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

rough solutions for the wave maps equation we consider the wave maps equation with values into a riemannian manifold which is isometrically embedded in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i our main result asserts that the cauchy problem is globally well posed for initial data which is small in the critical sobolev spaces this extends and completes recent work of tao and other authors

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

quadratic nonlinear derivative schrodinger equations part 2 in this paper we consider the local well posedness theory for the quadratic nonlinear schrodinger equation with low regularity initial data in the case when the nonlinearity contains derivatives we work in 2 1 dimensions and prove a local well posedness result for small initial data with low regularity

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

thermodynamics and statistical analysis of gaussian random fields gaussian fields are considered as gibbsian fields thermodynamic functions are calculated for them and the variational principle is proved as an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for gaussian markov fields

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

the cauchy problem for the ishimori equations abstract in 1984 y ishimori derived a nonlinear wave equation which is a two dimensional analogue of the classical isotropic spin chain in ferromagnetism in this paper we show that the cauchy problem associated to this equation is well posed in sobolev spaces when initial spins are almost parallel

global schrodinger maps in dimensions d 2 small data in the critical sobolev spaces we consider the schrodinger map initial value problem t on r d r 0 0 where r d r s 2 r 3 is a smooth function in all dimensions d 2 we prove that the schrodinger map initial value problem admits a unique global smooth solution 2 c r h 1 q q2 s 2 provided that the data 02 h 1 q is smooth and satises the smallness condition

finite and infinite systems of interacting diffusions we study the problem of relating the long time behavior of finite and infinite systems of locally interacting components we consider in detail a class of lincarly interacting diffusionsx t x i t i d in the regime where there is a one parameter family of nontrivial invariant measures for these systems there are naturally defined corresponding finite systems x n t left x i n t i in lambda n right with lambda n n n d cap mathbb z d our main result gives a comparison between the laws ofx t n andx n t n for timest n asn the comparison involves certain mixtures of the invariant measures for the infinite system

global existence and uniqueness of schrodinger maps in dimensions d 4 abstract in dimensions d 4 we prove that the schrodinger map initial value problem t s s s on r d r s 0 s 0 admits a unique solution s r d r s 2 r 3 s c r h q provided that s 0 h q and s 0 q h d 2 1 where q s 2

the range of stable random walks limit theorems are proved for the range of d dimensional random walks in the domain of attraction of a stable process of index p3 the range rn is the number of distinct sites of zd visited by the random walk before time n our results depend on the value of the ratio 3 d the most interesting results are obtained for 2 3 1 which can only occur if d 1 we prove the convergence in distribution of rn e rn toward some constant times the lebesgue measure of the range of the limiting stable process some of our results require regularity assumptions on the characteristic function of x

low regularity schrodinger maps ii global well posedness in dimensions d 3 in dimensions d 3 we prove that the schrodinger map initial value problem left begin array l partial ts s times delta x s hbox on mathbb r d times mathbb r s 0 s 0 end array right is globally well posed for small data s0 in the critical besov spaces dot b q d 2 mathbb r d mathbb s 2 q in mathbb s 2

a subspace algorithm for balanced state space system identification in an earlier paper by moonen et at 1989 an algorithm has been introduced for identifying multivariable linear systems directly from input output data this new algorithm falls in the class of so called subspace methods and resembles impulse response based realization algorithms in this paper we extend this algorithm by incorporating a balancing step such that the identified model is always in balanced coordinates with this modification one obtains a data driven counterpart for kung s realization algorithm

subspace model identification part 2 analysis of the elementary output error state space model identification algorithm the elementary moesp algorithm presented in the first part of this series of papers is analysed in this paper this is done in three different ways first we study the asymptotic properties of the estimated state space model when only considering zero mean white noise perturbations on the output sequence it is shown that in this case the moespl implementation yields asymptotically unbiased estimates an important constraint to this result is that the underlying system must have a finite impulse response and subsequently the size of the hankel matrices constructed from the input and output data at the beginning of the computations depends on the number of non zero markov parameters this analysis however leads to a second implementation of the elementary moesp scheme namely moesp2 the latter implementation has the same asymptotic properties without the finite impulse response constraint secondly we compare the moesp2 algorithm with a classical state space model identification scheme the latter

large data well posedness in the energy space of the chern simons schrodinger system abstract we consider the initial value problem for the chern simons schrodinger system which is a gauge covariant schrodinger system in r t r x 2 with a long range electromagnetic field we show that in the coulomb gauge it is locally well posed in h s for s 1 and the solution map satisfies a local in time weak lipschitz bound by energy conservation we also obtain a global regularity result the key is to retain the non perturbative part of the derivative nonlinearity in the principal operator and exploit the dispersive properties of the resulting paradifferential type principal operator using adapted u p and v p spaces

approximate realization based upon an alternative to the hankel matrix the page matrix abstract the ho kalman algorithm creates a minimum realization of a system when given a series of deterministic markov parameters however when such a truncated series of markov parameters has been disturbed with noise an approximating hankel matrix has to be constructed for applying the realization algorithm this approximating hankel matrix has either the improper rank or it lacks the hankel structure furthermore the markov parameters are not processed with a constant weighting factor which implies that the noise filtering is inadequate in this paper we propose to use an alternative matrix the page matrix it is shown that this method is better suited for handling the noisy markov parameters this holds with respect to three aspects order testing noise filtering and realization even in the deterministic case the page matrix offers the advantage of a considerable reduction in computation

critical large deviations in harmonic crystals with long range interactions we continue our study of large deviations of the empirical measures of a massless gaussian field on z d whose covariance is given by the green function of a long range random walk in this paper we extend techniques and results of bolthausen and deuschel to the nonlocal case of a random walk in the domain of attraction of the symmetric stable law with a 0 2 a d in particular we show that critical large deviations occur at the capacity scale n d with a rate function given by the dirichlet form of the embedded stable process we also prove that if we impose zero boundary conditions the rate function is given by the dirichlet form of the killed stable process

a technique for the identification of linear periodic state space models a non stochastic state space identification algorithm for linear time invariant systems is modified for use with periodic linear systems like its predecessor the modified algorithm forms block hankel matrices from input output data and uses the singular value decomposition of these hankel matrices to compute state vector sequences the state vector sequences are then used to compute system matrices associated with the periodic linear system by solving an overdetermined linear system the modifications to the original algorithm are as follows first the periodic system of period p is viewed as p separate time invariant period mapped systems this technique allows the structure of a periodic hankel matrix to be deduced which in turn allows a state vector sub sequence for the periodic system to be computed when a complete state vector sequence is computed it is used directly to construct the periodic state space models second similarities in the structure of each of the periodic hankel matrices ass

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

subspace algorithms for the identification of multivarible dynamic errors in variables models abstract we consider the problem of identifying multivariable finite dimensional linear time invariant systems from noisy input output measurements apart from the fact that both the measured input and output are corrupted by additive white noise the output may also be contaminated by a term which is caused by a white input process noise furthermore all these noise processes are allowed to be correlated with each other we shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open or closed loop two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms 1997 elsevier science ltd

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

long range scattering for the modified schrodinger map in two space dimensions we study the asymptotic behaviour in time of solutions and the theory of scattering for the modified schrodinger map in two space dimensions we solve the cauchy problem with large finite initial time up to infinity in time and we determine the asymptotic behaviour in time of the solutions thereby obtained as a by product we obtain global existence for small data in hk fhk with k 1 we also solve the cauchy problem with infinite initial time namely we construct solutions defined in a neighborhood of infinity in time with prescribed asymptotic behaviour of the previous type

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

on schrodinger maps we study the question of well posedness of the cauchy problem for schrodinger maps from 2 to the sphere 2 or to 2 the hyperbolic space the idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear schrodinger system of equations and then study this modified schrodinger map system msm we then prove local well posedness of the cauchy problem for the msm with minimal regularity assumptions on the data and outline a method to derive well posedness of the schrodinger map itself from it in proving well posedness of the msm the heart of the matter is resolved by considering truly quatrilinear forms of weighted l2 functions 2002 wiley periodicals inc

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

on the absence of spontaneous symmetry breaking and of crystalline ordering in two dimensional systems we develop a unified approach based on araki s relative entropy concept to proving absence of spontaneous breaking of continuous internal symmetries and translation invariance in two dimensional statistical mechanical systems more precisely we show that under rather general assumptions on the interactions all equilibrium states of a two dimensional system have all the symmetries compact internal and spatial of the dynamics except possibly rotation invariance rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance we also prove that two dimensional systems with a non compact internal symmetry group like anharmonic crystals typically do not have gibbs states

on and off line identification of linear state space models abstract a geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time invariant systems using possibly noisy input output i o measurements only as opposed to other mostly stochastic identification schemes no variance covariance information whatever is involved and only a limited number of i o data are required for the determination of the system matrices hence the algorithm can be best described and understood in the matrix formalism and consists of the following two steps first a state vector sequence is realized as the intersection of the row spaces of two block hankel matrices constructed with i o data then the system matrices are obtained at once from the least squares solution of a set of linear equations when dealing with noisy data this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition also the algorithm is easily applied to slowly time

state space identification of linear deterministic rainfall runoff models rainfall runoff models of the black box type abound in the water resources literature i e transfer function autoregressive moving average arma armax state space etc the corresponding system identification algorithms for such models are known to be numerically efficient and accurate leading in most cases to good parsimonious representations of the rainfall runoff process alternatively every model in transfer function arma and armax form has an equivalent state space representation however state space models do not necessarily have simple system identification algorithms unless the system matrices are restricted to some canonical form furthermore state space system identification algorithms that work with the rainfall runoff data directly i e covariance free require initial conditions and are inherently iterative and nonlinear in this paper we present a state space system identification theory which overcomes these limitations one advantage of such a theory is that the corresponding algorithms are highly robust to additive noise in the data they are referred to as subspace algorithms due to their ability to separate the signal subspace from the noise subspace the main advantages of the subspace algorithms are the automatic structure identification system order geometrical insights notions of angle between subspaces and the fact that they rely on robust numerical procedures singular value decomposition in this paper two algorithms are presented the first one is a two step procedure where the impulse response unit hydro graph ordinates for the single input single output case are computed from the input output data by solving a constrained deconvolution problem these impulse response ordinates are then used as inputs for identifying the system matrices by means of a hankel based realization algorithm the second approach uses the data directly to identify the system matrices bypassing the deconvolution step the algorithms are tested with real data from the voer catchment in belgium

large deviations and maximum entropy principle for interacting random fields on mathbb z d we present a new approach to the principle of large deviations for the empirical field of a gibbsian random field on the integer lattice zd this approach has two main features first we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities and thus obtain estimates which are finer and more widely applicable second we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field the limits being those predicted by the maximum entropy principle this result implies a general version of the equivalence of gibbs ensembles stating that every microcanonical limiting state is a grand canonical equilibrium state we also prove a converse to the last statement and discuss some applications 0 introduction as is well known the study of the asymptotic probabilities of large fluctuations of time averages or space averages away from the mean is based on two fundamental principles the principle of large deviations and the maximum entropy principle the former provides the exact rate of exponential decay of the fluctuation probabilities whereas the latter predicts the limiting conditional distribution under the condition that the fluctuations are large it is obvious that these principles are intimately related in this paper we investigate these principles in the case of interacting random fields on the integer lattice 7zd the setup is the following first we let e 4 be any measurable space we shall assume throughout that e a is standard borel but we shall avoid making any explicit topological assumptions on e so we do not assume that e is polish next we let s zd be the d dimensional integer lattice and q f e gas the associated product space in lattice models of statistical mechanics e is called the state space or single spin space and q f the configuration space we let j s f denote the set of all probability

entropic repulsion for the high dimensional gaussian lattice field between two walls the main aim of this paper is to discuss the entropic repulsion of random interfaces between two hard walls we consider the d 3 dimensional gaussian lattice field on n n n n d d and identify the repulsion of the field as n under the condition that the field lies between two hard walls at the height level 0 and l in n where l is large enough but finite we also study the same problem for two layered interfaces case

entropic repulsion and the maximum of the two dimensional harmonic crystal we consider the lattice version of the free field in two dimensions also called harmonic crystal the main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall the basic ingredient of the proof is the analysis of the maximum of the field which requires a multiscale analysis reducing the problem essentially to a problem on a field with a tree structure

subspace based identification algorithm for characterizing causal networks in resting brain the resting brain has been extensively investigated for low frequency synchrony between brain regions namely functional connectivity fc however the other main stream of brain connectivity analysis that seeks causal interactions between brain regions effective connectivity ec has been little explored inherent complexity of brain activities in resting state as observed in bold blood oxygenation level dependant fluctuations calls for exploratory methods for characterizing these causal networks on the other hand the inevitable effects that hemodynamic system imposes on causal inferences in fmri data lead us toward the methods in which causal inferences can take place in latent neuronal level rather than observed bold time series to simultaneously satisfy these two concerns in this paper we introduce a novel state space system identification approach for studying causal interactions among brain regions in the absence of explicit cognitive task this algorithm is a geometrically inspired method for identification of stochastic systems purely based on output observations using extensive simulations three aspects of our proposed method are investigated ability in discriminating existent interactions from non existent ones the effect of observation noise and downsampling on algorithm performance our simulations demonstrate that subspace based identification algorithm sia is sufficiently robust against above mentioned factors and can reliably uncover the underlying causal interactions of resting state fmri furthermore in contrast to previously established state space approaches in effective connectivity studies this method is able to characterize causal networks with large number of brain regions in addition we utilized the proposed algorithm for identification of causal relationships underlying anti correlation of default mode and dorsal attention networks during the rest using fmri we observed that default mode network places in a higher order in hierarchical structure of brain functional networks compared to dorsal attention networks

finite time blowup of multidimensional inhomogeneous isotropic landaulifshitz equation on a hyperbolic space blowup solutions for the multidimensional isotropic inhomogeneous landau lifshitz ill equation on a hyperbolic n space h2 are obtained if the inhomogeneous terms are carefully selected the ill will guarantee three types of singularity solutions type i and type ii blowup solutions of any dimensional ill on h2 can develop a finite time singularity from smooth initial data with finite l2 energy we prove the decay rate of energy density of the type iii implicit blowup solution if and c are non zero constants this solution develops a singularity in finite time with a local r c finite initial energy

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

qsvd approach to on and off line state space identification moonen et al 1989 a presented an svd based identification scheme for computing state space models for multivariable linear time invariant systems in the present paper this identification procedure is reformulated making use of the quotient singular value decomposition qsvd here the input output error co variance matrix can be taken into account explicitly thus extending the applicability of the identification scheme to the case were the input and output data are corrupted by coloured noise it turns out that in practice due to the use of various pre filtering techniques anti aliasing etc this latter case is most often encountered the extended identification scheme explicitly compensates for the filter characteristics and the consistency of the identification results follows from the consistency results for the qsvd the usefulness of this generalization is demonstrated the development is largely inspired by recent progress in total least squares solution techniques van huffel 1989 for

self similar solutions for the schrodinger map equation we study in this article the equivariant schrodinger map equation in dimension 2 from the euclidean plane to the sphere a family of self similar solutions is constructed this provides an example of regularity breakdown for the schrodinger map these solutions do not have finite energy and hence do not fit into the usual framework for solutions for data of infinite energy but small in some norm we build up associated global solutions

entropic repulsion of the lattice free field consider the massless free field on thed dimensional lattice d d 3 that is the centered gaussian field on open image in new window with covariances given by the green function of the simple random walk on d we show that the probability that all the spins are positive in a box of volumen d decays exponentially at a rate of ordernd 2 logn and compute explicitly the corresponding constant in terms of the capacity of the unit cube the result is extended to a class of transient random walks with transition functions in the domain of the normal and stable law

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

large deviations for stationary gaussian processes in their previous work on large deviations the authors always assumed the base process to be markovian whereas here they consider the base process to be stationary gaussian similar large deviation results are obtained under natural hypotheses on the spectral density function of the base process a rather explicit formula for the entropy involved is also obtained

almost sure local well posedness and scattering for the 4d cubic nonlinear schrodinger equation abstract we consider the cauchy problem for the defocusing cubic nonlinear schrodinger equation in four space dimensions and establish almost sure local well posedness and conditional almost sure scattering for random initial data in h x s r 4 with 1 3 s 1 the main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear schrodinger equation which is inspired by certain function spaces used in the study of the schrodinger maps problem and is based on strichartz spaces as well as variants of local smoothing inhomogeneous local smoothing and maximal function spaces additionally we prove an almost sure scattering result for randomized radially symmetric initial data in h x s r 4 with 1 2 s 1

suppression of bursting abstract we investigate the possibility of using a single small amplitude control input and feedback to stabilize equilibrium sets in a class of highly nonlinear o 2 symmetric dynamical systems possessing structurally stable heteroclinic cycles the leading order behavior near the equilibria is bilinear and homogeneous in the state variables while nonlinearities representing the symmetry breaking effect of the controller are crucial after a series of simplifying transformations we use ideas from optimal control theory to construct a stabilizing controller this study is motivated by the desire to stabilize the burst sweep cycle in low dimensional models of a turbulent boundary layer in the last two sections we apply the techniques to the 10 dimensional system of aubry et al 1988 kw bilinear systems dynamical systems heteroclinic cycles normal forms optimal control symmetry

subspace based methods for the identification of linear time invariant systems subspace based methods for system identification have attracted much attention during the past few years this interest is due to the ability of providing accurate state space models for multivariable linear systems directly from input output data the methods have their origin in classical state space realization theory as developed in the 1960s the main computational tools are the qr and the singular value decompositions here an overview of existing subspace based techniques for system identification is given the methods are grouped into the classes of realization based and direct techniques similarities between different algorithms are pointed out and their applicability is commented upon we also discuss some recent ideas for improving and extending the methods a simulation example is included for comparing different algorithms the subspace based approach is found to perform competitive with respect to prediction error methods provided the system is properly excited

global regularity of wave maps ii small energy in two dimensions we show that wave maps from minkowski space 1 n to a sphere sm 1 are globally smooth if the initial data is smooth and has small norm in the critical sobolev space in all dimensions n 5 this generalizes the results in the prequel 40 of this paper which addressed the high dimensional case n 5 in particular in two dimensions we have global regularity whenever the energy is small and global regularity for large data is thus reduced to demonstrating non concentration of energy

subspace model identification part 1 the output error state space model identification class of algorithms in this paper we present two novel algorithms to realize a finite dimensional linear time invariant state space model from input output data the algorithms have a number of common features they are classified as one of the subspace model identification schemes in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input output data this structure is then exploited in the calculation of a realization another common feature is their algorithmic organization an rq factorization followed by a singular value decomposition and the solution of an overdetermined set or sets of equations the schemes assume that the underlying system has an output error structure and that a measurable input sequence is available the latter characteristic indicates that both schemes are versions of the mimo output error state space model identification moesp approach the first algorithm is denoted in particular as the elementary moesp scheme

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc

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

entropic repulsion for the gaussian free field conditioned on disconnection by level sets we investigate level set percolation of the discrete gaussian free field on mathbb z d d ge 3 in the strongly percolative regime we consider the event that the level set of the gaussian free field below a level alpha disconnects the discrete blow up of a compact set a subseteq mathbb r d from the boundary of an enclosing box we derive asymptotic large deviation upper bounds on the probability that the local averages of the gaussian free field deviate from a specific multiple of the harmonic potential of a when disconnection occurs these bounds combined with the findings of the recent work by duminil copin goswami rodriguez and severo show that conditionally on disconnection the gaussian free field experiences an entropic push down proportional to the harmonic potential of a in particular due to the slow decay of correlations the disconnection event affects the field on the whole lattice furthermore we provide a certain profile description for the field in the presence of disconnection we show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of a it locally retains the structure of a gaussian free field shifted by a constant value our proofs rely crucially on the solidification estimates developed in nitzschner and sznitman to appear in j eur math soc arxiv 1706 07229

equivariant schrodinger maps in two spatial dimensions we consider equivariant solutions for the schrodinger map problem from 2 1 to s2 with energy less than 4 and show that they are global in time and scatter

critical large deviations for gaussian fields in the phase transition regime i we investigate large deviations for the empirical distribution functional of a gaussian random field on mathbb r mathbb z d d geq 3 in the phase transition regime we first prove that the specific entropy governs an n d volume order large deviation principle outside the gibbsian class within the gibbsian class we derive an n d 2 capacity order large deviation principle with exact rate function and we apply this result to the asymptotics of microcanonical ensembles we also give a spins profile description of the field and show that smooth profiles obey n d 2 order large deviations whereas discontinuous profiles obey n d 1 surface order large deviations

intrinsic ultracontractivity and conditional gauge for symmetric stable processes abstract it is shown in this paper that the conditional gauge theorem holds for symmetric stable processes on bounded c 1 1 domains in r n where 0 n 2 two of the major tools used to prove this conditional gauge theorem are logarithmic sobolev inequality and intrinsic ultracontractivity

local well posedness for hyperbolic elliptic ishimori equation abstract in this paper we consider the hyperbolic elliptic ishimori initial value problem with the form t s s x s b x 1 s x 2 x 2 s x 1 on r 2 1 1 2 s s x 1 s x 2 s 0 s 0 where s t r 2 s 2 r 3 denotes the wedge product in r 3 x x 1 2 x 2 2 b r we prove that such system is locally well posed for small data s 0 h q 0 r 2 s 2 0 3 2 q s 2 the new ingredient is that we develop the methods of ionescu and kenig 2006 6 and 2007 7 to approach the problem in a perturbative way

entropic repulsion of the lattice free field ii the 0 boundary case this paper is a continuation of 5 we consider the euclidean massless free field on a boxvn of volumend with o boundary condition that is the centered gaussian field with covariances given by the green function of the simple random walk on d d 3 killed as it exitsvn we show that the probability that all the spins are positive in the boxvn decays exponentially at a surface ratend 1 this is in contrast with the ratend 2 logn for the infinite field of 5

realization of stochastic systems with exogenous inputs and subspace identification methods a stochastic realization theory for a discrete time stationary process with an exogenous input is developed by extending the classical cca technique some stochastic subspace identification methods are derived by adapting the realization procedure to finite input output data