weakly gibbsian measures for lattice spin systems we present a weaker notion of gibbs measures by requiring that only an almost everywhere absolutely summable potential is given this has recently appeared in the context gibbsian versus non gibbsian measures we give a first exploration of the main features of this weaker notion we concentrate on the questions of where do weakly gibbsian measures appear and what remains of their thermodynamic description

a note on the projecton of gibbs measures we give an example of a projection which maps two gibbs measures for the same interaction into gibbs measures for different interactions as a corollary we find a case where by decimation a non gibbsian measure is transformed into a gibbs measure

renormalization group transformations under strong mixing conditions gibbsianness and convergence of renormalized interactions in this paper we study a renormalization group map the block averaging transformation applied to gibbs measures relative to a class of finite range lattice gases when suitable strong mixing conditions are satisfied using a block decimation procedure cluster expansion and detailed comparison between statistical ensembles we are able to prove gibbsianness and convergence to a trivial i e gaussian and product fixed point our results apply to the 2d standard ising model at any temperature above the critical one and arbitrary magnetic field

analogues of non gibbsianness in joint measures of disordered mean field models it is known that the joint measures on the product of spin space and disorder space are very often non gibbsian measures for lattice systems with quenched disorder at low temperature are there reflections of this non gibbsianness in the corresponding mean field models we study the continuity properties of the conditional probabilities in finite volume of the following mean field models a joint measures of random field ising b joint measures of dilute ising c decimation of ferromagnetic ising the conditional probabilities are functions of the empirical mean of the conditionings so we look at the large volume behavior of these functions to discover non trivial limiting objects for a we find 1 discontinuous dependence for almost any realization and 2 dependence of the conditional probabilities on the phase in contrast to that we see continuous behavior for b and c for almost any realization this is in complete analogy to the behavior of the corresponding lattice models in high dimensions it shows that non gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean field models

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

on the gibbsian nature of the random field kac model under block averaging we consider the kac ising model in an arbitrary configuration of local magnetic fields left eta kern 1pt i right i quad quad in quad kern 1pt quad mathbb z d in any dimension d at any inverse temperature we investigate the gibbs properties of the renormalized infinite volume measures obtained by block averaging any of the gibbs measures corresponding to fixed with block length small enough compared to the range of the kac interaction we show that these measures are gibbs measures for the same renormalized interaction potential this potential depends locally on the field configuration and decays exponentially uniformly in for which we give explicit bounds the construction of the potential is based on a high temperature type cluster expansion

gibbsian hypothesis in turbulence we show that kolmogorov multipliers in turbulence cannot be statistically independent of others at adjacent scales or even a finite range apart by numerical simulation of a shell model and by theory as the simplest generalization of independent distributions we suppose that the steady state statistics of multipliers in the shell model are given by a translation invariant gibbs measure with a short range potential when expressed in terms of suitable spin variables real valued spins that are logarithms of multipliers and xy spins defined by local dynamical phases numerical evidence is presented in favor of the hypothesis for the shell model in particular novel scaling laws and derivative relations predicted by the existence of a thermodynamic limit the gibbs measure appears to be in a high temperature unique phase regime with paramagnetic spin order

renormalization group in the uniqueness region weak gibbsianity and convergence we analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature we prove weak gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system the correlation length or the critical length for metastability or both thus to perturbatively treat the problem we have to use a scale adapted expansion moreover such a model below the critical temperature resembles a disordered system in the presence of griffiths singularity then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached

graphical representations for ising and potts models in general external fields this work is concerned with the theory of graphical representation for the ising and potts models over general lattices with non translation invariant external field we explicitly describe in terms of the random cluster representation the distribution function and consequently the expected value of a single spin for the ising and q state potts models with general external fields we also consider the gibbs states for the edwards sokal representation of the potts model with non translation invariant magnetic field and prove a version of the fkg inequality for the so called general random cluster model grc model with free and wired boundary conditions in the non translation invariant case adding the amenability hypothesis on the lattice we obtain the uniqueness of the infinite connected component and the almost sure quasilocality of the gibbs measures for the grc model with such general magnetic fields as a final application of the theory developed we show the uniqueness of the gibbs measures for the ferromagnetic ising model with a positive power law decay magnetic field with small enough power as conjectured in bissacot et al commun math phys 337 41 53 2015

relative entropy and variational properties of generalized gibbsian measures we study the relative entropy density for generalized gibbs measures we first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of almost gibbsian measures almost sure continuity of conditional probabilities for quasilocal measures we obtain a full variational principle for the joint measures of the random field ising model we show that the weak gibbs property holds with an almost surely rapidly decaying translation invariant potential for these measures we show that the variational principle fails as soon as the measures lose the almost gibbs property these examples suggest that the class of weakly gibbsian measures is too broad from the perspective of a reasonable thermodynamic formalism

almost gibbsian description of the sign fields of sos fields an example is presented of a measure on a lattice system which has a measure zero set of points configurations where some conditional probability can be discontinuous but does not become a gibbs measure under decimation or other transformations we also discuss some related issues

entropy for translation invariant random cluster measures we study translation invariant random cluster measures with techniques from large deviation theory and convex analysis in particular we prove a large deviation principle with rate function given by a specific entropy and a dobrushinlanfordruelle variational principle that characterizes translation invariant random cluster measures as the solutions of the variational equation for free energy consequences of these theorems include inequalities for edge and cluster densities of translationinvariant random cluster measures 1 introduction the random cluster model was introduced around 1972 by fortuin and kasteleyn 18 as a family of models that contains the bernoulli bond percolation model the ising model and the q state potts models as special cases it has developed into an important and very successful tool for studying the ising and potts models our paper focuses on the properties of the random cluster measures themselves our starting point is grimmett s 23 development of elements of a thermodynamic formalism for this model beginning with a definition of infinite volume random cluster measures in terms of a dlr dobrushinlanfordruelle equation just as for gibbsian systems natural next questions concern the validity of large deviation principles and variational characterizations of infinite volume measures these questions we answer in the present paper and derive some consequences here is a brief overview of the dlr approach of 23 the random cluster measures of interest live on the graph formed by putting edges between nearest neighbor vertices of the d dimensional hypercubic lattice d the edge that connects two vertices x and y that are l 1 distance one apart is

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

freezing transition in the ising model without internal contours we consider the low temperature ising model in a uniform magnetic field h 0 with minus boundary conditions and conditioned on having no internal contours this simple contour model defines a non gibbsian spin state for large enough magnetic fields h h c this state is concentrated on the single spin configuration of all spins up for smaller values h h c the spin state is non trivial at the critical point h c 0 the magnetization jumps discontinuously freezing provides also an example of a translation invariant weakly gibbsian state which is not almost gibbsian

weakly gibbsian measures and quasilocality a long range pair interaction counterexample we exhibit an example of a measure on a discrete and finite spin system whose conditional probabilities are given in terms of an almost everywhere absolutely summable potential but are discontinuous almost everywhere

the tightness of the kesten stigum reconstruction bound of symmetric model with multiple mutations it is well known that reconstruction problems as the interdisciplinary subject have been studied in numerous contexts including statistical physics information theory and computational biology to name a few we consider a 2q state symmetric model with two categories of q states in each category and 3 transition probabilities the probability to remain in the same state the probability to change states but remain in the same category and the probability to change categories we construct a nonlinear second order dynamical system based on this model and show that the kesten stigum reconstruction bound is not tight when q ge 4

group algebras with a unique c norm abstract call a locally compact group g c unique if l 1 g has exactly one separating c norm it is easy to see that a regular group g is c unique and that a c unique group is amenable for connected groups g it is proved that g is c unique if the interior r g 0 of a certain part r g of prim g called the regular part of prim g is dense in prim g and that c uniqueness of g implies the density of r g in prim g from this it is derived that a connected group of type i is c unique if and only if r g 0 is dense in prim g for exponential g a quite explicit version of this result in terms of the lie algebra of g is given as an easy consequence examples of amenable groups which are not c unique and c unique groups which are not regular are obtained furthermore it is shown that a connected locally compact group g is amenable if and only if l 1 g has exactly one c norm which is invariant under the isometric automorphisms of l 1 g

mathematical properties of renormalization group transformations abstract renormalization group transformations of the type introduced into statistical physics by k g wilson have been widely used with great success for a variety of many body calculations the basic approach consists in integrating out certain degrees of freedom and using a new image hamiltonian to describe the statistical properties of those that remain the mathematical foundations of the procedure however remain somewhat obscure p a pearce and the author have pointed out that there are serious questions about the existence and properties of a thermodynamic limit for renormalization group transformations in particular certain real space transformations show peculiarities in this limit these suggest that there may be serious mathematical problems with renormalization group procedures despite their considerable practical sucess

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

non gibbsian states and their gibbs description the driving principle behind this paper is the following thesis every physically reasonable random field has to be a gibbs random field in this paper the so called non gibbsian random fields are considered the usual definition of the gibbs field is generalized in such a way so as to include some of the discovered non gibbsian fields the new definition is then used to show that the projection of the two dimensional ising model onto the one dimensional sublattice 1 falls into the class of the generalized gibbs fields

measures for lattice systems abstract i review issues related with the presence or absence of gibbsianness in measures describing random fields in lattices after a brief exposition of the definition and properties of gibbs measures i discuss the phenomenon of non gibbsianness its examples characterization the proposed classification schemes and notions of generalized gibbsianness the review closes with comments on directions for further work

optimization of random searches on defective lattice networks we study the general problem of how to search efficiently for targets randomly located on defective lattice networks i e regular lattices which have some fraction of its nodes randomly removed we consider large but finite triangular lattices and assume for the search dynamics that the walker chooses steps lengths lj from the power law distribution p lj approximately lj mu with the exponent mu regulating the strategy of the search process at each step lj the searcher moves in straight lines and constantly looks within a detection radius of vision rv for the targets along the way if there is contact with a defect the movement stops and a new step length is chosen hence the presence of defects decreases the efficiency of the overall process we study numerically how three different aspects of the lattice influence the optimization of the search efficiency i the type of boundary conditions ii the concentration of targets and defects and iii the category or class of search destructive nondestructive or regenerative motivated by the results we develop a type of mean field model for the problem and obtain an analytical approximation for the search efficiency function finally we discuss in the context of searches how defective lattices compare with perfect lattices and with continuous environments

second order large deviation estimates for ferromagnetic systems in the phase coexistence region we consider thed dimensional ising model with ferromagnetic nearest neighbor interaction at inverse temperature let m lambda lambda 1 sum limits i in lambda sigma i be the magnetization inside ad dimensional hyper cube be the gibbs state andm be the spontaneous magnetization for such thatm 0 we find a sufficient condition easily verified to hold for large for m a b to decay exponentially with d 1 d when m b m 1 a b ford 2 this sufficient condition is the exponential decay of a connectivity function we also prove a partial converse to this result obtain a sharper result for the magnetization ond 1 dimensional cross sections of the model and prove a similar result ford 2 m a b m and large when free boundary conditions are chosen outside

almost gibbsianness and parsimonious description of the decimated 2d ising model in this paper we complete and provide details for the existing characterizations of the decimation of the ising model on mathbb z 2 in the generalized gibbs context we first recall a few features of the dobrushin program of restoration of gibbsianness and present the construction of global specifications consistent with the extremal decimated measures we use them to prove that these renormalized measures are almost gibbsian at any temperature and to analyse in detail its convex set of dlr measures we also recall the weakly gibbsian description and complete it using a potential that admits a quenched correlation decay i e a well defined configuration dependent length beyond which this potential decays exponentially we use these results to incorporate these decimated measures in the new framework of parsimonious random fields that has been recently developed to investigate probability aspects related to neurosciences

levy flights in random searches we review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager and discuss its quantitative description using the theory of random walks we illustrate levy flight foraging by comparison to brownian random walks and discuss experimental observations of levy flights in biological foraging we review recent findings suggesting that an inverse square probability density distribution p l l 2 of step lengths l can lead to optimal searches finally we survey the explanations put forth to account for these unexpected findings

disagreement percolation in the study of markov fields recently one of the authors van den berg has obtained a uniqueness condition for gibbs measures in terms of disagreement percolation involving two independent realizations in the present paper we study the dependence of markov fields on boundary conditions by taking a more suitable coupling this coupling leads to a new uniqueness condition which improves the one mentioned above we also compare it with the dobrushin uniqueness condition in the case of the ising model our coupling shares certain properties with the fortuin kasteleyn representation it gives an explicit expression of the boundary effect on a certain vertex in terms of percolation like probabilities

propagation of gibbsianness for infinite dimensional gradient brownian diffusions we study the strong gibbsian character on mathbb r mathbb z d of the law at time t of an infinite imensional gradient brownian diffusion when the initial distribution is gibbsian

weakly gibbsian representations for joint measures of quenched lattice spin models can the joint measures of quenched disordered lattice spin models with finite range on the product of spin space and disorder space be represented as suitably generalized gibbs measures of an annealed system we prove that there is always a potential depending on both spin and disorder variables that converges absolutely on a set of full measure w r t the joint measure weak gibbsianness this positive result is surprising when contrasted with the results of a previous paper k6 where we investigated the measure of the set of discontinuity points of the conditional expectations investigation of a s gibbsianness in particular we gave natural negative examples where this set is even of measure one including the random field ising model further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations we apply them to various examples from this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero our proof uses a martingale argument that allows to cut an infinite volume analogue of the quenched free energy into local pieces along with generalizations of kozlov s constructions

optimal search behavior and classic foraging theory random walk methods and diffusion theory pervaded ecological sciences as methods to analyze and describe animal movement consequently statistical physics was mostly seen as a toolbox rather than as a conceptual framework that could contribute to theory on evolutionary biology and ecology however the existence of mechanistic relationships and feedbacks between behavioral processes and statistical patterns of movement suggests that beyond movement quantification statistical physics may prove to be an adequate framework to understand animal behavior across scales from an ecological and evolutionary perspective recently developed random search theory has served to critically re evaluate classic ecological questions on animal foraging for instance during the last few years there has been a growing debate on whether search behavior can include traits that improve success by optimizing random stochastic searches here we stress the need to bring together the general encounter problem within foraging theory as a mean for making progress in the biological understanding of random searching by sketching the assumptions of optimal foraging theory oft and by summarizing recent results on random search strategies we pinpoint ways to extend classic oft and integrate the study of search strategies and its main results into the more general theory of optimal foraging

ill defined block spin transformations at arbitrarily high temperatures examples are presented of block spin transformations which map the gibbs measures of the ising model in two or more dimensions at temperature intervals extending to arbitrarily high temperatures onto non gibbsian measures in this way we provide the first example of this kind of pathology for very high temperatures and as a corollary also the first example of such a pathology happening at a critical point

regularity and uniqueness of c norm for tensor products of algebras abstract in this paper we study regularity and uniqueness of c norm for tensor products of algebras in relation to the corresponding properties of the factors these two concepts are involved with the ideal theory of a algebra and its enveloping c algebra our results are applied to direct products of locally compact groups

variational principle for some renormalized measures we show that some measures suffering from the so called renormalization group pathologies satisfy a variational principle and that the corresponding limit of the pressure with boundary conditions in a set of measure 1 is equal to the pressure of the ising model module a scale factor

some permanence properties of c unique groups abstract we will study some permanence properties of c unique groups in details in particular normal subgroups and extensions will be considered among other interesting results we prove that every second countable amenable group with an injective finite dimensional representation not necessarily unitary is a retract of a c unique group moreover any amenable discrete group is a retract of a discrete c unique group

global specifications and nonquasilocality of projections of gibbs measures reference gr pf article 1997 001doi 10 1214 aop 1024404514view record in web of science record created on 2008 11 27 modified on 2017 05 12

graded cluster expansion for lattice systems in this paper we develop a general theory which provides a unified treatment of two apparently different problems the weak gibbs property of measures arising from the application of renormalization group maps and the mixing properties of disordered lattice systems in the griffiths phase we suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely large regions are widely separated we then show how it is possible to construct a convergent multi scale cluster expansion

the restriction of the ising model to a layer we discuss the status of recent gibbsian descriptions of the restriction projection of the ising phases to a layer we concentrate on the projection of the two dimensional low temperature ising phases for which we prove a variational principle

variational principle and almost quasilocality for renormalized measures we study the variational principle for some non gibbsian measures we give a necessary and sufficient condition for the validity of the implication zero relative entropy density implies common version of conditional probabilities so called second part of the variational principle applying this to noisy decimations of the low temperature phases of the ising model we obtain almost sure quasilocality for these measures and the second part of the variational principle for the projection of low temperature ising phases on a one dimensional layer we also obtain the second part of the variational principle

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

robustness of the non gibbsian property some examples we discuss some examples of measures on lattice systems which lack the property of being a gibbs measure in a rather strong sense

loss and recovery of gibbsianness for xy models in external fields we consider planar rotors xy spins in zd starting from an initial gibbs measure and evolving with infinite temperature stochastic diffusive dynamics at intermediate times if the system starts at low temperature gibbsianness can be lost due to the influence of the external initial field gibbsianness can be recovered after large finite times we prove some results supporting this picture

large deviations of the dynamical activity in the east model analysing structure in biased trajectories we consider large deviations of the dynamical activity in the east model we bias this system to larger than average activity and investigate the structure that emerges to best characterize this structure we exploit the fact that there are effective interactions that would reproduce the same behaviour in an equilibrium system we combine numerical results with linear response theory and variational estimates of these effective interactions giving the first insights into such interactions in a many body system across a wide range of biases the system exhibits a hierarchy of responses to the bias remaining quasi equilibrated on short length scales but deviating far from equilibrium on large length scales we discuss the connection between this hierarchy and the hierarchical ageing behaviour of the system

transformations of gibbs measures we study local transformations of gibbs measures we establish sufficient conditions for the quasilocality of the images and obtain results on the existence and continuity properties of their relative energies general results are illustrated by simple examples

the influence of the environment on levy random search efficiency fractality and memory effects an open problem in the field of random searches relates to optimizing the search efficiency in fractal environments here we address this issue through a systematic study of levy searches in landscapes encompassing several degrees of target aggregation and fractality for scarce resources non destructive searches with unrestricted revisits to targets are shown to present universal optimal behavior irrespective of the general scaling properties of the spatial distribution of targets in contrast no such universal behavior occurs in the destructive case with forbidden revisits in which the optimal strategy strongly depends on the degree of target aggregation we also investigate how the presence of memory and learning skills of the searcher affect the search efficiency by considering a limiting model in which the searcher learns through recent experience to recognize food rich areas we find that a statistical memory of previous encounters does not necessarily increase the rate of target findings in random searches instead there is an optimal extent of memory dependent on specific details of the search space and stochastic dynamics which maximizes the search efficiency this finding suggests a more general result namely that in some instances there are actual advantages to ignoring certain pieces of partial information while searching for objects

large deviations and variational principle for harmonic crystals we consider massless gaussian fields with covariance related to the green function of a long range random walk on d these are viewed as gibbs measures for a linear quadratic interaction we establish thermodynamic identities and prove a version of gibbs variational principle showing that translation invariant gibbs measures are characterized as minimizers of the relative entropy density we then study the large deviations of the empirical field of a gibbs measure we show that a weak large deviation principle holds at the volume order with rate given by the relative entropy density

invariant ideals of twisted crossed products we will prove a result concerning the inclusion of non trivial invariant ideals inside non trivial ideals of a twisted crossed product we will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on c algebras

d dimensional levy flights exact and asymptotic the analytic and asymptotic properties of the spherically symmetric d dimensional levy stable probability density function p d r are discussed in detail these isotropic stable probability density functions pdfs are analogous to the one dimensional symmetric levy stable pdfs previously studied by the present authors j math phys 43 2670 2002 we construct a hypergeometric representation of p d r when is rational and find a number of new representations of p d r in terms of special functions for various values of d and a recursion relation is found between p d r and p d 2 r which in particular implies there exists a simple map between p 1 r and p 3 r as in our previous paper we discuss the properties of p d r for both the cases 2 and 2 we demonstrate the existence of intricate exponentially small series in the large r asymptotics of p d r when is an integer which are dominant when is even we explicitly construct this beyond all orders expansion of p d r for arbitr

pathological behavior of renormalization group maps at high fields and above the transition temperature we show that decimation transformations applied to high q potts models result in non gibbsian measures even for temperatures higher than the transition temperature we also show that majority transformations applied to the ising model in a very strong field at low temperatures produce non gibbsian measures this shows that pathological behavior of renormalization group transformations is even more widespread than previous examples already suggested

optimization of random searches on regular lattices we investigate random searches on isotropic and topologically regular square and triangular lattices with periodic boundary conditions and study the efficiency of search strategies based on a power law distribution p ensuremath ell ensuremath sim ensuremath ell ensuremath ensuremath mu of step lengths ensuremath ell we consider both destructive searches in which a target can be visited only once and nondestructive searches when a target site is always available for future visits we discuss i the dependence of the search efficiency on the choice of the lattice topology ii the relevance of the periodic boundary conditions iii the behavior of the optimal power law exponent ensuremath mu mathrm opt as a function of target site density iv the differences between destructive and nondestructive environments and finally v how the results for the discrete searches differ from the continuous cases previously studied

the interaction potential of the stationary measure of a high noise spinflip process a continuous time spinflip process started from an arbitrary state is proven to yield a gibbs state after any time t 0 if the noise is high enough this is done by approximating the process by discrete time processes the relative energies of those gibbs states are constructed explicitly via a high noise cluster expansion

some remarks on pathologies of renormalization group transformations for the ising model the results recently obtained by van enter fernandez and sokal on non gibbsianness of the measurev t b h arising from the application of a single decimation transformationt b with spacingb to the gibbs measure h of the ising model for suitably chosen large inverse temperature and nonzero external fieldh are critically analyzed in particular we show that if keeping fixed the same values of h andb one iterates a sufficiently large number of timesn the transformationt b one obtains a new measurev t b n h which is gibbsian and moreover very weakly coupled

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

non reversible stationary measures for infinite interacting particle systems two results concerning the local conditional distributions of a stationary measure for a spin flip process with strictly positive and continuous rates are obtained 1 the local conditional distributions and the rates of the reversed process determine each other 2 either all shift invariant stationary measures are gibbs with the same potential or no shift invariant stationary gibbs measure exist

improvements in the statistical approach to random levy flight searches recently it has been shown that the most e0cient strategy for searching randomly located objects when the sites are randomly distributed and can be revisited any number of times leads to a power law distribution p of the ights with 2 we show analytically that the incorporation of energy considerations limits the possible range for the l evy exponent however 2 still emerges as the optimal foraging condition furthermore we show that the probability distribution of ight lengths for the short and intermediate ight length regimes depends on the details of the system c 2001 elsevier science b v all rights reserved

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

percolation path large deviations and weakly gibbs states we present a unified approach to establishing the gibbsian character of a wide class of non gibbsian states arising in the renormalisation group theory inside the realm of the pirogov sinai theory for lattice spin systems we prove that rg transformations applied to low temperature phases give rise to weakly gibbsian measures in other words we show that the griffiths pearce israel scenario of rg pathologies is carried by atypical configurations the renormalized measures are described by an effective interaction with relative energies well defined on a full measure set of configurations in this way we complete the first part of the dobrushin restoration program to give a gibbsian description to non gibbsian states a disagreement percolation estimate is used in the proof to bound the decay of quenched correlations through which the interaction potential is constructed the percolation is controlled via a novel type of pathwise large deviation theory

taming griffiths singularities infinite differentiability of quenched correlation functions we prove infinite differentiability of the magnetization and of all quenched correlation functions for disordered spin systems at high temperature or strong magnetic field in the presence of griffiths singularities we also show uniqueness of the gibbs state and exponential decay of truncated correlation functions with probability one our results are obtained through new simple modified high temperature or low activity expansions whose convergence can be displayed by elementary probabilistic arguments our results require no assumptions on the probability distributions of the random parameters except for the obvious one of no percolation of infinite couplings and in the strong field situation for the also obvious requirement that zero magnetic fields do not percolate

from pca s to equilibrium systems and back stationary measures for probabilistic cellular automata pca s ind dimensions give rise to space time histories whose statistics may naturally be described by gibbs states ind 1 dimensions for an interaction energy obtained from the pca in this note we study the converse question do all gibbs states for this correspond to statistical space time histories for the pca our main result states that the answer is yes at least for translation invariant or periodic gibbs states thus ergodicity questions for pca s can at least partially be formulated as questions of uniqueness of gibbs states

existence of compatible families of proper regular conditional probabilities let be a perfect probability space with countably generated and let ib be a family of sub fields of under a countability condition on the family ib i show that there exists a family ib of regular conditional probabilities which are everywhere compatible under a more stringent condition on ib i show that the can furthermore be chosen to be everywhere proper it follows that in the dobrushin lanford ruelle formulation of the statistical mechanics of classical lattice systems every perfect probability measure is a gibbs measure for some specification

a note on the renormalization group and contour expansions abstract we show how to apply the usual renormalization group prescription to some statistical mechanical systems that can be described in terms of a polymer model

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large