partial eigenstructure assignment for the quadratic pencil abstract it is shown in this paper that by the appropriate choice of gain and input influence matrices certain eigenpairs of a vibrating system may be assigned while the other eigenpairs remain unchanged the system under consideration is modelled by a set of second order differential equations and the assignment is carried by multi input state feedback control the solution may be of particular interest in the stabilization and control of flexible structures using smart materials where only a small part of the eigenstructure is to be reassigned and the rest is required to remain unchanged the method presented is illustrated with a numerical example

a receptance based method for predicting latent roots and critical points in friction induced vibration problems of asymmetric systems this paper studies the latent roots and critical points of friction induced vibration problems in which the stiffness matrix is asymmetric the asymmetric terms are represented by a parameter or parameters related to the friction coefficient as the parameter value increases some latent roots of the asymmetric system change and even become complex with positive real parts at a critical point indicating flutter instability a method is put forward for computing the latent roots and predicting the critical value of this parameter at the flutter instability boundary of the asymmetric system based on the receptance of the symmetric system when measured receptances of the symmetric system at those locations where friction forces would be acting in the corresponding asymmetric system are available the simulated numerical example shows that this method is efficient

large deviations for finite state markov jump processes with mean field interaction via the comparison principle for an associated hamilton jacobi equation we prove the large deviation principle ldp for the trajectory of a broad class of finite state mean field interacting markov jump processes via a general analytic approach based on viscosity solutions examples include generalized ehrenfest models as well as curie weiss spin flip dynamics with singular jump rates the main step in the proof of the ldp which is of independent interest is the proof of the comparison principle for an associated collection of hamilton jacobi equations additionally we show that the ldp provides a general method to identify a lyapunov function for the associated mckean vlasov equation

statistical mechanics of classical particles with logarithmic interactions the inhomogeneous mean field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions the results apply to various physical models of translation invariant plasmas gravitating systems as well as to planar fluid vortex motion for attractive interactions a critical behavior occurs which can be classified as an extreme case of a second order phase transition to include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic geometric mean inequality the method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions in addition it also provides us with a new proof of the trudinger moser inequality known from differential geometry in its sharp form

travelling fronts in nonlocal models for phase separation in an external field we consider the one dimensional nonlocal evolution equation derived by de masi et al 1995 for ising systems with glauber dynamics kac potentials and magnetic field we prove the existence of travelling fronts their uniqueness modulo translations among the monotone profiles and their linear stability for all the admissible values of the magnetic field for which the underlying spin system exhibits a stable and metastable phase

structural modification part 2 assignment of natural frequencies and antiresonances by an added beam in this paper the inverse structural modification problem is solved in order to determine the dimensions of the cross section of a beam that when added to an original structure will assign natural frequencies or antiresonances as specified in order for this to be accomplished rotational receptances must be measured as presented in the companion paper when the modification is cast as an additional forcing term on the original structure a system of multivariate polynomials in the parameters of the beam cross section are revealed the solution of this system yields the beam parameters that assign the specified vibration behaviour

improved perturbation expansion for disordered systems beating griffiths singularities we introduce a new expansion to prove exponential clustering of connected correlations in a large class of disordered systems our expansion converges for values of the temperature and magnetic field where standard cluster expansions diverge due to the presence of griffiths type singularities it is organized inductively over an infinite sequence of increasing distance scales in each induction step one redefines what is means by the unperturbed system a procedure somewhat reminiscent of k a m theory our techniques may be useful in dealing with the so called large field problem in real space renormalization group schemes

infinite differentiability for one dimensional spin system with long range random interaction we consider one dimensional spin systems with hamiltonian h left sigma lambda right sum limits t t in lambda frac varepsilon tt left t t right alpha sigma t sigma t h sum limits t in lambda sigma t where tt are independent random variables and using decimation and the cluster expansion we show that when 3 2 ande tt 0 for any magnetic fieldh and inverse temperature the correlation functions and the free energy arec both inh and

complete analysis of phase transitions and ensemble equivalence for the curie weiss potts model using the theory of large deviations we analyze the phase transition structure of the curie weiss potts spin model which is a mean field approximation to the nearest neighbor potts model it is equivalent to the potts model on the complete graph on n vertices the analysis is carried out both for the canonical ensemble and the microcanonical ensemble besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates relating these to concavity and support properties of the microcanonical entropy the curie weiss potts model is the first statistical mechanical model for which such a detailed and rigorous analysis has been carried out

the order parameter in a spin glass various possible precise definitions of an edwards anderson type of order parameter for an ising model spin glass are considered using boundary conditions for a finite system states of an infinite system and a duplicate system approach several of these definitions are shown to yield identical results

wigner function evolution in a self kerr medium derived by entangled state representation by introducing the thermo entangled state representation we convert the calculation of wigner function wf of density operator to an overlap between two pure states in a two mode enlarged fock space furthermore we derive a new wf evolution formula of any initial state in a self kerr medium with photon loss and find that the photon number distribution for any initial state is independent of the coupling factor with the kerr medium where the number state is not affected by the kerr nonlinearity and evolves into a density operator of binomial distribution

multidimensional random subshifts of finite type and their large deviations i introduce random multidimensional subshifts of finite type which generalize models of spin glasses and establish the almost sure large deviations bounds for gibbs measures there the paper is sequel to ekw where the corresponding results were obtained for deterministic multidimensional subshifts of finite type

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

partial eigenstructure assignment for undamped vibration systems using acceleration and displacement feedback abstract a new method for partial eigenstructure assignment using acceleration and displacement feedback for undamped vibration systems is presented in this paper firstly a necessary and sufficient condition is proposed for the incremental mass and stiffness matrices that modify some eigenpairs while keeping other eigenpairs unchanged secondly based on this condition an algorithm for determining the required control gain matrices of acceleration and displacement feedback which assign the desired eigenstructure is developed this algorithm is easy to implement and works directly on the second order system model more importantly the algorithm allows the control matrix to be specified beforehand and also leads naturally to a small norm solution of the feedback gain matrices finally some numerical examples are given to demonstrate the effectiveness and accuracy of the proposed algorithm

effect of random field disorder on the first order transition in p spin interaction model abstract we study the random field p spin model with ising spins on a fully connected graph using the theory of large deviations in this paper this is a good model to study the effect of quenched random field on systems which have a sharp first order transition in the pure state for p 2 the phase diagram of the model for bimodal distribution of the random field has been well studied and is known to undergo a continuous transition for lower values of the random field h and a first order transition beyond a threshold h t p 0 439 we find the phase diagram of the model for all p 2 with bimodal random field distribution using large deviation techniques we also look at the fluctuations in the system by calculating the magnetic susceptibility for p 2 beyond the tricritical point in the regime of first order transition we find that for h t p h 0 447 magnetic susceptibility increases rapidly even though it never diverges as one approaches the transition from the high temperature side on the other hand for 0 447 h 0 5 the high temperature behaviour is well described by the curie weiss law for all p 2 we find that for larger magnitudes of the random field h h o 1 p the system does not show ferromagnetic order even at zero temperature we find that the magnetic susceptibility for p 3 is discontinuous at the transition point for h h o

on the absence of breakdown of symmetry for the plane rotator model with long range random interaction we study the plane rotator model with hamiltonian frac 1 2 sum limits x ne y frac jxy cos mathcal o x mathcal o y left x y right 3 mathcal e where jxy for different pair x y are independent symmetric unbounded random variables it is proved that for almost all j all gibbs states p j are invariant by rotation

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

renormalization group and analyticity in one dimension a proof of dobrushin s theorem we consider one dimensional systems described by many body potentials with finite first moment and prove that the correlation functions are analytic in the interaction parameters this result is not new dobrushin 1973 but our proof is simpler and physically more transparent we show that by introducing suitable blocks and averaging over the variables associated to a subset of the blocks decimation procedure the resulting effective interaction is such that the system can always be dealt with as a high temperature system

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

pole assignment of friction induced vibration for stabilisation through state feedback control abstract friction induced self excited linear vibration is often governed by a second order matrix differential equation of motion with an asymmetric stiffness matrix the asymmetric terms are product of friction coefficient and the normal stiffness at the contact interface when the friction coefficient becomes high enough the resultant vibration becomes unstable as frequencies of two conjugate pairs of complex eigenvalues poles coalesce when viscous damping is low this short paper presents a receptance based inverse method for assigning complex poles to second order asymmetric systems through active state feedback control of a combination of active stiffness active damping and active mass which is capable of assigning negative real parts to stabilise an unstable system

variational description of gibbs non gibbs dynamical transitions for spin flip systems with a kac type interaction we continue our study of gibbs non gibbs dynamical transitions in the present paper we consider a system of ising spins on a large discrete torus with a kac type interaction subject to an independent spin flip dynamics infinite temperature glauber dynamics we show that in accordance with the program outlined in van enter et al moscow math j 10 687 711 2010 in the thermodynamic limit gibbs non gibbs dynamical transitions are equivalent to bifurcations in the set of global minima of the large deviation rate function for the trajectories of the empirical density conditional on their endpoint more precisely the time evolved measure is non gibbs if and only if this set is not a singleton for some value of the endpoint a partial description of the possible scenarios of bifurcation is given leading to a characterization of passages from gibbs to non gibbs and vice versa with sharp transition times our analysis provides a conceptual step up from our earlier work on gibbs non gibbs dynamical transitions for the curie weiss model where the mean field interaction allowed us to focus on trajectories of the empirical magnetization rather than the empirical density

the legendre transform of two replicas of the sherrington kirkpatrick spin glass model we prove a variational inequality linking the values of the free energy per site at different temperatures this inequality is based on the legendre transform of the free energy of two replicas of the system we prove that equality holds when 1 sqrt 2 and fails when 1 sqrt 2 1 sqrt 2 we also use our inequality numerically within the low temperature region to improve slightly the best previously known lower bounds for the free energy and the ground state energy per site

gibbs states of the hopfield model in the regime of perfect memory we study the thermodynamic properties of the hopfield model of an autoassociative memory ifn denotes the number of neurons andm n the number of stored patterns we prove the following results ifm n 0 asn then there exists an infinite number of infinite volume gibbs measures for all temperaturest 1 concentrated on spin configurations that have overlap with exactly one specific pattern moreover the measures induced on the overlap parameters are dirac measures concentrated on a single point and the gibbs measures on spin configurations are products of bernoulli measures ifm n asn for small enough we show that for temperaturest smaller than somet 1 the induced measures can have support only on a disjoint union of balls around the previous points but we cannot construct the infinite volume measures through convergent sequences of measures

on secondary bifurcations for some nonlinear convolution equations etude de quelques equations de convolution non lineaires sur le tore a d dimensions conditions d existence de bifurcations secondaires

surface tension in ising systems with kac potentials we consider an ising spin system with kac potentials in a torus of d d 2 and fix the temperature below its lebowitz penrose critical value we prove that when the kac scaling parameter vanishes the log of the probability of an interface becomes proportional to its area and the surface tension related to the proportionality constant converges to the van der waals surface tension the results are based on the analysis of the rate functionals for gibbsian large deviations and on the proof that they converge to the perimeter functional of geometric measure theory which extends the notion of area our considerations include nonsmooth interfaces proving that the gibbsian probability of an interface depends only on its area and not on its regularity

on the free energy of the hopfield model the general theory of inhomogeneous mean field systems of raggio and werner provides a variational expression for the almost sure limiting free energy density of the hopfield model h n p xi s frac 1 2n sum limits i j 1 n sum limits mu 1 n xi i mu xi j mu s i s j for ising spinssi andp random patterns 1 2 n under the assumption that mathop lim limits n to gamma n 1 sum limits i 1 n delta xi i lambda xi i xi i 1 xi i 2 xi i p exists almost surely in the space of probability measures overp copies of 1 1 including an external field term p h i 1 n i si we give a number of general properties of the free energy density and compute it for a p 2 in general and b p arbitrary when is uniform and at most the two componentsh 1 andh 2 are nonzero obtaining the almost sure formula f beta h tfrac 1 2 f ew beta h mu 1 h mu 2 tfrac 1 2 f ew beta h mu 1 h mu 2 for the free energy wherefcw denotes the limiting free energy density of the curie weiss model with unit interaction constant in both cases we obtain explicit formulas for the limiting almost sure values of the so called overlap parameters m n mu beta h n 1 sum limits i 1 n xi i mu left langle s i right rangle in terms of the curie weiss magnetizations for the general i i d case with prob i 1 1 2 we obtain the lower bound 1 4 2 p 1 for the temperaturetc separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero this lower bound is exact forp 2 or e 0 or e 1 2 forp 2 we identify an intermediate temperature region between t 1 4 2 and tc 1 4 2 where the overlap vector is homogeneous i e all its components are equal and nonzero t marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct we conjecture that the homogeneous nonzero regime exists forp 3 and that t max 1 4 2 p 1 0

one dimensional random ising systems with interaction decay r 1 epsilon a convergent cluster expansion we consider a one dimensional random ising model with hamiltonian h sum limits i ddag j frac j ij left i j right 1 varepsilon s i s j h sum limits i s i where e 0 andj ij are independent identically distributed random variables with distributiondf x such that int xdf left x right 0 i int e tx df left x right infty forall t in mathbb r ii we construct a cluster expansion for the free energy and the gibbs expectations of local observables this expansion is convergent almost surely at every temperature in this way we obtain that the free energy and the gibbs expectations of local observables arec functions of the temperature and of the magnetic fieldh moreover we can estimate the decay of truncated correlation functions in particular for every e 0 there exists a random variablec m finite almost everywhere such that left left langle s 0 s j right rangle h left langle s 0 right rangle h left langle s j right rangle h right leqq frac c left omega right left j right 1 varepsilon varepsilon where h denotes the gibbs average with respect to the hamiltonianh

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

on equivalence of spin and field pictures of lattice systems we investigate the spin and field systems on a lattice connected by the kac siegert transform it is shown that the structures of corresponding theories are equivalent in the sense of isomorphy of space of gibbs states and order parameters using the idea of equivalence of spin and field pictures we exhibit a class of lattice systems possessing infinitely uncountably many ground states the systems of this type with infinite range slow decaying interactions are expected to have a spin glass phase transition

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

statistics of complex eigenvalues in friction induced vibration abstract self excited vibrations appear in many mechanical systems with sliding contacts there are several mechanisms whereby friction can cause the self excited vibration to become unstable of these mechanisms mode coupling is thought to be responsible for generating annoying high frequency noise and vibration in brakes conventionally in order to identify whether a system is stable or not complex eigenvalue analysis is performed however what has recently received much attention of researchers is the variability and uncertainty of input variables in the stability analysis of self excited vibrations for this purpose a second order perturbation method is extended and employed in the current study the moments of the output distribution along with its joint moment generating function are used for quantifying the statistics of the complex eigenvalues moreover the eigen derivatives required for the perturbation method are presented in a way that they can deal with the asymmetry of the stiffness matrix and non proportional damping since the eigen derivatives of such systems are complex valued numbers it is mathematically more informative and convenient to derive the statistics of the eigenvalues in a complex form without decomposing them into two real valued real and imaginary parts then the variance and pseudo variance of the complex eigenvalues are used for determining the statistics of the real and imaginary parts the reliability and robustness of the system in terms of stability can also be quantified by the approximated output distribution

quasi additive estimates on the hamiltonian for the one dimensional long range ising model in this work we study the problem of getting quasi additive bounds for the hamiltonian of the long range ising model when the two body interaction term decays proportionally to 1d2 0 1 we revisit the paper by cassandro et al j math phys 46 053305 2005 where they extend to the case 0 ln3ln2 1 the result of the existence of a phase transition by using a peierls argument given by frohlich and spencer commun math phys 84 87 101 1982 for 0 the main arguments of cassandro et al j math phys 46 053305 2005 are based in a quasi additive decomposition of the hamiltonian in terms of hierarchical structures called triangles and contours which are related to the original definition of contours introduced by frohlich and spencer commun math phys 84 87 101 1982 in this work we study the existence of a quasi additive decomposition of the hamiltonian in terms of the contours defined in the work of cassandro et al j math phys 46 053305 2005 the most relevant r

quenched large deviation principle for words in a letter sequence when we cut an i i d sequence of letters into words according to an independent renewal process we obtain an i i d sequence of words in the annealed large deviation principle ldp for the empirical process of words the rate function is the specific relative entropy of the observed law of words w r t the reference law of words in the present paper we consider the quenched ldp i e we condition on a typical letter sequence we focus on the case where the renewal process has an algebraic tail the rate function turns out to be a sum of two terms one being the annealed rate function the other being proportional to the specific relative entropy of the observed law of letters w r t the reference law of letters with the former being obtained by concatenating the words and randomising the location of the origin the proportionality constant equals the tail exponent of the renewal process earlier work by birkner considered the case where the renewal process has an exponential tail in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere in a companion paper the annealed and the quenched ldp are applied to the collision local time of transient random walks and the existence of an intermediate phase for a class of interacting stochastic systems is established

metastates in disordered mean field models ii the superstates we continue to investigate the size dependence of disordered mean field models with finite local spin space in more detail illustrating the concept of superstates as recently proposed by bovier and gayrard we discuss various notions of convergence for the behavior of the paths t tn t 0 1 in the thermodynamic limit n here n is the gibbs measure in the finite volume 1 n and is the disorder variable in particular we prove refined convergence statements in our concrete examples the hopfield model with finitely many patterns having continuous paths and the curie weiss random field ising model having singular paths

ordered overlaps in disordered mean field models let m n be a sequence of integers with m as n and m o n for bounded i i d r v i k and bounded i i d r v i we study the large deviation of the family of ordered scalar products x k n 1 i 1 n i i k k m under the distribution conditioned on the i k s to get a full large deviation principle it is necessary to specify also the total norm k m x k 2 1 2 which turns to be associated with some extra gaussian distribution our results apply to disordered mean field systems including generalized hopfield models in the regime of a sublinear number of patterns we build also a class of examples where this norm is the crucial order parameter

asymptotic dynamics non critical and critical fluctuations for a geometric long range interacting model we study the dynamics of geometric spin system on the torus with long range interaction as the number of particles goes to infinity the process converges to a deterministic dynamical magnetization field that satisfies an euler equation law of large numbers its stable steady states are related to the limits of the equilibrium measures gibbs states of the finite particle system a related equation holds for the magnetization densities for which the property of propagation of chaos also is established we prove a dynamical central limit theorem with an infinite dimensional ornstein uhlenbeck process as a limiting fluctuation process at the critical temperature of a ferromagnetic phase transition both a tighter quantity scaling and a time scaling is required to obtain convergence to a one dimensional critical fluctuation process with constant magnetization fields which has a non gaussian invariant distribution similarly at the phase transition to an antiferromagnetic state with frequencyp0 the fluctuation process with critical scaling converges to a two dimensional critical fluctuation process which consists of fields with frequencyp0 and has a non gaussian invariant distribution on these fields finally we compute the critical fluctuation process in the infinite particle limit at a triple point where a ferromagnetic and an antiferromagnetic phase transition coincide

efficient numerical path following beyond critical points the paper presents a numerical pathfollowing technique based on an implicit change of parametrization using qr decomposition an automatic steplength control for tangent continuation is derived on a theoretical basis in the suggested approach turning points do not play any exceptional role for the actual computation of simple bifurcation points a special algorithm based on the augmented system of moore is developed this treatment includes imperfect bifurcations and permits an easy way of branch switching an efficient linear equation solver is worked out in connection with the qr decomposition finally numerical comparisons are included that clearly document the efficiency of the newly developed algorithms

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

metastates in disordered mean field models random field and hopfield models we rigorously investigate the size dependence of disordered mean field models with finite local spin space in terms of metastates thereby we provide an illustration of the framework of metastates for systems of randomly competing gibbs measures in particular we consider the thermodynamic limit of the empirical metastate 1 n sum nolimits n 1 n delta mu eta eta where n is the gibbs measure in the finite volume 1 n and the frozen disorder variable is fixed we treat explicitly the hopfield model with finitely many patterns and the curie weiss random field ising model in both examples in the phase transition regime the empirical metastate is dispersed for largen moreover it does not converge for a e but rather in distribution for whose limits we given explicit expressions we also discuss another notion of metastates due to aizenman and wehr

self excited vibrations induced by dry friction part 2 stability and limit cycle analysis abstract some mechanical systems have frictional joints where one surface slides on another with dry friction under certain conditions the steady state sliding motion becomes unstable the instability leads to the occurrence of stable limit cycle type self excited vibrations this study is concerned with the analysis of the experimental results reported in a companion paper dealing with the instability of the steady state sliding motion and the resulting self excited vibrations the conditions which cause the steady state sliding motion to become unstable are investigated with a linear stability theory a parametric study is conducted to assess the influence of the system parameters on the stability of the steady state sliding motion the method of triple input describing functions is used with a non linear model to analyze the resulting self excited vibrations

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

fluctuation results for general block spin ising models we study a block spin mean field ising model i e a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices and where pair interactions are given according to their blocks for the vector of block magnetizations we prove large deviation principles and central limit theorems under general assumptions for the block interaction matrix using the exchangeable pair approach of stein s method we establish a rate of convergence in the central limit theorem for the block magnetization vector in the high temperature regime

correction added in proof we extend sanov s theorem on i i d large deviations to independent but not identically distributed random variables and study the generalization of relative entropy that appears as the rate function

spatializing random measures doubly indexed processes and the large deviation principle the main theorem is the large deviation principle for the doubly indexed sequence of random measures w r q dx dy dx 1 dr k x l q k dy k 1 here is a probability measure on a polish space x d r k k 1 2 r is a dyadic partition of x hence the use of 2 r summands satisfying d r k 1 2 r and l q 1 l q 2 l q 2 r is an independent identically distributed sequence of random probability measures on a polish space y such that l q k q n satisfies the large deviation principle with a convex rate function a number of related asymptotic results are also derived the random measures w have important applications to the statistical mechanics of turbulence in a companion paper the large deviation principle presented here is used to give a rigorous derivation of maximum entropy principles arising in the well known miller robert theory of two dimensional turbulence as well as in a modification of that theory recently proposed by turkington

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

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

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

continuous spin mean field models limiting kernels and gibbs properties of local transforms we extend the notion of gibbsianness for mean field systems to the setup of general possibly continuous local state spaces we investigate the gibbs properties of systems arising from an initial mean field gibbs measure by application of given local transition kernels this generalizes previous case studies made for spins taking finitely many values to the first step in the direction to a general theory containing the following parts 1 a formula for the limiting conditional probability distributions of the transformed system it holds both in the gibbs and in the non gibbs regime and invokes a minimization problem for a constrained rate function 2 a criterion for gibbsianness of the transformed system for initial lipschitz hamiltonians involving concentration properties of the transition kernels and 3 a continuity estimate for the single site conditional distributions of the transformed system while 2 and 3 have provable lattice counterparts the characterization of 1 is stronger in m

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

general hermite and laguerre two dimensional polynomials general hermite and laguerre two dimensional 2d polynomials which form a complex three parameter unification of the special hermite and laguerre 2d polynomials are defined and investigated the general hermite 2d polynomials are related to the two variable hermite polynomials but are not the same the advantage of the newly introduced hermite and laguerre 2d polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the two variable hermite polynomials two different sets of such polynomials are introduced which are biorthonormal to each other the matrix which plays a role in the new definition of hermite and laguerre 2d polynomials is in a considered sense the square root of the matrix which plays a role in the definition of two variable hermite polynomials two essentially different explicit representations of the hermite and laguerre 2d polynomials are derived where the first involves jacobi polynomials as coefficients in superpositions of special hermite or laguerre 2d polynomials and the second is a superposition of products of two hermite polynomials with decreasing indices and with coefficients related to the special laguerre 2d polynomials generating functions are derived for the hermite and laguerre 2d polynomials

large deviation principles for the hopfield model and the kac hopfield model we study the kac version of the hopfield model and prove a lebowitz penrose theorem for the distribution of the overlap parameters at the same time we prove a large deviation principle for the standard hopfield model with infinitely many patterns

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

interfaces and typical gibbs configurations for one dimensional kac potentials we consider a one dimensional ising spin system with a ferromagnetic kac potential j r j having compact support we study the system in the limit 0 below the lebowitz penrose critical temperature where there are two distinct thermodynamic phases with different magnetizations we prove that the empirical spin average in blocks of size 1 for any positive converges as 0 to one of the two thermodynamic magnetizations uniformly in the intervals of size p for any given positivep 1 we then show that the intervals where the magnetization is approximately constant have lengths of the order of exp c 1 c 0 and that when normalized they converge to independent variables with exponential distribution we show this by proving large deviation estimates and applying the ventsel and friedlin methods to gibbs random fields finally if the temperature is low enough we characterize the interface namely the typical magnetization pattern in the region connecting the two phases

absence of phase transitions in certain one dimensional long range random systems an ising chain is considered with a potential of the formj i j i j where thej i j are independent random variables with mean zero the chain contains both randomness and frustration and serves to model a spin glass a simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if 1 this is to be contrasted with a ferromagnetic interaction which requires 2 the basic idea is to prove that the surface free energy between two half lines is finite although the surface energy may be unbounded ford dimensional systems it is shown that the free energy does not depend on the specific boundary conditions if 1 2 d

quenched large deviations for interacting diffusions in random media the aim of the paper is to establish a large deviation principle ldp for the empirical measure of mean field interacting diffusions in a random environment the point is to derive such a result once the environment has been frozen quenched model the main theorem states that a ldp holds for every sequence of environment satisfying appropriate convergence condition with a rate function that does not depend on the disorder and is different from the rate function in the averaged model similar results concerning the empirical flow and local empirical measures are provided

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

on the gaussian fluctuations of the critical curie weiss model in statistical mechanics it has been known for some time that the fluctuations of the curie weiss mean field model of ferromagnetism are non gaussian at the critical temperature here we establish the presence of a substantial gaussian element in the critical model by showing that the internal fluctuations i e the fluctuations within the fluctuating field are gaussian we show for instance that as the number n of sites increases to infinity the system with the appropriate space and spin scalings behaves asymptotically as a brownian motion with randomised drift and that the distribution of the drift has the non gaussian form familiar in this theory with the more conventional space scaling 1 n it is known that the system degenerates asymptotically to a straight line with random slope we prove here that the error from the straight line when appropriately amplified converges in distribution to a brownian bridge

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

robust partial eigenvalue assignment problem for the second order system in this paper we study the partial eigenvalue assignment problem for the second order system where only a small part of eigenvalues of the open loop system is to be reassigned and the rest are required to remain unchanged it is desirable that the feedback controller not only assigns specific eigenvalues to the second order closed loop system but also that the system is robust or insensitive to perturbations we propose a numerical method such that the condition number of the matrix of the eigenvectors of the closed loop system is minimized in the method we only need the knowledge of the eigenvalues to be altered and the corresponding eigenvectors while we do not need the knowledge of the eigenstructures that are required to remain unchanged and are often unknown numerical examples show that the present method often leads to better conditioned closed loop system

hermite and laguerre 2d polynomials abstract we define hermite 2d polynomials hm n u x y and laguerre 2d polynomials l m n u z z as functions of two variables with an arbitrary 2d matrix u as parameter and discuss their properties and their explicit representation recursion relations and generating functions for these polynomials are derived the advantage of the introduced hermite and laguerre 2d polynomials in comparison to the related usual two variable hermite polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the last one has to introduce two different kinds of such polynomials which are biorthogonal to each other