My BibTeX file.
My profiles on arXiv, MathSciNet, zbMATH, dblp, Google Scholar and MS Academic Search.
Articles

Limits and limitations of cluster expansion in the hardsphere model
In preparation
abstract
To come.

GraphBased Lossless Markov Lumpings
Submitted
abstract
arXiv
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.

Disagreement percolation for the hardsphere model
Preprint
abstract
arXiv
We generalise disagreement percolation to the hardsphere model. This yields uniqueness of the infinite volume Gibbs measure for low fugacities. The lower bounds on the fugacity improve upon previous bounds obtained by cluster expansion techniques.

Shearer's point process and the hardsphere model in one dimension
Preprint
abstract
arXiv
We revisit the smallest nonphysical singularity of the hardsphere model in one dimension, also known as Tonks gas. We give an explicit expression of the free energy and reduced correlations at negative real fugacity and elaborate the nature of the singularity: the free energy is rightcontinuous, but its derivative diverges. We derive these results in several novel ways: First, by scaling up the discrete solution. Second, by an inductive argument on the partition function \`a la Dobrushin. Third, by a perfect cluster expansion counting the Penrose trees in the Mayer expansion perfectly. Fourth, by an explicit construction of Shearer's point process, the unique Rdependent point process with an Rhardcore. The last connection yields explicit and optimal lower bounds on the avoidance function of Rdependent point processes on the real line.

Shearer's point process, the hardsphere gas and a continuum Lovász Local Lemma
Submitted
abstract
arXiv
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.

Clique trees of infinite, locally finite chordal graphs
Submitted
abstract
arXiv
We investigate clique trees of infinite, locally finite chordal graphs. Our key tool is a bijection between the set of clique trees and the product of local families of finite trees. This enables us to enumerate all clique trees of a chordal graph. It also induces a local projection onto clique trees of finite chordal graphs, allowing us to lift various classic properties of clique trees of finite graphs to infinite clique trees.

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.

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.

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.

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.
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).

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.