Tomasz Schreiber

Random dynamics and thermodynamic limits for polygonal Markov fields in the plane

We construct random dynamics for collections of nonintersecting planar contours, leaving invariant the distributions of length- and area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the dynamic construction of consistent polygonal fields, as presented in the original articles by Arak (1983) and Arak and Surgailis (1989), (1991), and it provides an easy-to-implement Metropolis-type simulation algorithm. The second dynamics leads to a graphical construction in the spirit of Fernández et al. (1998), (2002) and yields a perfect simulation scheme in a finite window in the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of the thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include, in the class of infinite-volume Gibbs measures without infinite contours, the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low-temperature region. Outside this class, we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.

Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane

The point process of vertices of an iteration infinitely divisible or, more specifically, of an iteration stable random tessellation in the Euclidean plane is considered. We explicitly determine its covariance measure and its pair-correlation function, as well as the cross-covariance measure and the cross-correlation function of the vertex point process and the random length measure in the general nonstationary regime. We also give special formulae in the stationary and isotropic setting. Exact formulae are given for vertex count variances in compact and convex sampling windows, and asymptotic relations are derived. Our results are then compared with those for a Poisson line tessellation having the same length density parameter. Moreover, a functional central limit theorem for the joint process of suitably rescaled total edge counts and edge lengths is established with the process (ξ, tξ), t > 0, arising in the limit, where ξ is a centered Gaussian variable with explicitly known variance.

Brownian limits, local limits and variance asymptotics for convex hulls in the ball

Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball Bd, d≥2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.

LIMIT THEOREMS FOR THE TYPICAL POISSON-VORONOI CELL AND THE CROFTON CELL WITH A LARGE INRADIUS

In this paper, we are interested in the behaviour of the typical Poisson-Voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to innit y. We prove a law of large numbers for its number of vertices and the area of the cell outside the disk. Moreover, for the latter, we establish a central limit theorem as well as moderate deviation type results. The proofs deeply rely on precise connections between Poisson-Voronoi tessellations, convex hulls of Poisson samples and germ-grain models in the unit ball. Besides, we derive analogous facts for the Crofton cell of a stationary Poisson line process in the plane.

VARIANCE ASYMPTOTICS AND CENTRAL LIMIT THEOREMS FOR VOLUMES OF UNIONS OF RANDOM CLOSED SETS

Let X, X 1, X 2, … be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X 1 ∪ ∙ ∙ ∙ ∪ X n )) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in ℝ d with centres distributed according to a spherically-symmetric heavy-tailed law.

An algorithm for binary image segmentation using polygonal markov fields

We present a novel algorithm for binary image segmentation based on polygonal Markov fields. We recall and adapt the dynamic representation of these fields, and formulate image segmentation as a statistical estimation problem for a Gibbsian modification of an underlying polygonal Markov field. We discuss briefly the choice of Hamiltonian, and develop Monte Carlo techniques for finding the optimal partition of the image. The approach is illustrated by a range of examples.

Limit theorems for iteration stable tessellations

The intent of this paper is to describe the large scale asymptotic geometry of iteration stable (STIT) tessellations in Rd, which form a rather new, rich and flexible class of random tessellations considered in stochastic geometry. For this purpose, martingale tools are combined with second-order formulas proved earlier to establish limit theorems for STIT tessellations. More precisely, a Gaussian functional central limit theorem for the surface increment process induced a by STIT tessellation relative to an initial time moment is shown. As second main result, a central limit theorem for the total edge length/facet surface is obtained, with a normal limit distribution in the planar case and, most interestingly, with a nonnormal limit showing up in all higher space dimensions.

Spontaneous scale-free structure of spike flow graphs in recurrent neural networks

In this paper we introduce a simple and mathematically tractable model of an asynchronous spiking neural network which to some extent generalizes the concept of a Boltzmann machine. In our model we let the units contain a certain (possibly unbounded) charge, which can be exchanged with other neurons under stochastic dynamics. The model admits a natural energy functional determined by weights assigned to neuronal connections such that positive weights between two units favor agreement of their states whereas negative weights favor disagreement. We analyze energy minima (ground states) of the presented model and the graph of charge transfers between the units in the course of the dynamics where each edge is labeled with the count of unit charges (spikes) it transmitted. We argue that for independent Gaussian weights in low enough temperature the large-scale behavior of the system admits an accurate description in terms of a winner-take-all type dynamics which can be used for showing that the resulting graph of charge transfers, referred to as the spike flow graph in the sequel, has scale-free properties with power law exponent gamma=2. Whereas the considered neural network model may be perceived to some extent simplistic, its asymptotic description in terms of a winner-take-all type dynamics and hence also the scale-free nature of the spike flow graph seem to be rather universal as suggested both by a theoretical argument and by numerical evidence for various neuronal models. As establishing the presence of scale-free self-organization for neural models, our results can also be regarded as one more justification for considering neural networks based on scale-free graph architectures.

Theoretical Model for Mesoscopic-Level Scale-Free Self-Organization of Functional Brain Networks

In this paper, we provide theoretical and numerical analysis of a geometric activity flow network model which is aimed at explaining mathematically the scale-free functional graph self-organization phenomena emerging in complex nervous systems at a mesoscale level. In our model, each unit corresponds to a large number of neurons and may be roughly seen as abstracting the functional behavior exhibited by a single voxel under functional magnetic resonance imaging (fMRI). In the course of the dynamics, the units exchange portions of formal charge, which correspond to waves of activity in the underlying microscale neuronal circuit. The geometric model abstracts away the neuronal complexity and is mathematically tractable, which allows us to establish explicit results on its ground states and the resulting charge transfer graph modeling functional graph of the network. We show that, for a wide choice of parameters and geometrical setups, our model yields a scale-free functional connectivity with the exponent approaching 2, which is in agreement with previous empirical studies based on fMRI. The level of universality of the presented theory allows us to claim that the model does shed light on mesoscale functional self-organization phenomena of the nervous system, even without resorting to closer details of brain connectivity geometry which often remain unknown. The material presented here significantly extends our previous work where a simplified mean-field model in a similar spirit was constructed, ignoring the underlying network geometry.

Dobrushin-Kotecký-Shlosman Theorem for Polygonal Markov Fields in the Plane

We consider the so-called length-interacting Arak-Surgailis polygonal Markov fields with V-shaped nodes — a continuum and isometry invariant process in the plane sharing a number of properties with the two-dimensional Ising model. For these polygonal fields we establish a low-temperature phase separation theorem in the spirit of the Dobrushin-Kotecký-Shlosman theory, with the corresponding Wulff shape deteremined to be a disk due to the rotation invariant nature of the considered model. As an important tool replacing the classical cluster expansion techniques and very well suited for our geometric setting we use a graphical construction built on contour birth and death process, following the ideas of Férnandez, Ferrari and Garcia.