2010 Volume 16 Issue 1

V.A. Malyshev
Foreword - After 15 Years pp. 1-2

15 years of MPRF is a good occasion to express our biggest gratitude to the editorial board, to the authors, to the referees and to everyone else without whom the journal could not exist. We thank also the authors who submitted invited papers for the anniversary issues (there will be two) of the journal. It is a moment to say how the journal appeared. Once I wondered who are the owners and the real bosses (those who periodically change main editors) of the existing mathematical journals. Instead of pursuing this direction, the idea to launch a new journal appeared to be more useful. However, I still think that any journal should have one who is personally responsible for the scientific policy. The authors of the rejected papers should know personally one who rejected their papers. Seemingly, it is the main editor, often he has to play mediator between the author and some unknown referees. But as far as I know, often the main editor is not completely free in his decisions. That is why I prefer smaller journals, owned and managed by a working mathematician. It could be useful also that the author knew the names of his referees. But now the world is still far from this. MPRF was the first and still remains the unique international mathematical journal, which is owned, edited and published in Russia. International means that it is published only in English, there is no Russian translation of MPRF. Moreover, it is really a private journal, all other journals are owned by grand institutions and societies. While launching the journal, I underwent big pressure from some institutions who wanted to own it and from others who did not want such kind of journals to exist. Next important remark. Now there appeared the so called ratings, impact factors etc. of papers, journals, and even authors. MPRF did not make any effort to get these scores for the only reason - when the science wants to be similar to a business enterprise and to use PR agencies, the science starts degenerate, and scientific politics prevails. I want to thank personally my colleagues and former colleagues whose efforts allowed the journal to appear. Vadim Scherbakov was the first general director of our small publishing company POLYMAT, followed now by Elena Petrova. Elena Petrova is also the executive editor who governs all current activity of the journal. Without Flora Spieksma we could hardly establish connections with the distributors, that allowed to get first subscriptions (for some years I was the only financial sponsor of the journal). She also, for some period, took responsibility for the English language. Sergej Popov, Alexander Gajrat, Anatoli Manita designed the journal style, its cover etc. At the beginning we suffered from the lack of submissions and in the first issues we had to publish mostly the papers of my Moscow laboratory. Soon we received some papers which used words "stochastic, Markov" etc. but were so ridiculous that I thought it was the test for our editorial board. I rejected them immediately without any refereeing process. Now a difficult era for the academic science seems to come. Too many fields of mathematics, too many interesting problems, but no global goals. Insufficient number of scientists to cover even most interesting fields. There are several big communities which aggressively propagate their field of interest, making no attention to others. That is why wider papers and reviews, explaining connections between different areas of mathematics and other disciplines, are welcome to MPRF, and we ask the authors to spare no pains to write them.

A. van Enter and E. Verbitskiy Erasure Entropies and Gibbs Measures pp. 3-14

Recently Verdu and Weissman introduced erasure entropies, which are meant to measure the information carried by one or more symbols given all of the remaining symbols in the realization of a random process or field. A natural relation to Gibbs measures has also been observed. In this short note we study this relation further, review a few earlier contributions from statistical mechanics, and provide the formula for the erasure entropy of a Gibbs measure in terms of the corresponding potential. For some 2-dimensional Ising models, for which Verdu and Weissman suggested a numerical procedure, we show how to obtain an exact formula for the erasure entropy.
Keywords: erasure entropy, Gibbs measures, solvable Ising models, conditional entropy

G. Basile and A. Bovier
Convergence of a Kinetic Equation to a Fractional Diffusion Equation pp. 15-44

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $(K(t)$, $Y(t))$ on $(\T\times\R)$, where $\T$ is the one-dimensional torus. $K(t)$ is a autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y(t)$ is an additive functional of $K$, defined as $\int_0^t v(K(s)) ds$, where $|v|\sim 1$ for small $k$. We prove that the rescaled process $N^{-2/3}Y(Nt)$ converges in distribution to a symmetric Levy process, stable with index $\alpha=3/2$.
Keywords: anomalous diffusion, Levy process, Boltzmann equation, coupled oscillators, kinetic limit, heat conductance

C. Maes and B. Wynants
On a Response Formula and Its Interpretation pp. 45-58

We present a physically inspired generalization of equilibrium response formulae, the fluctuation-dissipation theorem, to Markov jump processes possibly describing interacting particle systems out-of-equilibrium, following the recent work of Baiesi, Maes and Wynants [M. Baiesi, C. Maes and B. Wynants, Fluctuations and response of nonequilibrium states. Phys. Rev. Lett., 2009, v. 103, 010602. Nonequilibrium linear response for Markov dynamics, I. Jump processes and overdamped diffusions. J. Stat. Phys., v. 137, 1094-1116]. Here, the time-dependent perturbation adding a potential $V$ with small amplitude $h_t$ changes the rates $W(x,y)$ for the transition $x\rightarrow y$ into \[ W_t(x,y) = W(x,y) \exp \big\{ h_t \big( bV(y)-aV(x) \big) \big\} \] as first considered by Diezemann [G. Diezemann, Fluctuation-dissipation relations for Markov processes. Phys. Rev. E, 2005, v. 72, 011104] $a$, $b$ are constants. We observe that the linear response relation shows a reciprocity symmetry in the nonequilibrium stationary regime and we interpret the connection with dynamical fluctuation theory.
Keywords: fluctuation response relation, nonequilibrium dynamics

V. Jaksic, Y. Pautrat and C.-A. Pillet
A Non-commutative Levy - Cramer Continuity Theorem pp. 59-78

The classical Levy - Cramer continuity theorem asserts that the convergence of the characteristic functions implies the weak convergence of the corresponding probability measures. We extend this result to the setting of non-commutative probability theory and discuss some applications.
Keywords: non-commutative probability, Levy - Cramer theorem

L.T. Rolla, V. Sidoravicius, D. Surgailis and M.E. Vares
The Discrete and Continuum Broken Line Process pp. 79-116

In this work we introduce the discrete-space broken line process (with discrete and continuous parameter values) and derive some of its properties. We explore polygonal Markov field techniques developed by Arak - Surgailis. The discrete version is presented first and a natural generalization to a continuous object living on the discrete lattice is then proposed and studied The broken lines also resemble the Young diagram and the Hammersley process and are useful for computing last passage percolation values and finding maximal oriented paths. For a class of passage time distributions there is a family of boundary conditions that make the process stationary and self-dual. For such distributions there is a law of large numbers and the process extends to the infinite lattice. A proof of Burke's theorem emerges from the construction. We present a simple proof of the explicit law of large numbers for last passage percolation as an application. Finally we show that the exponential and geometric distributions are the only non-trivial ones that yield self-duality.
Keywords: spatial random processes, Hammersley process, last passage percolation, time constant, broken line process

J. Fritz
Hyperbolic Scaling Limits: The Method of Compensated Compactness pp. 117-138

Hydrodynamic limit of various models with hyperbolic (Euler) scaling law is discussed, we are mainly interested in the limiting behavior of the microscopic systems in a regime of shocks. In the absence of an effective coupling anadvanced method of PDE theory: compensated compactness is required. We consider some deterministic and Ginzburg - Landau models of classical statistical mechanics; the proof of several recent results is outlined. Microscopic systems living on the infinite line are preferred.
Keywords: interacting exclusions, hyperbolic scaling, Lax entropy pairs, compensated compactness, logarithmic Sobolev inequalities, relaxation schemes

L. Avena, F. den Hollander and F. Redig
Large Deviation Principle for One-Dimensional Random Walk in Dynamic Random Environment: Attractive Spin-Flips and Simple Symmetric Exclusion pp. 139-168

Consider a one-dimensional shift-invariant attractive spin-flip system in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on vacant sites has a local drift to the left. In [L. Avena, F. den Hollander and F. Redig, Law of large numbers for a class of random walks in dynamic random environments. EURANDOM Report 2009-032] we proved a law of large numbers for dynamic random environments satisfying a space-time mixing property called cone-mixing. If an attractive spin-flip system has a finite average coupling time at the origin for two copies starting from the all-occupied and the all-vacant configuration, respectively, then it is cone-mixing. In the present paper we prove a large deviation principle for the empirical speed of the random walk, both quenched and annealed, and exhibit some properties of the associated rate functions. Under an exponential space-time mixing condition for the spin-flip system, which is stronger than cone-mixing, the two rate functions have a unique zero, i.e., the slow-down phenomenon known to be possible in a static random environment does not survive in a fast mixing dynamic random environment. In contrast, we show that for the simple symmetric exclusion dynamics, which is not cone-mixing (and which is not a spin-flip system either), slow-down does occur.
Keywords: dynamic random environment, random walk, quenched vs. annealed large deviation principle, slow-down

A.M. Vershik
Dynamics of Metrics in Measure Spaces and Their Asymptotic Invariants pp. 169-184

We discuss the Kolmogorov's entropy and Sinai's definition of it; and then define a deformation of the entropy, called scaling entropy; this is also a metric invariant of the measure preserving actions of the group, which is more powerful than the ordinary entropy. To define it, we involve the notion of the $\epsilon$-entropy of a metric in a measure space, also suggested by A.N. Kolmogorov slightly earlier. We suggest to replace the techniques of measurable partitions, conventional in entropy theory, by that of iterations of metrics or semi-metrics. This leads us to the key idea of this paper which as we hope is the answer on the old question: what is the natural context in which one should consider the entropy of measure-preserving actions of groups? the same question about its generalizations - scaling entropy, and more general problems of ergodic theory. Namely, we propose a certain research program, called asymptotic dynamics of metrics in a measure space, in which, for instance, the generalized entropy is understood as the asymptotic Hausdorff dimension of a sequence of metric spaces associated with dynamical system. As may be supposed, the metric isomorphism problem for dynamical systems as a whole also gets a new geometric interpretation.
Keywords: scaling entropy, metric compact with measure, asymptotic geometry, filtrations

W. Kager, H. Liu and R. Meester
Existence and Uniqueness of the Stationary Measure in the Continuous Abelian Sandpile pp. 185-204

Let $\Lambda \subset \mathbb{Z}^d$ be finite. We study the following sandpile model on $\Lambda$. The height at any given vertex $x \in \Lambda$ is a positive real number, and additions are uniformly distributed on some interval $[a,b] \subset [0,1]$. The threshold value is $1$; when the height at a given vertex exceeds $1$, it topples, that is, its height is reduced by $1$, and the heights of all its neighbours in $\Lambda$ increase by $1/2d$. We first establish that the uniform measure $\mu$ on the so called `allowed configurations' is invariant under the dynamics. When $a < b$, we show with coupling ideas that starting from any initial configuration of heights, the process converges in distribution to $\mu$, which therefore is the unique invariant measure for the process. When $a=b$, that is, when the addition amount is non-random, and $a \notin \mathbb{Q}$, it is still the case that $\mu$ is the unique invariant probability measure, but in this case we use random ergodic theory to prove this; this proof proceeds in a very different way. Indeed, the coupling approach cannot work in this case since we also show the somewhat surprising fact that when $a=b\notin \mathbb{Q}$, the process does not converge in distribution at all starting from any initial configuration.
Keywords: continuous Abelian sandpile model, invariant measures, measure-preserving transformations, ergodic theory

F. Dunlop
Space-time Correlations of a Gaussian Interface pp. 205-222

The serial harness introduced by Hammersley [J.M. Hammersley, Harnesses. In: Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability, 1966, Vol. III, 89-117] is equivalent, in the Gaussian case, to the Gaussian Solid-On-Solid interface model with parallel heat bath dynamics. Here we consider sub-lattice parallel dynamics, and give exact results about relaxation dynamics, based on the equivalence to the infinite time limit of a time periodic random field. We also give a numerical comparison to the harness process in continuous time studied by Hsiao [C.-T. Hsiao, Stochastic processes with Gaussian interaction of components. Z. Wahrsch. Verw. Geb., 1982, v. 59, 39-53] and by Ferrari, Niederhauser and Pechersky [P.A. Ferrari and B.M. Niederhauser, Harness processes and harmonic crystals. Stoch. Process. Appl., 2006, v. 116, 939-956, math.PR/0312402; P.A. Ferrari, B.M. Niederhauser and E.A. Pechersky, Harness processes and non-homogeneous crystals. J. Stat. Phys., 2007, v. 128, 1159-1176, math.PR/0409301].
Keywords: random surface, interface dynamics, harness.

2010 Volume 16 Issue 2

The second part of invited papers of the 15th anniversary issue of Markov Processes and Related Fields
M. Campanino, D. Ioffe and O. Louidor
Finite Connections for Supercritical Bernoulli Bond Percolation in 2D pp. 225-266

Two vertices x and y are said to be finitely connected if they belong to the same cluster and this cluster is finite. We derive sharp asymptotics of finite connections for super-critical Bernoulli bond percolation on $Z^2$. These asymptotics are based on a detailed fluctuation analysis of long finite super-critical clusters or, more precisely, of dual open (sub-critical) loops which surround such clusters.
Keywords: Bernoulli bond percolation, random walk representation, interacting random walks, Ornstein - Zernike decay of correlations, local limit theorems

A. Rybko, S. Shlosman and A. Vladimirov
Absence of Breakdown of the Poisson Hypothesis. I. Closed Networks at Low Load pp. 267-285

We prove that the general mean-field type networks at low load behave in accordance with the Poisson Hypothesis. That means that the network equilibrates in time independent of its size. This is a 'high-temperature' counterpart of our earlier result, where we have shown that at high load the relaxation time can diverge with the size of the network ('low-temperature'). In other words, the phase transitions in the networks can happen at high load, but cannot take place at low load.
Keywords: coupled dynamical systems, non-linear Markov processes, stable attractor, phase transition, long-range order

S. Miracle-Sole
On the Theory of Cluster Expansions pp. 287-294

A short exposition with complete proofs of the theory of cluster expansions for an abstract polymer system is presented.
Keywords: lattice systems, cluster expansions, cluster properties

K.R. Duffy
Mean Field Markov Models of Wireless Local Area Networks pp. 295-328

In 1998, Giuseppe Bianchi introduced a mean field Markov model of the fundamental medium access control protocol used in Wireless Local Area Networks (WLANs). Due to the model's intuitive appeal and the accuracy of its predictions, since then there has been a vast body of material published that extends and analyzes models of a similar character. As the majority of this development has taken place within the culture and nomenclature of the telecommunications community, the aim of the present article is to review this work in a way that makes it accessible to probabilists. In doing so, we hope to illustrate why this modeling approach has proved so popular, to explain what is known rigorously, and to draw attention to outstanding questions of a mathematical nature whose solution would be of interest to the telecommunications community. For non-saturated WLANs, these questions include rigorous support for its fundamental decoupling approximation, determination of the properties of the self-consistent equations and the identification of the queueing stability region.
Keywords: wireless local area networks, mean field models, Markov chains

C. Boeinghoff, E.E. Dyakonova, G. Kersting and V.A. Vatutin
Branching Processes in Random Environment which Extinct at a Given Moment pp. 329-350

Let $\{Z_{n}, n\geq 0\}$ be a critical branching process in random environment and let $T$ be its moment of extinction. Under the annealed approach we prove, as $n\rightarrow \infty$, a limit theorem for the number of particles in the process at moment $n$ given $T=n+1$ and a functional limit theorem for the properly scaled process $\{ Z_{nt}, \delta \leq t \leq 1-\delta \} $ given $T = n+1$ and $\delta \in (0,1/2)$.
Keywords: branching process, random environment, random walk, change of measure, survival probability, functional limit theorem

I.M. MacPhee, M.V. Menshikov and A.R. Wade
Angular Asymptotics for Multi-Dimensional Non-Homogeneous Random Walks with Asymptotically Zero Drift pp. 351-388

We study the first exit time $\tau$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at $\bx \in \Z^d$ is of magnitude $O(\| \bx\|^{-1})$, we show that $\tau<\infty$ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude $\| \bx\|^{-\beta}$, $\beta \in (0,1)$, we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on $2$nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model.
Keywords: asymptotic direction, exit from cones, inhomogeneous random walk, perturbed random walk, random walk in random environment

P. Bremaud and S. Foss
Ergodicity of a Stress Release Point Process Seismic Model with Aftershocks pp. 389-408

We prove ergodicity of a point process earthquake model combining the classical stress release model for primary shocks with the Hawkes model for aftershocks.
Keywords: earthquakes, secondary earthquakes, point process, stochastic intensity, ergodicity, Harris chains, Foster's theorem

O.J. Boxma, H. Jonsson, J.A.C. Resing and S. Shneer
An Alternating Risk Reserve Process - Part I pp. 409-424

We consider an alternating risk reserve process with a threshold dividend strategy. The process can be in two different states and the state of the process can only change just after claim arrival instants. If at such an instant the capital is below the threshold, the system is set to state $1$ (paying no dividend), and if the capital is above the threshold, the system is set to state~$2$ (paying dividend). Our interest is in the survival probabilities. In the case of exponentially distributed claim sizes, survival probabilities are found by solving a system of integro-differential equations. In the case of generally distributed claim sizes, they are expressed in the survival probabilities of the corresponding standard risk reserve processes.
Keywords: insurance risk models, alternating risk reserve process, survival probabilities

O.J. Boxma, H. Jonsson, J.A.C. Resing and S. Shneer
An Alternating Risk Reserve Process - Part II pp. 425-446

We consider an alternating risk reserve process with a threshold dividend strategy. The process can be in two different states and the state of the process can only change at the arrival instants of an independent Poisson observer. Whether or not a change then occurs depends on the value of the risk reserve w.r.t. the barrier. If at such an instant the capital is below the threshold, the system is set to state 1 (paying no dividend), and if the capital is above the threshold, the system is set to state 2 (paying dividend). In each of the two states, the process is described by different premium rates, Poisson claim arrival intensities, and claim size distributions. For this model we determine the survival probabilities, distinguishing between the initial state being $1$ or 2, and the process starting below or above the barrier. In the case of exponentially distributed claim sizes, survival probabilities are found by solving a system of integro-differential equations. In the case of generally distributed claim sizes, they are expressed in the survival probabilities of the corresponding standard risk reserve processes. We perform several numerical experiments, including a comparison with the case in which state changes can only occur just after claim arrival instants; that case is treated in Part I.
Keywords: insurance risk models, alternating risk reserve process, survival probabilities

2010 Volume 16 Issue 3

I. Papageorgiou
The Logarithmic Sobolev Inequality in Infinite Dimensions for Unbounded Spin Systems on the Lattice with Non-Quadratic Interactions pp. 447-484

We are interested in the Logarithmic Sobolev inequality for the infinite volume Gibbs measure with no quadratic interactions. We consider unbounded spin systems on the one-dimensional lattice with interactions that go beyond the usual strict convexity and without uniform bound on the second derivative. We assume that the one-dimensional single-site measure with boundaries satisfies the Log-Sobolev inequality uniformly in the boundary conditions and we determine conditions under which the Log-Sobolev inequality can be extended to the infinite volume Gibbs measure.
Keywords: Logarithmic Sobolev inequality, Gibbs measure, infinite dimensions, spin systems

G. Fayolle and K. Raschel
On the Holonomy or Algebraicity of Generating Functions Counting Lattice Walks in the Quarter-Plane
pp. 485-496

In two recent works [M. Bousquet-Melou and M. Mishna, Walks with small steps in the quarter plane. In: "Algorithmic Probability and Combinatorics", special volume of the Contemporary Mathematics series of the Amer. Math. Soc., 2010, v. 520, pp. 1-40; A. Bostan and M. Kauers, The complete generating function for Gessel walks is algebraic. Proc. Amer. Math. Soc., 2009, v. 138, N9, 3063-3078], it has been shown that the counting generating functions (CGF) for the 23 walks with small steps confined in a quarter-plane and associated with a finite group of birational transformations are holonomic, and even algebraic in 4 cases - in particular for the so-called Gessel's walk. It turns out that the type of functional equations satisfied by these CGF appeared in a probabilistic context almost 40 years ago. Then a method of resolution was proposed in [G. Fayolle, R. Iasnogorodski and V. Malyshev, Random Walks in the Quarter-Plane, Applications of Mathematics (New York), vol. 40, 1999, Springer-Verlag, Berlin], involving at once algebraic techniques and reduction to boundary value problems. Recently this method has been developed in a combinatorics framework in [K. Raschel, Counting walks in a quadrant: a unified approach via boundary value problems. Preprint http://arxiv.org/abs/1003.1362, 2010], where a thorough study of the explicit expressions for the CGF is proposed. The aim of this paper is to derive the nature of the bivariate CGF by a direct use of some general theorems given in [G. Fayolle, R. Iasnogorodski and V. Malyshev].
Keywords: generating function, piecewise homogeneous lattice walk, quarter-plane, universal covering, Weierstrass elliptic functions, automorphism

A. Faggionato, D. Gabrielli and M.R. Crivellari
Averaging and Large Deviation Principles for Fully-Coupled Piecewise Deterministic Markov Processes and Applications to Molecular Motors pp. 497-548

We consider Piecewise Deterministic Markov Processes (PDMPs) with a finite set of discrete states. In the regime of fast jumps between discrete states, we prove a law of large numbers and a large deviation principle. In the regime of fast and slow jumps, we analyze a coarse-grained process associated to the original one and prove its convergence to a new PDMP with effective force fields and jump rates. In all the above cases, the continuous variables evolve slowly according to ODEs. Finally, we discuss some applications related to the mechanochemical cycle of macromolecules, including strained-dependent power-stroke molecular motors. Our analysis covers the case of fully-coupled slow and fast motions.
Keywords: piecewise deterministic Markov process, averaging principle, large deviations, molecular motors

N. Berglund and B. Gentz
The Eyring - Kramers Law for Potentials with Nonquadratic Saddles pp. 549-598

The Eyring - Kramers law describes the mean transition time of an overdamped Brownian particle between local minima in a potential landscape. In the weak-noise limit, the transition time is to leading order exponential in the potential difference to overcome. This exponential is corrected by a prefactor which depends on the principal curvatures of the potential at the starting minimum and at the highest saddle crossed by an optimal transition path. The Eyring - Kramers law, however, does not hold whenever one or more of these principal curvatures vanishes, since it would predict a vanishing or infinite transition time. We derive the correct prefactor up to multiplicative errors that tend to one in the zero-noise limit. As an illustration, we discuss the case of a symmetric pitchfork bifurcation, in which the prefactor can be expressed in terms of modified Bessel functions, as well as bifurcations with two vanishing eigenvalues. The corresponding transition times are studied in a full neighbourhood of the bifurcation point. These results extend work by Bovier, Eckhoff, Gayrard and Klein [A. Bovier, M. Eckhoff, V. Gayrard and M. Klein, Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Euro. Math. Soc., 2004, v. 6, N 4, pp. 399-424], who rigorously analysed the case of quadratic saddles, using methods from potential theory.
Keywords: stochastic differential equations, exit problem, transition times, most probable transition path, large deviations, Wentzell - Freidlin theory, metastability, potential theory, capacities, subexponential asymptotics, pitchfork bifurcation

P.J. Fitzsimmons and D.M. Wroblewski
Martingale Functions of Brownian Motion and its Local Time pp. 599-608

We characterize the class of local martingales of the form $H(B_t,L_t)$ for a standard one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$ and its local time at $0$, $L=(L_t)_{t\ge 0}$. The main result is closely related to work of J. Obloj, who studied the local martingales of the form $H(B_t,\ov B_t)$, where $\ov B_t = \sup_{0\le s\le t} B_s$.
Keywords: Brownian motion, local martingale, local time

2010 Volume 16 Issue 4

Inhomogeneous Random Systems. Concentration Inequalities. Random Matrices pp. 609-611

The present issue of "Markov Processes and Related Fields" contains papers presented at the meeting "Inhomogeneous Random Systems" held at the Institut Henri Poincare, Paris, on January 27-28, 2009. These meetings, which concentrate each year on different topics, bring together an interdisciplinary audience of mathematicians and physicists.

M. Ledoux
Introduction to Concentration Inequalities pp. 613-614

This conference was the opportunity to present some of the most recent developments in the area, emphasizing in particular the multiple scopes of measure concentration. The speakers indeed presented aspects of measure concentration and its applications in various settings, from isoperimetric and functional aspects to random matrices and dynamical systems. The contributions gathered in this issue reflect this variety and the lively recent developments in the area.

C. Roberto
Isoperimetry for Product of Probability Measures: Recent Results pp. 617-634

We present recent results on the isoperimetric problem for product of probability measures. For distributions with tails between exponential and Gaussian we state a dimension free result. We sketch its proof that relies on a functional inequality of Poincar\'e type and on a semi-group argument. Also, we give isoperimetric and concentration results, depending on the dimension, for measures with heavy tails.
Keywords: isoperimetry, Poincare type inequality, concentration of measure phenomenon, heavy tails distributions, Gaussian measure

N. Gozlan and C. Leonard
Transport Inequalities. A Survey pp. 635-736

This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory.
Keywords: transport inequalities, optimal transport, relative entropy, Fisher information, concentration of measure, deviation inequalities, logarithmic Sobolev inequalities, inf-convolution inequalities, large deviations

F. Redig and F. Wang
Transformations of One-Dimensional Gibbs Measures with Infinite Range Interaction pp. 737-752

We study single-site stochastic and deterministic transformations of one-dimensional Gibbs measures in the uniqueness regime with infinite-range interactions. We prove conservation of Gibbsianness and give quantitative estimates on the decay of the transformed potential. As examples, we consider exponentially decaying potentials, and potentials decaying as a power-law.
Keywords: Gibbs measures, potential, Kozlov theorem, house-of-cards coupling, renormalization group transformation

B. Zegarlinski
Linear and Nonlinear Concentration Phenomena pp. 753-782

In this work we consider variety of themes including Linear versus Nonlinear, Extensive versus Intensive (i.e. dimension independent and where dependence on dimension is relevant), conjugation of measures, interplay of Entropy and Nonlinearity, and others.
Keywords: linear and nonlinear Markov semigroups, coercive inequalities, nonlinear probability, entropic switch

A. Edelman
The Random Matrix Technique of Ghosts and Shadows pp. 783-790

We propose to abandon the notion that a random matrix exists only if it can be sampled. Much of today's applied finite random matrix theory concerns real or complex random matrices ($\beta=1,2$). The "threefold way" so named by Dyson in 1962 [F.J. Dyson, The threefold way. Algebraic structures of symmetry groups and ensembles in Quantum Mechanics. J. Math. Phys., 1963, v. 3, pp. 1199-1215] adds quaternions ($\beta=4$). While it is true there are only three real division algebras ($\beta$="dimension over the reals"), this mathematical fact while critical in some ways, in other ways is irrelevant and perhaps has been over interpreted over the decades. We introduce the notion of a "ghost" random matrix quantity that exists for every beta, and a shadow quantity which may be real or complex which allows for computation. Any number of computations have successfully given reasonable answers to date though difficulties remain in some cases. Though it may seem absurd to have a "three and a quarter" dimensional or "pi" dimensional algebra, that is exactly what we propose and what we compute with. In the end $\beta$ becomes a noisiness parameter rather than a dimension.
Keywords: random matrix theory, ghost random variables

P. Seba
Parking in the City: An Example of Limited Resource Sharing pp. 793-802

During the attempt to park a car in the city the drivers have to share limited resources (the available roadside). We show that this fact leads to a predictable distribution of the distances between the cars that depends on the length of the street segment used for the collective parking. We demonstrate in addition that the individual parking maneuver is guided by generic psychophysical perceptual correlates. Both predictions are compared with the actual parking data collected in the city of Hradec Kralove (Czech Republic).
Keywords: parking process, distances between cars distribution

F. Bornemann
On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review pp. 803-866

In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painleve transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) $\beta$-ensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the k-th largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.
Keywords: random matrix theory, numerical approximation, Painleve transcendents, Fredholm determinants
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