Publications

My BibTeX file.

My profiles on arXiv, MathSciNet, zbMATH, dblp, Google Scholar and MS Academic Search.

Articles

  1. Limits and limitations of cluster expansion in the hard-sphere model In preparation abstract

    To come.

  2. Graph-Based Lossless Markov Lumpings Submitted With Bernhard C. Geiger abstract arXiv

    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.

  3. Disagreement percolation for the hard-sphere model Preprint abstract arXiv

    We generalise disagreement percolation to the hard-sphere 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.

  4. Shearer's point process and the hard-sphere model in one dimension Preprint abstract arXiv

    We revisit the smallest non-physical singularity of the hard-sphere 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 right-continuous, 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 R-dependent point process with an R-hard-core. The last connection yields explicit and optimal lower bounds on the avoidance function of R-dependent point processes on the real line.

  5. Shearer's point process, the hard-sphere gas and a continuum Lovász Local Lemma Submitted abstract arXiv

    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.

  6. Clique trees of infinite, locally finite chordal graphs With Florian Lehner 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.

  7. 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.

  8. 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.

  9. 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.

  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. 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.

Theses

  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.

Other

  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).

  2. 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.