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We study a ferromagnetic Ising model with a staggered cell-board magnetic field previously proposed for image processing [Maruani et al., Markov Processes Relat. Fields 1, 419–442 (1995)]. We complement previous results on the existence of phase transitions at low temperature [Gonzalez-Navarrete et al., J. Stat. Phys. 162, 139–161 (2016)] by determining bounds to the region of uniqueness of Gibbs measures. We establish sufficient rigorous uniqueness conditions derived from three different criteria: (1) Dobrushin criterion [R. Dobrushin, Theory Probab. Appl. 13, 197–224 (1968)], (2) disagreement percolation [J. van den Berg and C. Maes, Ann. Probab. 22, 749–763 (1994)], and (3) Dobrushin–Shlosman criteria [R. Dobrushin and S. Shlosman, in Statistical Physics and Dynamical Systems: Rigorous Results, edited by J. Fritz, A. Jaffe, and D. Szasz (Birkhauser, Basel, 1985)]. These conditions are subsequently solved numerically and the resulting uniqueness regions are compared.

We study a class of random processes on $N$ particles which can be interpreted as stochastic model of luminescence. Each particle can stay in one of two states: Excited state or ground state. Any particle at ground state is excited with a constant rate (pumping). The number of excited particles decreases by means of photon emission through interactions of the particles. We analyse the rare event of flashes, i.e., the emission of a very large number of photons $B$ during a fixed time interval $T$. We employ the theory of large deviations to provide the asymptotics of the probability of such event when the total number of particles $N$ tends to infinity. This theory gives us also the optimal trajectory of scaled process corresponding to this event. The stationary regime of this process we call the large emission regime. In several cases we prove that in the large emission regime a share of excited particles in a system is stable under the changes of the pumping and emission rates., 14 pages, 0 figures

We propose a new model based on the Ising model with the aim to study synaptic plasticity phenomena in neural networks. It is today well established in biology that the synapses or connections between certain types of neurons are strengthened when the neurons are co-active, a form of the so called synaptic plasticity. Such mechanism is believed to mediate the formation and maintenance of memories. The proposed model describes some features from that phenomenon. Together with the spin-flip dynamics, in our model the coupling constants are also subject to stochastic dynamics, so that they interact with each other. The evolution of the system is described by a continuous-time Markov jump process.

We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The tool is the detailed balance equations., 15 pages

International audience

We consider the Harmonic crystal, a measure on \(\mathbb{R}^{\mathbb{Z}^{d}}\) with Hamiltonian H(x)=∑i,jJi,j(x(i)−x(j))2+h∑i(x(i)−d(i))2, where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤd. The configuration d is given and considered as observations. The ‘couplings’ Ji,j are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration m corresponding to the Hamiltonian.

In this work we consider the Wulff construction at zero temperature for a class of Gibbs models and study the shape of the obtained droplets. Considering zero temperature we avoid all difficulties connected with the competition between energy and entropy. It allows us to study a quite wide class of models which provides a variety of shapes. The motivations of the study come from attempts to describe isotropic properties of some models on 2D lattice at zero temperature. The studied models are binary (the spin space is 0,1) with a ferromagnetic behavior such that the potential functions are not equal to zero only for some tiles with size 3×3. In fact, we study herein droplet shapes of a subclass of the ferromagnetic models with potential functions as mentioned above. This subclass of models is defined by a condition called regularity. We call the model classified here as having regular micro-boundaries. Several examples of non-regular models are also presented.

We study the large fluctuations of emitted radiations in the system of $N$ non-interacting two-level atoms. Two methods are used to calculate the probability of the large fluctuations and the time dependence of the excitation and emission. The first method is based on the large deviation principle for Markov processes. The second one uses an analogue of the quantum formalism for classical probability problems. Particularly we prove that in a large fluctuation limit approximately a half of the atoms are excited. This fact is independent on the fraction of the excited atoms in the stationary state., Comment: 22 pages, no figures

We show the presence of a first-order phase transition for a ferromagnetic Ising model on $\mathbb{Z}^2$ with a periodical external magnetic field. The external field takes two values $h$ and $-h$, where $h>0$. The sites associated with positive and negative values of external field form a cell-board configuration with rectangular cells of sides $L_1\times L_2$ sites, such that the total value of the external field is zero. The phase transition holds if $h, Comment: 24 pages, 4 figures

International audience; In this work we propose a 2D discrete stochastic model for the simulation of axonal biogenesis. The model is defined by a third order Markov Chain. The model considers two main processes: the growth process that models the elongation and shape of the neurites and the bifurcation process that models the generation of branches. The growth process depends, among other variables, on the external attraction field generated by a chemoattractant molecule secreted by the target area. We propose an estimation scheme of the involved parameters from real fluorescent confocal microscopy images of single neurons within intact adult Drosophila fly brains. Both normal neurons and neurons in which certain genes were inactivated have been considered (two mutations). In total, 53 images (18 normal, 21 type 1 mutant and 14 type 2 mutant) were used. The model parameters allow us to describe pathological characteristics of the mutated populations.

We consider a classical gas of particles in ℝ d interacting via a pair potential. We prove that in a given region of the (β, μ) plane, where β is the inverse temperature, and μ is the chemical potential, either the Gibbs state is unique or it does not exist. Our method uses a version of the well-known Dobrushin uniqueness theorem adapted for lattice systems with a noncompact spin space and proved by Dobrushin and Pechersky. The advantage of this version is that using it one needs to check Dobrushin's contraction condition not for all boundary configurations, but only for those that have spin values in some compact subset of the spin space.

The authors propose a two-step algorithm for almost unsupervised detection of linear structures, in particular, main axes in road networks, as seen in synthetic aperture radar (SAR) images. The first step is local and is used to extract linear features from the speckle radar image, which are treated as road-segment candidates. The authors present two local line detectors as well as a method for fusing information from these detectors. In the second global step, they identify the real roads among the segment candidates by defining a Markov random field (MRF) on a set of segments, which introduces contextual knowledge about the shape of road objects. The influence of the parameters on the road detection is studied and results are presented for various real radar images.

In this paper, the large deviations at the trajectory level for ergodic Markov processes are studied. These processes take values in the non-negative quadrant of the two-dimensional lattice and are concentrated on step-wise functions. The rates of jumps towards the axes (downward jumps) depend on the position of the process - the higher the position, the greater the rate. The rates of jumps going in the same direction as the axes (upward jumps) are constants. Therefore the processes are ergodic. The large deviations are studied under equal scalings of both space and time. The scaled versions of the processes converge to 0. The main result is that the probabilities of excursions far from 0 tend to 0 exponentially fast with an exponent proportional to the square of the scaling parameter. The proportionality coefficient is an integral of a linear combination of path components. A rate function of the large deviation principle is calculated for continuous functions only.

International audience

The crude asymptotics of the large delay probability in a tandem queueing system is considered. The main result states that one of the two channels in the tandem system defines the crude asymptotics. The constant that determines the crude asymptotics is given. The results obtained are based on the large deviation principle for random processes with independent increments on an infinite interval recently established by the authors.

We study a generalisation of the notion of Gibbs model in a lattice in which the interactions have variable range. We prove the uniqueness of the measure at high temperature.

Consider a discrete locally finite subset G of Rd and the complete graph (G, E), with vertices G and edges E. We consider Gibbs measures on the set of sub-graphs with vertices G and edges E´ ⊂ E. The Gibbs interaction acts between open edges having a vertex in common. We study percolation properties of the Gibbs distribution of the graph ensemble. The main results concern percolation properties of the open edges in two cases: (a) when G is sampled from a homogeneous Poisson process; and (b) for a fixed G with sufficiently sparse points. Fil: Ferrari, Pablo Augusto. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Pechersky, Eugene A.. Russian Academy of Sciences. Dobrushin Laboratory of Institute for Information Transmission Problems; Rusia Fil: Sisko, Valentin V.. Universidade Federal Fluminense; Brasil Fil: Yambartsev, Anatoly. Universidade de Sao Paulo; Brasil

We study a Markov random process describing a muscle molecular motor behavior. Every motor is either bound up with a thin filament or unbound. In the bound state the motor creates a force proportional to its displacement from the neutral position. In both states the motor spend an exponential time depending on the state. The thin filament moves at its velocity proportional to average of all displacements of all motors. We assume that the time which a motor stays at the bound state does not depend on its displacement. Then one can find an exact solution of a non-linear equation appearing in the limit of infinite number of the motors., Comment: 10 pages

In this chapter, we study isotropic properties of some Gibbs fields used for image segmentation. We consider ferromagnetic models defined by 3 × 3 interactions. We compute the Wulff shape of these models at zero temperature. A classification of the considered models with respect to this shape is given. We also give some conjectures which provide conditions necessary to obtain a regular shape. Finally, the influence of the Wulff shape of a given model is shown on real data in the context of magnetic resonance image segmentation.

A system with two infinite-buffer FCFS servers is analyzed and the arrival is formed by three independent Poisson flows /spl Xi//sub i/, of rates /spl lambda//sub i/, i = 0,1,2, each with IID task service times. The tasks from /spl Xi//sub i/ are directed to server i, i = 1,2. The tasks from /spl Xi//sub 0/ are directed to the server that has the shorter workload in the buffer at the time of arrival. We analyze the large deviation probabilities for the virtual waiting time in flow /spl Xi//sub 0/ in the stationary regime.

We discuss two queueing system models from the large deviation theory point of view. The asymptotics of the logarithm of large delay probability are derived. We also explain what performance dynamics in the system cause the large delay for a particular protocol.

In this paper we investigate several dynamics to optimize a posterior distribution defined to solve segmentation problems. We first consider the Metropolis and the Kawasaki dynamics. We also compare the associated Bayesian cost functions. The Kawasaki dynamic appears to provide better results but requires the exact values of the class ratios. Therefore, we define alternative dynamics which conserve the properties of the Kawasaki dynamic and require only an estimation of the class ratios. We show on synthetic data that these new dynamics can improve the segmentation results by incorporating some information on the class ratios. Results are compared using a Potts model as prior distribution.

The life and research work of Professor R.L. Dobrushin (1929-1995) had a profound influence on several areas of probability theory, information theory, and mathematical physics. The paper contains a biographical note, a review of Dobrushin's results, and a list of his publications.

R.L. Dobrushin (1929-1995) made substantial contributions to Queueing Network Theory (QNT). A review of results from QNT which arose from his ideas or were connected to him in other ways is given. We also comment on various related open problems.

We consider a system of ordered cars moving in $\R$ from right to left. Each car is represented by a point in $\R$; two or more cars can occupy the same point but cannot overpass. Cars have two possible velocities: either 0 or 1. An unblocked car needs an exponential random time of mean 1 to pass from speed 0 to speed 1 (\emph{slow-to-start}). Car $i$, say, travels at speed 1 until it (possibly) hits the stopped car $i-1$ to its left. After the departure of car $i-1$, car $i$ waits an exponential random time to change its speed to 1, travels at this speed until it hits again stopped car $i-1$ and so on. Initially cars are distributed in $\R$ according to a Poisson process of parameter $\lambda, Comment: 14 pages, 2 figures