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The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand-canonical partition function of the hard-sphere model in d dimensions. Up to a fugacity of λ < e/2^d, the runtime of our algorithm is polynomial in the volume of the system. This covers the entire known real-valued regime for the uniqueness of the Gibbs measure. Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance we use clique dynamics, a Markov chain that was recently introduced in the setting of abstract polymer models. We prove rapid mixing of clique dynamics up to the tree threshold of the univariate hard-core model. This is achieved by relating clique dynamics to block dynamics and adapting the spectral expansion method, which was recently used to bound the mixing time of Glauber dynamics within the same parameter regime., LIPIcs, Vol. 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), pages 66:1-66:15

We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with sub-exponential neighborhood growth like Z𝑑, our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer-dimer models in such graphs.

Theoretical Computer Science, 745, ISSN:0304-3975

We propose a numerical method to learn Maximum Entropy (MaxEnt) distributions with spatio-temporal constraints from experimental spike trains. This is an extension of two papers [10] and [4] who proposed the estimation of parameters where only spatial constraints were taken into account. The extension we propose allows to properly handle memory effects in spike statistics, for large sized neural networks., Comment: 34 pages, 33 figures

We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution, which favours trajectories with low interference, measured in terms of signal-to-interference ratio. This model was introduced in our earlier paper [KT18], where we expressed, in the limit of a high density of users, the typical distribution of the family of trajectories in terms of a law of large numbers. In the present work, we derive its qualitative properties. We analytically identify the emerging typical scenarios in three extreme regimes. We analyse the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory approaches a straight line quickly, in regime (1) with equal hop lengths. Interestingly, in regime (1), the typical length of a hop diverges logarithmically in the distance of the transmitter to the base station. We further analyse (3) local and global repulsive effects of a densely populated subarea on the trajectories., Comment: 36 pages, 5 figures

We prove that a stochastic process of pure coagulation has at any time t ≥ 0 a time-dependent Gibbs distribution if and only if the rates ψ(i, j) of single coagulations are of the form ψ(i; j) = if(j) + jf(i), where f is an arbitrary nonnegative function on the set of positive integers. We also obtain a recurrence relation for weights of these Gibbs distributions that allow us to derive the general form of the solution and the explicit solutions in three particular cases of the function f. For the three corresponding models, we study the probability of coagulation into one giant cluster by time t > 0.

We construct an integrated probabilistic model to capture interactions between atoms of a nanocomponent. We then use this model to assess reliabilities of nanocomponents with different structures. Several properties of our proposed model are also described under a sparseness condition. The model is an extension of our previous model based on Markovian random field theory. The proposed integrated model is flexible in that pairwise relationship information among atoms as well as features of individual atoms can be easily incorporated. An important feature that distinguishes the integrated probabilistic model from our previous model is that the integrated approach uses all available sources of information with different weights for different types of interaction. In this paper we consider the nanocomponent at a fixed moment of time, say the present moment, and we assume that the present state of the nanocomponent depends only on the present states of its atoms.

A vast literature has recently been concerned with the analysis of variation in disease counts recorded across geographical areas with the aim of detecting clusters of regions with homogeneous behavior. Most of the proposed modeling approaches have been discussed for the univariate case and only very recently spatial models have been extended to predict more than one outcome simultaneously. In this paper we extend the standard finite mixture models to the analysis of multiple, spatially correlated, counts. Dependence among outcomes is modeled using a set of correlated random effects and estimation is carried out by numerical integration through an EM algorithm without assuming any specific parametric distribution for the random effects. The spatial structure is captured by the use of a Gibbs representation for the prior probabilities of component membership through a Strauss-like model. The proposed model is illustrated using real data.

In this paper, we propose a model for image segmentation based on a finite mixture of Gaussian distributions. For each pixel of the image, prior probabilities of class memberships are specified through a Gibbs distribution, where association between labels of adjacent pixels is modeled by a class-specific term allowing for different interaction strengths across classes. We show how model parameters can be estimated in a maximum likelihood framework using Mean Field theory. Experimental performance on perturbed phantom and on real benchmark images shows that the proposed method performs well in a wide variety of empirical situations.

Image modeling using Gibbs priors was shown to be effective in image reconstruction problems. This motivated us to evaluate three techniques for estimating the priors: the heuristic method, the histogram method and Borges' method. We found that both the histogram and Borges' method accurately recovered the parameters needed to specify four different Gibbs distributions from training sets consisting of random sample images from those distributions. This was not the case for the heuristic method. We evaluated the usefulness of the estimated distributions as priors for binary tomography in two experiments: in one the images for the training set were taken from a Gibbs distribution determined by five parameters, and they were typical cardiac phantom images for the second one. We estimated the parameters using both the heuristic and Borges' method in both experiments. We used a modified Metropolis algorithm for the reconstructions. In the first experiment both estimation methods gave rise to excellent reconstructions, but this was not the case for the second experiment; however, even in this case typically less than four percent of the pixels were mis-classified in the reconstructions.

Wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients. This is basically a local procedure, because wavelet coefficients characterize the local regularity of a function. Although a wavelet transform has decorrelating properties, structures in images, like edges, are never decorrelated completely, and these structures appear in the wavelet coefficients: a classification based on a local criterion-like coefficient magnitude is not the perfect method to distinguish important, uncorrupted coefficients from coefficients dominated by noise. We therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a Bayesian framework. The local characterization is incorporated into the conditional model, whereas the prior model describes only configurations, not coefficient values. More precisely, local characterization favors configurations with clusters of important coefficients. In this way, we can compute, for each coefficient, the posterior probability of being "sufficiently clean." This article proposes and motivates the particular and original choice of the conditional model. Instead of introducing this Bayesian framework, we could also apply heuristic image processing techniques to find clustered configurations of large coefficients. This article also explains the benefits of the Bayesian approach compared to these simple techniques. The parameter of the prior model is estimated on an empirical basis using a pseudolikelihood criterion.

We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous' beta-splitting model, which has an extended parameter range $\beta>-2$ with respect to the ${\rm beta}(\beta+1,\beta+1)$ probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson--Dirichlet models for exchangeable random partitions of $\mathbb {N}$, with an extended parameter range $0\le\alpha\le1$, $\theta\ge-2\alpha$ and $\alpha, Comment: Published in at http://dx.doi.org/10.3150/08-BEJ134 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

It is expected that nano-scale devices and interconnections will introduce unprecedented level of defects in the substrates, and architectural designs need to accommodate the uncertainty inherent at such scales. This consideration motivates the search for new architectural paradigms based on redundancy based defect-tolerant designs. However, redundancy is not always a solution to the reliability problem, and often too much or too little redundancy may cause degradation in reliability. The key challenge is in determining the granularity at which defect tolerance is designed, and the level of redundancy to achieve a specific level of reliability. Analytical probabilistic models to evaluate such reliability-redundancy trade-offs are error prone and cumbersome, and do not scalewell for complex networks of gates. In this thesiswe develop different tools and techniques that can evaluate the reliability measures of combinational circuits, and can be used to analyze reliability-redundancy trade-offs for different defect-tolerant architectural configurations. In particular, we have developed two tools, one of which is based on probabilistic model checking and is named NANOPRISM, and another MATLAB based tool called NANOLAB. We also illustrate the effectiveness of our reliability analysis tools by pointing out certain anomalies which are counter-intuitive but can be easily discovered by these tools, thereby providing better insight into defecttolerant design decisions. We believe that these tools will help furthering research and pedagogical interests in this area, expedite the reliability analysis process and enhance the accuracy of establishing reliability-redundancy trade-off points. Master of Science