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A Markovian single-server queue is studied in an interactive random environment. The arrival and service rates of the queue depend on the environment, while the transition dynamics of the random environment depend on the queue length. We consider in detail two types of Markov random environments: a pure jump process and a reflected jump diffusion. In both cases, the joint dynamics are constructed so that the stationary distribution can be explicitly found in a simple form (weighted geometric). We also derive an explicit estimate for the exponential rate of convergence to the stationary distribution via coupling.

This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered: a continuous-time Markov chain (finite or countably infinite) and a reflected (jump) diffusion process. The background is determined by a joint Markov process carrying a specific interactive mechanism, with an explicit invariant measure whose structure is similar to a product form. We discuss a number of queueing and population-growth models and establish conditions under which the above-mentioned invariant measure can be derived. Next, an analysis of the rate of convergence to stationarity is performed for the models under consideration. We consider two settings leading to either an exponential or a polynomial convergence rate. In both cases we assume that the underlying environmental Markov process has an exponential rate of convergence, but the convergence rate of the joint Markov process is determined by certain conditions on the birth and death rates. To prove these results a coupling method turns out to be useful., Comment: 32 pages

This paper focuses on infinite-volume bosonic states for a quantum particle system (a quantum gas) in Rd. The kinetic energy part of the Hamiltonian is the standard Laplacian (with a boundary condition at the border of a `box&rsquo, ). The particles interact with each other through a two-body finite-range potential depending on the distance between them and featuring a hard core of diameter a&gt, 0. We introduce a class of so-called FK-DLR functionals containing all limiting Gibbs states of the system. As a justification of this concept, we prove that for d=2, any FK-DLR functional is shift-invariant, regardless of whether it is unique or not. This yields a quantum analog of results previously achieved by Richthammer.

Following a series of works on capital growth investment, we analyse log-optimal portfolios where the return evaluation includes ‘weights’ of different outcomes. The results are twofold: (A) under certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional betting. We focus on properties of the optimal portfolios and discuss a number of simple examples extending the well-known Kelly betting scheme. An important restriction is that the investment does not exceed the current capital value and allows the trader to cover the worst possible losses. The paper deals with a class of discrete-time models. A continuous-time extension is a topic of an ongoing study.

Contact inhibition is a central feature orchestrating cell proliferation in culture experiments; its loss is associated with malignant transformation and tumorigenesis. We performed a co-culture experiment with human metastatic melanoma cell line (SKMEL- 147) and immortalized keratinocyte cells (HaCaT). After 8 days a spatial pattern was detected, characterized by the formation of clusters of melanoma cells surrounded by keratinocytes constraining their proliferation. In addition, we observed that the proportion of melanoma cells within the total population has increased. To explain our results we propose a spatial stochastic model (following a philosophy of the Widom-Rowlinson model from Statistical Physics and Molecular Chemistry) which considers cell proliferation, death, migration, and cell-to-cell interaction through contact inhibition. Our numerical simulations demonstrate that loss of contact inhibition is a sufficient mechanism, appropriate for an explanation of the increase in the proportion of tumor cells and generation of spatial patterns established in the conducted experiments.

We study multi-particle interactive quantum disordered systems on a polynomially-growing countable connected graph (Z,E). The novelty is to give localization bounds uniform in finite or infinite volumes (subgraphs) in Z^N as well as for the whole of Z^N. Such bounds are proved here by means of a comprehensive fixed-energy multi-particle multi-scale analysis. Another feature of the paper is that we consider -- for the first time in the literature -- an infinite-range (although fast-decaying) interaction between particles. For the models under consideration we establish (1) exponential spectral localization, and (2) strong dynamical localization with sub-exponential rate of decay of the eigenfunction correlators.

The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function. In this paper, we establish a number of simple inequalities for the weighted entropies (general as well as specific), mirroring similar bounds on standard (Shannon) entropies and related quantities. The required assumptions are written in terms of various expectations of the weight functions. Examples are weighted Ky Fan and weighted Hadamard inequalities involving determinants of positive-definite matrices, and weighted Cram\'{e}r-Rao inequalities involving the weighted Fisher information matrix., Comment: arXiv admin note: substantial text overlap with arXiv:1409.4102

The paper considers a continuous-time birth–death process where the jump rate has an asymptotically polynomial dependence on the process position. We obtain a rough exponential asymptotic for the probability of trajectories of a re-scaled process contained within a neighborhood of a given continuous nonnegative function.

The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function. We analyse analogs of the Fisher information inequality and entropy-power inequality for the weighted entropy and discuss connections with weighted Lieb’s splitting inequality. The concepts of rates of the weighted entropy and information are also discussed.

The Shannon Noiseless coding theorem (the data-compression principle) asserts that for an information source with an alphabet \(\mathcal {X}=\{0,\ldots ,\ell -1\}\) and an asymptotic equipartition property, one can reduce the number of stored strings \((x_0,\ldots ,x_{n-1})\in \mathcal {X}^{n}\) to \(\ell ^{nh}\) with an arbitrary small error-probability. Here h is the entropy rate of the source (calculated to the base \(\ell \)). We consider further reduction based on the concept of utility of a string measured in terms of a rate of a weight function. The novelty of the work is that the distribution of memory is analyzed from a probabilistic point of view. A convenient tool for assessing the degree of reduction is a probabilistic large deviation principle. Assuming a Markov-type setting, we discuss some relevant formulas and examples.

We consider a nearest-neighbor hard-core model, with three states, on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate A at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on 'splitting' Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, ∀ λ > 0 and k ≥ 1, there exists a unique translation-invariant splitting Gibbs measure μ*. We also study periodic splitting Gibbs measures and show that the above model admits only translation - invariant and periodic with period two (chess-board) Gibbs measures. We discuss some open problems and state several related conjectures.

We introduce a transfer matrix formalism for the (annealed) Ising model coupled to two-dimensional causal dynamical triangulations. Using the Krein-Rutman theory of positivity preserving operators we study several properties of the emerging transfer matrix. In particular, we determine regions in the quadrant of parameters beta, mu >0 where the infinite-volume free energy converges, yielding results on the convergence and asymptotic properties of the partition function and the Gibbs measure., 28 pages, 4 figures, minor changes in main text, title changed, as published

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

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We establish lower-edge spectral and dynamical localization for a multi-particle Anderson model in a Euclidean space , d ≥ 1, in the presence of a non-trivial short-range interaction and an alloy-type random external potential.

Contact inhibition of proliferation can be described as the decrease of proliferation rates when the cell density increases. Its loss is a key feature of tumor development. Cell density of human metastatic melanoma cell line (SK-MEL-147) and immortalized keratinocyte cells (HaCaT) was evaluated daily in culture experiments. Keratinocytes reach lower cell density than melanoma cells at confluence, indicating that the population growth arrest is associated with contact inhibition (KHaCaT = 1779.6 ± 130.5 cells/mm2; KSK-MEL-147 = 5043.5 ± 316.5 cells/mm2). We performed a coculture experiment to assess the proliferation restriction between both cell lines. After 8 days a spatial pattern was detected, characterized by the formation of clusters of melanoma cells surrounded by keratinocytes constraining their proliferation. Despite showing approximately the same proliferation rates when data were fitted with a logistic model (ρHaCaT = 1.1 ± 0.1 days-1; ρSK-MEL-147 = 1.1 ± 0.3 days-1), the proportion of melanoma cells within the total population has increased from 1:10 to 1:4 approximately. To explain our results we propose a spatial stochastic model where we consider cell proliferation, death, migration, and cell-to-cell interaction through contact inhibition. The agreement between experimental results and numerical simulation indicates that loss of contact inhibition is important for the analysis of the development of melanoma in situ, and explains the increase in the proportion of tumor cells and generation of spatial patterns established in conducted experiments. In addition, the distribution of the distances between cells of the same line indicates the higher proximity of melanoma cells and allowed us to identify their clustering pattern. This work provides results of a quantitative study of the impact of contact inhibition upon formation and growth of tumor-like melanoma cell clusters in a coculture with keratinocytes. Citation Format: Mauro Cesar C. Morais, Izabella Stuhl, Alan U. Sabino, Willian W. Lautenschlager, Alexandre S. Queiroga, Tharcisio C. Tortelli, Jr., Roger Chammas, Yuri Suhov, Alexandre F. Ramos. Stochastic model of contact inhibition and the proliferation of melanoma in situ [abstract]. In: Proceedings of the AACR International Conference held in cooperation with the Latin American Cooperative Oncology Group (LACOG) on Translational Cancer Medicine; May 4-6, 2017; São Paulo, Brazil. Philadelphia (PA): AACR; Clin Cancer Res 2018;24(1_Suppl):Abstract nr B71.

We consider an N -particle quantum systems in R d , with interaction and in pres- ence of a random external alloy-type potential (a continuous N -particle Anderson model). We establish Wegner-type bounds (inequalities) for such models, giving upper bounds for the probability that random spectra of Hamiltonians in finite volumes intersect a given set.

We establish the phenomenon of Anderson localisation for a quantum two-particle system on a d-dimensional lattice with short-range interaction and in presence of an IID external potential with sufficiently regular marginal distribution.

We propose a model of a loss averse investor who aims to maximize his expected wealth under certain constraints. The constraints are that he avoids, with high probability, incurring an (suitably defined) unacceptable loss. The methodology employed comes from the theory of large deviations. We explore a number of fundamental properties of the model and illustrate its desirable features. We demonstrate its utility by analyzing assets that follow some commonly used financial return processes: Fractional Brownian Motion, Jump Diffusion, Variance Gamma and Truncated Levy.

Consider a branching diffusion process on R1 starting at the origin. Take a high level u>0 and count the number R(u,n) of branches reaching u by generation n. Let Fk,n(u) be the probability P(R(u,n)

A system of interacting qubits can be viewed as a non-i.i.d quantum information source. A possible model of such a source is provided by a quantum spin system, in which spin-1/2 particles located at sites of a lattice interact with each other. We establish the limit for the compression of information from such a source and show that asymptotically it is given by the von Neumann entropy rate. Our result can be viewed as a quantum ana-logue of Shannon's noiseless coding theorem for a class of non-i.i.d. quantum informa-tion sources. From the probabilistic point of view it is an analog of the Shannon-McMillan-Breiman theorem considered as a cornerstone of modern Information Theory. PACS: 03.67-a; 03.67.Lx

Probability and statistics are as much about intuition and problem solving as they are about theorem proving. Consequently, students can find it very difficult to make a successful transition from lectures to examinations to practice because the problems involved can vary so much in nature. Since the subject is critical in so many applications from insurance to telecommunications to bioinformatics, the authors have collected more than 200 worked examples and examination questions with complete solutions to help students develop a deep understanding of the subject rather than a superficial knowledge of sophisticated theories. With amusing stories and historical asides sprinkled throughout, this enjoyable book will leave students better equipped to solve problems in practice and under exam conditions.

The paper extends earlier results from Suhov and Kelbert [“FK-DLR states of a quantum Bose-gas with a hardcore interaction,” arXiv:1304.0782] and Suhov et al. [“Shift-invariance for FK-DLR states of a 2D quantum Bose-gas,” arXiv:1304.4177] about infinite-volume quantum bosonic states (FK-DLR states) to the case of multi-type particles with non-negative interactions. (An example is a quantum Widom–Rowlinson model.) Following the strategy from Suhov and Kelbert and Suhov et al., we establish that, for the values of fugacity z ∈ (0, 1) and inverse temperature β > 0, finite-volume Gibbs states form a compact family in the thermodynamic limit. Next, in dimension two we show that any limit-point state (an FK-DLR state in the terminology adopted in Suhov and Kelbert and Suhov et al.) is translation-invariant.

This fundamental monograph introduces both the probabilistic and algebraic aspects of information theory and coding. It has evolved from the authors' years of experience teaching at the undergraduate level, including several Cambridge Maths Tripos courses. The book provides relevant background material, a wide range of worked examples and clear solutions to problems from real exam papers. It is a valuable teaching aid for undergraduate and graduate students, or for researchers and engineers who want to grasp the basic principles.

In this chapter we prove Theorems 3.1.1 and 3.1.2. Accordingly, in Sects 4.2–4.6 we suppose that Assumptions A and 3.1.1 are fulfilled.

Chapter 2 prepares the ground for the analysis of interacting particle systems carried out in Part II. The authors introduce here the principal technical tools of the single-particle multi-scale analysis (MSA), developed over the last thirty years by the mathematical community. The analytical tools of the so-called variable-energy MSA, developed in late 1980s, are streamlined and complemented by a simpler, fixed-energy approach. A simple and comprehensive derivation of the spectral and strong dynamical localization from the fixed-energy MSA, suitable for adaptations to interacting systems, is presented for the first time in mathematical literature.

This chapter outlines physical origins and the development of rigorous mathematical methods of the Anderson localization theory, describing unusual propagation properties of quantum particles (as well as electromagnetic and acoustic waves) in disordered media. While the main scope of the book is restricted to the analysis of Anderson localization in a strongly disordered environment, Chap. 1 gives the reader a broad perspective and indicates directions for possible future research in the area of multi-particle localization theory.

In this chapter we begin our analysis of localization in multi-particle Anderson tight-binding models with interaction. We already mentioned that the principal difficulty encountered when working with multi-particle systems is the structure of the external random potential term (3.3) in the Hamiltonian combined with the presence of interaction between particles (see Eq. (3.8) below). To tackle this obstacle, we develop a multi-particle version of the MSA (in short, the MPMSA) by scrutinizing and—when necessary—modifying the subsequent steps of the single-particle MSA scheme.

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This article addresses the issue of the proof of the entropy power inequality (EPI), an important tool in the analysis of Gaussian channels of information transmission, proposed by Shannon. We analyse continuity properties of the mutual entropy of the input and output signals in an additive memoryless channel and discuss assumptions under which the entropy-power inequality holds true.

The Markov property and its immediate consequences Mathematics cannot be learned by lectures alone, anymore than piano playing can be learned by listening to a player. C. Runge (1856–1927), German applied mathematician Typically, the subject of Markov chains represents a logical continuation from a basic course of probability. We will study a class of random processes describing a wide variety of systems of theoretical and practical interest (and sometimes simply amusing). The fact that deep insight into the subject is possible without using sophisticated mathematical tools may also be an explanation of why Markov chains are popular in so many different disciplines which are seemingly remote from pure mathematics. The basic model for the first half of the book will be a system which changes state in discrete time, according to some random mechanism. The collection of states is called a state space and throughout the whole book will be assumed finite or countable; we will denote it by I . Each i ∈ I is called a state; our system will always be in one of these states. Sometimes we will know what state the system occupies and sometimes only that the system is in state i with some probability.

This paper is a follow-up of our recent papers \cite{CS08} and \cite{CS09} covering the two-particle Anderson model. Here we establish the phenomenon of Anderson localisation for a quantum $N$-particle system on a lattice $\Z^d$ with short-range interaction and in presence of an IID external potential with sufficiently regular marginal cumulative distribution function (CDF). Our main method is an adaptation of the multi-scale analysis (MSA; cf. \cite{FS}, \cite{FMSS}, \cite{DK}) to multi-particle systems, in combination with an induction on the number of particles, as was proposed in our earlier manuscript \cite{CS07}. Similar results have been recently obtained in an independent work by Aizenman and Warzel \cite{AW08}: they proposed an extension of the Fractional-Moment Method (FMM) developed earlier for single-particle models in \cite{AM93} and \cite{ASFH01} (see also references therein) which is also combined with an induction on the number of particles. An important role in our proof is played by a variant of Stollmann's eigenvalue concentration bound (cf. \cite{St00}). This result, as was proved earlier in \cite{C08}, admits a straightforward extension covering the case of multi-particle systems with correlated external random potentials: a subject of our future work. We also stress that the scheme of our proof is \textit{not} specific to lattice systems, since our main method, the MSA, admits a continuous version. A proof of multi-particle Anderson localization in continuous interacting systems with various types of external random potentials will be published in a separate papers.

In this paper we consider a general hyperbolic branching diffusion on a Lobachevsky space ${\bf H}^d$. The question is to evaluate the Hausdorff dimension of the limiting set on the boundary (i.e., absolute) $\partial{\bf H}^d$. In the case of a homogeneous branching diffusion, an elegant formula for the Hausdorff dimension was obtained by Lalley and Sellke [Probab. Theory Related Fields, 108 (1997), pp. 171–192] for $d=2$ and by Karpelevich, Pechersky, and Suhov [Commun. Math. Phys., 195 (1998), pp. 627–642] for a general d. Later on, Kelbert and Suhov [Probab. Theory Appl., 51 (2007), pp. 155–167] extended the formula to the case where the branching diffusion was in a sense asymptotically homogeneous (i.e., its main relevant parameter, the fission potential, approached a constant limiting value near the absolute). In this paper we show that the Hausdorff dimension of the limiting set can be bounded from above and below in terms of the maximum and minimum points of the fission potential. As in [M. Kelber...

Publié dans Communications in Mathematical Physics, 2008; We consider a quantum two-particle system on a d-dimensional lattice with interaction and in presence of an IID external potential. We establish Wegner-typer estimates for such a model. The main tool used is Stollmann's lemma.

We describe relations between SLE processes, certain replacements of the Haar measure on the Virasoro group (which we call Malliavin measures), and domain walls in small massive perturbations of a CFT (like in Ising model)., Comment: 75 pages, 22 figures