a unifying convergence analysis of second order methods for secular equations existing numerical methods of second order are considered for a so called secular equation we give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of newton s method for an equivalent problem for which newton s method exhibits convergence from any point in a given interval this interpretation unifies the convergence analysis of these methods provides convergence proofs where they were lacking and furnishes ways to construct improved methods in addition we show that some of these methods are in fact equivalent a second secular equation is also briefly considered

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

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

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

almost sure quasilocality fails for the random cluster model on a tree we study the random cluster model on a homogeneous tree and show that the following three conditions are equivalent for a random cluster measure quasilocality almost sure quasilocality and the almost sure nonexistence of infinite clusters as a consequence of this we find that the plus measure for the ising model on a tree at sufficiently low temperatures can be mapped via a local stochastic transformation into a measure which fails to be almost surely quasilocal

some numerical results on the block spin transformation for the 2d ising model at the critical point we study the block spin transformation for the 2d ising model at the critical temperaturetc we consider the model with the constraint that the total spin in each block is zero an old argument by cassandro and gallavotti strongly supports the gibbsianness of the transformed measure provided that such model has a critical temperaturet c lower thantc after describing a possible rigorous approach to the problem we present numerical evidence that indeedt c tc and study the dobrushin shlosman uniqueness condition

universal teichmuller space and diff s 1 s 1 we point out that the coset space diffs1 s1 is a dense complex submanifold of the universal teichmuller spaces of compact riemann spaces of genus g 1 a holomorphic map ofs into the inifinite dimensional segal diskd1 is constructed this is the universal analogue of the map of teichmuller spaces into the siegel disk provided by the period matrix the kahler potential for the general homogenous metric on diffs1 s1 is computed explicitly using the map intod1 some applications to string theory are discussed

the theory of large deviations from boltzmann s 1877 calculation to equilibrium macrostates in 2d turbulence abstract after presenting some basic ideas in the theory of large deviations this paper applies the theory to a number of problems in statistical mechanics these include deriving the form of the gibbs state for a discrete ideal gas describing probabilistically the phase transition in the curie weiss model of a ferromagnet and deriving variational formulas that describe the equilibrium macrostates in models of two dimensional turbulence a general approach to the large deviation analysis of models in statistical mechanics is also formulated

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

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

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

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

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

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

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

continuity properties of the real complex structured singular value the structured singular value function mu is defined with respect to a given uncertainty set this function is continuous if the uncertainties are allowed to be complex however if some uncertainties are required to be real then it can be discontinuous it is shown that mu is always upper semi continuous and conditions are derived under which it is also lower semi continuous with these results the real parameter robustness problem is reexamined a related although not equivalent problem is formulated which is always continuous and the relationship between the new problem and the original real mu m problem is made explicit a numerical example and results obtained via this related problem are presented

self similarity of brownian motion and a large deviation principle for random fields on a binary tree using self similarity of brownian motion and its representation as a product measure on a binary tree we construct a random sequence of probability measures which converges to the distribution of the brownian bridge we establish a large deviation principle for random fields on a binary tree this leads to a class of probability measures with a certain self similarity property the same construction can be carried out forc 0 1 valued processes and we can describe for instance ac 0 1 valued ornstein uhlenbeck process as a large deviation of brownian sheet

entropy limit theorems and variational principles for disordered lattice systems we study infinite volume limits and gibbs states of disordered lattice systems with bounded and continuous potentials our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes we find that many familiar results of invariant potentials such as large deviation theorems variational principles and equivalence of ensembles continue to hold for disordered models with suitably modified statements

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

minimum entropy of error estimation for discrete random variables the principle of minimum entropy of error estimation meee is formulated for discrete random variables in the case when the random variable to be estimated is binary we show that the meee is given by a neyman pearson type strictly monotonous test in addition the asymptotic behavior of the error probabilities is proved to be equivalent to that of the bayesian test

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

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

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

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

a maximum modulus theorem for linear fractional transformations abstract a main result in the structured singular value theory is a maximum modulus like theorem for a special case of linear fractional transformations we extend that result to general linear fractional transformations

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

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

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

on the second order feasibility cone primal dual representation and efficient projection we study the second order feasibility cone mathcal f y in mathbb r n my leq g ty for given data m g we construct a new representation for this cone and its dual based on the spectral decomposition of the matrix m tm gg t this representation is used to efficiently solve the problem of projecting an arbitrary point x in mathbb r n onto mathcal f min y y x my leq g ty which aside from theoretical interest also arises as a necessary subroutine in the rescaled perceptron algorithm we develop a method for solving the projection problem to an accuracy varepsilon whose computational complexity is bounded by o mn 2 n ln ln 1 varepsilon n ln ln 1 min operatorname width mathcal f operatorname width mathcal f operations here operatorname width mathcal f and operatorname width mathcal f denote the width of mathcal f and mathcal f respectively we also perform computational tests that indicate that the method is extremely efficient in practice

pseudo free energies and large deviations for non gibbsian fkg measures a large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on 0 1 zd is proven in this way a pseudo free energy and pressure is defined for ergodic systems the large deviations property holds with the usual scaling the case of non ergodic systems is also discussed a similar result holds for occupation times the perturbation by an external field is treated

uniqueness of minimizer for countable markov shifts and equidistribution of periodic points for a finitely irreducible countable markov shift and a potential with summable variations we provide a condition on the associated pressure function which ensures that bowen s gibbs state the equilibrium state and the minimizer of the level 2 large deviations rate function are all unique and they coincide from this we deduce that the set of periodic points weighted with the potential equidistributes with respect to the gibbs equilibrium state as the periods tend to infinity applications are given to the gauss map and the bowen series map associated with a finitely generated free fuchsian group with parabolic elements

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

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

microcanonical distributions for lattice gases in this article a large deviation principle cf theorem 1 3 for the empirical distribution functional is applied to prove a rather general version of boltzmann s principle cf theorem 3 5 for models with shift invariant finite range potentials the final section contains an application of these considerations to the two dimensional ising model at sub critical temperature

large deviations and concentration properties for ϕ interface models we consider the massless field with zero boundary conditions outside d n d d n n d a suitable subset of d i e the continuous spin gibbs measure n on d n with hamiltonian given by h x y x y 1 v x y and x 0 for x d n c the interaction v is taken to be strictly convex and with bounded second derivative this is a standard effective model for a d 1 dimensional interface represents the height of the interface over the base d n due to the choice of scaling of the base we scale the height with the same factor by setting n n we study various concentration and relaxation properties of the family of random surfaces n and of the induced family of gradient fields n n as the discretization step 1 n tends to zero n in particular we prove a large deviation principle for n and show that the corresponding rate function is given by d u x dx where is the surface tension of the model this is a multidimensional version of the sample path large deviation principle we use this result to study the concentration properties of n under the volume constraint i e the constraint that 1 n d x dn n x stays in a neighborhood of a fixed volume v 0 and the hard wall constraint i e n x 0 for all x this is therefore a model for a droplet of volume v lying above a hard wall we prove that under these constraints the field n of rescaled heights concentrates around the solution of a variational problem involving the surface tension as it would be predicted by the phenomenological theory of phase boundaries our principal result however asserts local relaxation properties of the gradient field n n to the corresponding extremal gibbs states thus our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments the proofs have both probabilistic and analytic aspects essential analytic tools are p estimates for elliptic equations and the theory of young measures on the side of probability tools a central role is played by the helffer sjostrand 31 pde representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of t funaki and h spohn 25 on the structure of gradient fields

two connections between random systems and non gibbsian measures in this contribution we discuss the role disordered or random systems have played in the study of non gibbsian measures this role has two main aspects the distinction between which has not always been fully clear 1 from disordered systems disordered systems can be used as a tool analogies with as well as results and methods from the study of random systems can be employed to investigate non gibbsian properties of a variety of measures of physical and mathematical interest 2 of disordered systems non gibbsianness is a property of various joint measures describing quenched disordered systems we discuss and review this distinction and a number of results related to these issues moreover we discuss the mean field version of the non gibbsian property and present some ideas how a kac limit approach might connect the finite range and the mean field non gibbsian properties

large deviation principle for empirical fields of log and riesz gases we study a system of n particles with logarithmic coulomb or riesz pairwise interactions confined by an external potential we examine a microscopic quantity the tagged empirical field for which we prove a large deviation principle at speed n the rate function is the sum of an entropy term the specific relative entropy and an energy term the renormalized energy introduced in previous works coupled by the temperature we deduce a variational property of the sine beta processes which arise in random matrix theory we also give a next to leading order expansion of the free energy of the system proving the existence of the thermodynamic limit

large deviations for lattice systems i parametrized independent fields we prove large deviation theorems for empirical measures of independent random fields whose distributions depend measurably on an auxiliary parameter this dependence respects the action of the shift group and a large deviation principle holds whenever a certain ergodicity condition is satisfied we also investigate the entropy functions for these processes especially in relation to the usual relative entropy

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

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

on the closedness of the linear image of a closed convex cone when is the linear image of a closed convex cone closed we present very simple and intuitive necessary conditions that 1 unify and generalize seemingly disparate classical sufficientconditions such as polyhedrality of the cone and slater type conditions 2 are necessary and sufficient when the dual cone belongs to a class that we call nice cones nice cones subsume all cones amenable to treatment by efficient optimization algorithms for instance polyhedral semidefinite and p cones and 3 provide similarly attractive conditions for an equivalent problem the closedness of the sum of two closed convex cones

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

renormalization group at criticality and complete analyticity of constrained models a numerical study we study the majority rule transformation applied to the gibbs measure for the 2d ising model at the critical point the aim is to show that the renormalized hamiltonian is well defined in the sense that the renormalized measure is gibbsian we analyze the validity of dobrushin shlosman uniqueness dsu finite size condition for the constrained models corresponding to different configurations of the image system it is known that dsu implies in our 2d case complete analyticity from which as recently shown by haller and kennedy gibbsianness follows we introduce a monte carlo algorithm to compute an upper bound to vasserstein distance appearing in dsu between finite volume gibbs measures with different boundary conditions we get strong numerical evidence that indeed the dsu condition is verified for a large enough volumev for all constrained models

entropy production and information gain in axiom a systems anosov systems are mathematical models for chaotic systems in statistical mechanics and fluid dynamics most of these systems enjoy the property of positive entropy production we introduce the concept of specific information gain or specific relative entropy h in anosov systems and prove that it is identical to the entropy production rate e p defined by ruelle and gallavotti in anosov systems from this point of view the entropy production rate e p characterizes the degree of macroscopic irreversibility of the system

large deviation estimates for a conditional probability distribution applications to random interaction gibbs measures let x i y i d be independent identically distributed random variables with arbitrary distribution we show that for almost every y i i the conditional law of the empirical field given y i i satisfies to large deviation inequalities this applies to the study of gibbs measures with random interaction in the case of some mean field models as well as of short range summable interaction we show that the pressure is nonrandom and is given by a variational formula these random gibbs measures have the same large deviation rate which does not depend on the particular realization of the interaction their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula

chaos in cellular automaton systems with toom rule abstract the critical behaviour of cellular automata can be studied like in chaotic systems as the sensitivity to initial conditions we find the fractal dimension for the basin boundary set of the attracting configurations morever obtained data allow us to qualify configurations from the attracting set