order sorted model theory for temporal executable specifications abstract an order sorted temporal programming paradigm is presented it consists of a typed modular declarative language its associated order sorted temporal horn clause logic basis and a model theory generalizing order sorted algebras with predicates to temporal order sorted structures the essence of this generalization is in time dependent interpretation of predicates so that a temporal order sorted model amounts to a sequence of order sorted equational models with predicates one per each state the main advantage of the presented paradigm in comparison with paradigms based on horn clause logic with equality is that it is more expressive particularly so in representing properly state transitions and other event oriented temporal behavioral properties of objects at the same time the generalized paradigm is proved to have the initial model semantics the rules for temporal order sorted deduction are established as an appropriate generalization of the rules for order sorted horn clause logic with equality the initial model is a quotient temporal order sorted structure constructed from the initial temporal order sorted structure and a congruence relation derived from a given set of temporal constraints temporal order sorted model theoretic properties are also naturally established for temporal queries the temporal constraint language has an execution model and it is intended to be a basis for a prototyping tool for complex typed modular software systems

modified schmidt games and diophantine approximation with weights abstract we show that the sets of weighted badly approximable vectors in r n are winning sets of certain games which are modifications of games introduced by w m schmidt in 1966 the latter winning property is stable with respect to countable intersections and is shown to imply full hausdorff dimension

random infinite volume gibbs states for the curie weiss random field ising model an approach to the definition of infinite volume gibbs states for the quenched random field ising model is considered in the case of a curie weiss ferromagnet it turns out that these states are random quasi free measures they are random convex linear combinations of the free product measures shifted by the corresponding effective mean fields the conditional self averaging property of the magnetization related to this randomness is also discussed

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

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

equidistribution of expanding translates of curves and dirichlet s theorem on diophantine approximation we show that for almost all points on any analytic curve on k which is not contained in a proper affine subspace the dirichlet s theorem on simultaneous approximation as well as its dual result for simultaneous approximation of linear forms cannot be improved the result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in k 1 the proof involves ratner s theorem on ergodic properties of unipotent flows on homogeneous spaces

mean field theory of random site q state potts models a class of random site mean field potts models is introduced and solved exactly the bifurcation properties of the resulting mean field equations are analysed in detail particular emphasis is put on the relation between the solutions and the underlying symmetries of the model it turns out that in contrast to the ising case the introduction of randomness in the mattis potts model can change the order of the transition for q or 6 the transition becomes second order

on the distribution of the effective field for a spin glass on a cayley tree at zero temperature an ansatz for the probability distribution of the effective field for a spin glass on a cayley tree at zero temperature is proposed the continuous part of the distribution is not differentiable at the origin in contrast to an ansatz by katsura and in qualitative agreement with numerical results of de oliveira possible implications are discussed

order sorted algebra i equational deduction for multiple inheritance overloading exceptions and partial operations abstract this paper generalizes many sorted algebra msa to order sorted algebra osa by allowing a partial ordering relation on the set of sorts this supports abstract data types with multiple inheritance in roughly the sense of object oriented programming several forms of polymorphism and overloading partial operations as total on equationally defined subsorts exception handling and an operational semantics based on term rewriting we give the basic algebraic constructions for osa including quotient image product and term algebra and we prove their basic properties including quotient homomorphism and initiality theorems the paper s major mathematical results include a notion of osa deduction a completeness theorem for it and an osa birkhoff variety theorem we also develop conditional osa including initiality completeness and mckinsey malcev quasivariety theorems and we reduce osa to conditional msa which allows lifting many known msa results to osa retracts which intuitively are left inverses to subsort inclusions provide relatively inexpensive run time error handling we show that it is safe to add retracts to any osa signature in the sense that it gives rise to a conservative extension a final section compares and contrasts many different approaches to osa this paper also includes several examples demonstrating the flexibility and applicability of osa including some standard benchmarks like stack and list as well as a much more substantial example the number hierarchy from the naturals up to the quaternions

rounding effects of quenched randomness on first order phase transitions frozen in disorder in an otherwise homogeneous system is modeled by interaction terms with random coefficients given by independent random variables with a translation invariant distribution for such systems it is proven that ind 2 dimensions there can be no first order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter discontinuities which would amount to a continuous symmetry breaking in systems which are stochastically invariant under the action of a continuous subgroup ofo n are suppressed by the randomness in dimensionsd 4 specific implications are found in the random field ising model for which we conclude that ind 2 dimensions at all h the gibbs state is unique for almost all field configurations and in the random bond potts model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point the results are explained by the argument of imry and ma 1 the proofs involve the analysis of fluctuations of free energy differences which are shown using martingale techniques to be gaussian on the suitable scale

local conjugacy classes for analytic torus flows abstract if a real analytic flow on the multidimensional torus close enough to linear has a unique rotation vector which satisfies an arithmetical condition y then it is analytically conjugate to linear we show this by proving that the orbit under renormalization of a constant y vector field attracts all nearby orbits with the same rotation vector

some remarks on p spin interaction models in a random field abstract we present some calculations for the thermodynamic behavior of mean field ferromagnetic p spin interaction models in the presence of quenched random fields for both ising and spherical spin variables we use the law of large numbers without recourse to the replica trick to obtain a free energy functional in terms of the order parameter and an extra non ordering density in the spherical limit we show that the transition is continuous for p 2 but turns into first order for p 3 regardless of the probability distribution of the random fields in the ising case for p 2 we recover previously known results the free energy functional obtained in this treatment can be used as a starting point for a dynamical study of these models

moderate deviations for the overlap parameter in the hopfield model we derive moderate deviation principles for the overlap parameter in the hopfield model of spin glasses and neural networks if the inverse temperature is different from the critical inverse temperature c 1 and the number of patterns m n satisfies m n n 0 the overlap parameter multiplied by n 1 2 1 obeys a moderate deviation principle with speed n1 2 and a quadratic rate function i e the gaussian limit for 1 2 remains visible on the moderate deviation scale at the critical temperature we need to multiply the overlap parameter by n 1 4 1 if then m n satisfies m n 6 log n m n 2n4 log n n 0 the rescaled overlap parameter obeys a moderate deviation principle with speed n1 4 and a rate function that is basically a fourth power the random term occurring in the central limit theorem for the overlap at c 1 is no longer present on a moderate deviation scale if the scaling is even closer to n1 4 e g if we multiply the overlap parameter by n1 4 log log n the moderate deviation principle breaks down the case of variable temperature converging to one is also considered if n converges to c fast enough i e faster than open image in new window the non gaussian rate function persists whereas for n converging to one slower than open image in new window the moderate deviations principle is given by the gaussian rate at the borderline the moderate deviation rate function is the one at criticality plus an additional gaussian term

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

spin glass order parameter on a bethe lattice an order parameter function for the spin glass state on a bethe lattice of coordination number z is calculated by the use of hierarchical boundary conditions the order parameter is discrete for finite z and becomes a continuous function in the limit z to infinity numerical calculations are performed at zero temperature and zero external field for several values of z

on the mean field ising model in a random external field we use a method developed by van hemmen to obtain the free energy of the mean field ising model in a random external magnetic field some results of previous mean field calculations are confirmed and generalized the tricritical point in the global phase diagram is discussed in detail we also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point

stochastic symmetry breaking in a gaussian hopfield model we study a two pattern hopfield model with gaussian disorder we find that there are infinitely many pure states at low temperatures in this model and that the metastate is supported on an infinity of symmetric pairs of pure states the origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the disorder variables

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

one dimensional random field ising model gibbs states and structure of ground states we consider the random gibbs field formalism for the ferromagnetic id dichotomous random field ising model as the simplest example of a quenched disordered system we prove that for nonzero temperatures the gibb state is unique for any realization of the external field then we prove that ast 0 the gibbs state converges to a limit a ground state for almost all realizations of the external field the ground state turns out to be a probability measure concentrated on an infinite set of configurations and we give a constructive description of this measure

2 dimensional badly approximable vectors and schmidt s game we prove that for any pair s t of nonnegative numbers with s t 1 the set of s t badly approximable vectors in r 2 is 34 p 2 1 winning for schmidt s game

bounded orbits of diagonalizable flows on finite volume quotients of products of sl2 r abstract we prove a number field analogue of a result of c mcmullen that the set of badly approximable numbers is absolute winning we also prove a weighted version we use this to prove an instance of a conjecture of an guan and kleinbock 4 namely let g sl 2 r sl 2 r and be a lattice in g we show that the set of points on g whose orbits under a one parameter ad semisimple subgroup of g are bounded form a hyperplane absolute winning set

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

low temperature dynamics of the curie weiss model periodic orbits multiple histories and loss of gibbsianness we consider the curie weiss model at initial temperature 0 1 in vanishing external field evolving under a glauber spin flip dynamics with temperature 0 1 we study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity bad points we provide a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending earlier work for the case of independent spin flip dynamics

approach to equilibrium for a class of random quantum models of infinite range we consider random generalizations of a quantum model of infinite range introduced by emch and radin the generalizations allow a neat extension from the class l 1 of absolutely summable lattice potentials to the optimal class l 2 of square summable potentials first considered by khanin and sinai and generalised by van enter and van hemmen the approach to equilibrium in the case of a gaussian distribution is proved to be faster than for a bernoulli distribution for both short range and long range lattice potentials while exponential decay to equilibrium is excluded in the nonrandom l 1 case it is proved to occur for both short and long range potentials for gaussian distributions and for potentials of class l 2 in the bernoulli case open problems are discussed

second order phase transitions and tricritical points in the random field ising models abstract a method that applies to a wide class of the random field ising models is used to investigate the influence of random field on the critical phenomena the 3d random field ising models with nearest neighbour interactions exhibit tricritical behavior the obtained results express rather large deviations from the mean field theory for particular cases the formulation gives some well known exact results

metastates in finite type mean field models visibility invisibility and random restoration of symmetry we consider a general class of disordered mean field models where both the spin variables and disorder variables take finitely many values to investigate the size dependence in the phase transition regime we construct the metastate describing the probabilities to find a large system close to a particular convex combination of the pure infinite volume states we show that under a non degeneracy assumption only pure states j are seen with non random probability weights w j for which we derive explicit expressions in terms of interactions and distributions of the disorder variables we provide a geometric construction distinguishing invisible states having w j 0 from visible ones as a further consequence we show that in the case where precisely two pure states are available these must necessarily occur with the same weight even if the model has no obvious symmetry relating the two

semantics of temporal classes a model theory of a typed declarative temporal object oriented language system is presented the declarative nature of the language makes it very different from the dominating procedural strongly typed object oriented programming languages in this declarative system methods are specified in a high level temporal constraint language two fundamental properties of these constraints are that they have an execution model and algebraic semantics the model theory is based on temporal order sorted algebras with predicates a variety of orderings are explored in order to represent various types of inheritance as well as the subtyping discipline temporal classes are viewed as temporal theories and some inheritance relationships as morphisms of temporal theories a model of a temporal class is a temporal order sorted structure with predicates which satisfies a set of temporal constraints specified in that class morphisms of those models are naturally required to preserve type coercions a distinguished model of a temporal theory is constructed as a colimit of a suitably defined functor this colimit construction reflects the temporal nature of the paradigm and generalizes the classical initial algebra semantics in contradistinction to major difficulties in developing a model theory for full fledged typed procedural object oriented languages this paper shows that such a task becomes possible for a suitably defined declarative object oriented language this in particular leads to model theoretic results on the preservation of the behavioral properties in the inheritance hierarchies

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

badly approximable systems of affine forms fractals and schmidt games this is the final version of the article available from de gruyter via the doi in this record

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

hausdorff dimension of singular vectors we prove that the set of singular vectors in rd d 2 has hausdorff dimension d2d 1 and that the hausdorff dimension of the set of e dirichlet improvable vectors in rd is roughly d2d 1 plus a power of e between d2 and d as a corollary the set of divergent trajectories of the flow by diag et et e dt acting on sld 1 r sld 1 z has hausdorff codimension dd 1 these results extend the work of the first author

on mean field equations for spin glasses in this paper we study rigorously the random ising model on a cayley tree in the limit of infinite coordination numberz 8 an iterative scheme is developed relating mean magnetizations and mean square magnetizations of successive shells far removed from the surface of the lattice in this way we obtain local properties of the model in the thermodynamic limit of an infinite number of shells when the coupling constants are independent gaussian random variables the sk expressions emerge as stable fixed points of our scheme and provide a valid local mean field theory of spin glasses in which negative local entropy at low temperatures while perfectly possible mathematically may still perhaps be physically undesirable finally we examine the tap equations and show that if the average over bond disorder and the limitz 8 are actually performed one recovers our iterative scheme and hence the sk equations also in the thermodynamic limit

nearest neighbor distances on a circle multidimensional case we study the distances called spacings between pairs of neighboring energy levels for the quantum harmonic oscillator specifically we consider all energy levels falling between e and e 1 and study how the spacings between these levels change for various choices of e particularly when e goes to infinity primarily we study the case in which the spring constant is a badly approximable vector we first give the proof by boshernitzan dyson that the number of distinct spacings has a uniform bound independent of e then if the spring constant has components forming a basis of an algebraic number field we show that when normalized up to a unit the spacings are from a finite set moreover in the specific case that the field has one fundamental unit the probability distribution of these spacings behaves quasiperiodically in loge we conclude by studying the spacings in the case that the spring constant is not badly approximable providing examples for which the number of distinct spacings is unbounded

moderate deviations for random field curie weiss models the random field curie weiss model is derived from the classical curie weiss model by replacing the deterministic global magnetic field by random local magnetic fields this opens up a new and interestingly rich phase structure in this setting we derive moderate deviations principles for the random total magnetization sn which is the partial sum of dependent spins a typical result is that under appropriate assumptions on the distribution of the local external fields there exist a real number m a positive real number and a positive integer k such that sn nm n satisfies a moderate deviations principle with speed n1 2k 1 and rate function x2k 2k where 1 1 2 2k 1 1

the fluctuations of the overlap in the hopfield model with finitely many patterns at the critical temperature we investigate the limiting fluctuations of the order parameter in the hopfield model of spin glasses and neural networks with finitely many patterns at the critical temperature 1 c 1 at the critical temperature the measure valued random variables given by the distribution of the appropriately scaled order parameter under the gibbs measure converge weakly towards a random measure which is non gaussian in the sense that it is not given by a dirac measure concentrated in a gaussian distribution this remains true in the case of n c 1 as n provided n converges to c 1 fast enough i e at speed 1 the limiting distribution is explicitly given by its random density

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

local properties of an ising model on a cayley tree the ising model on a cayley tree displays a peculiar continuous order phase transition with zero long range order at all finite temperatures when one studies expection values of spins far removed from the surface which contains a finite fraction of the total number of spins in the thermodynamic limit however one obtains the so called bethe approximation here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface in the ferromagnetic case the local magnetization is a fixed point of the iterative transformation while in the antiferromagnetic case the fixed point bifurcates to a two cycle of the transformation for low temperatures and fields giving rise to local sublattice magnetizations in both cases local thermodynamical properties are obtained by integration

closed orbits for actions of maximal tori on homogeneous spaces let g be a real algebraic group defined over mathbb q let gamma be an arithmetic subgroup and let t be any torus containing a maximal mathbb r split torus we prove that the closed orbits for the action of t on g gamma admit a simple algebraic description in particular we show that if g is reductive an orbit tx gamma is closed if and only if x 1 tx is a product of a compact torus and a torus defined over mathbb q and it is divergent if and only if the maximal mathbb r split subtorus of x 1 tx is defined over mathbb q and mathbb q split our analysis also yields the following there is a compact k subset g gamma which intersects every t orbit if rank mathbb q g rank mathbb r g there are no divergent orbits for t

fluctuations in dilute antiferromagnets curie weiss models the authors compute the fluctuations of the order parameter in the curie weiss version of a site dilute antiferromagnet their results show i gaussian fluctuations away from criticality or at a first order critical point with sample and thermal fluctuations contributing in same order ii non gaussian fluctuations with critical exponents modified by the presence of dilution at the second order critical point in this case sample induced fluctuations are enhanced to dominate over the thermal ones critical exponents are the same as in the curie weiss random field ising model

dimension estimates for the set of points with non dense orbit in homogeneous spaces let x g gamma where g is a lie group and gamma is a lattice in g and let u be a subset of x whose complement is compact we use the exponential mixing results for diagonalizable flows on x to give upper estimates for the hausdorff dimension of the set of points whose trajectories miss u this extends a recent result of kadyrov et al dyn syst 30 2 149 157 2015 and produces new applications to diophantine approximation such as an upper bound for the hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms generalizing an estimate due to broderick and kleinbock int j number theory 11 7 2037 2054 2015

badly approximable points on manifolds this paper is motivated by two problems in the theory of diophantine approximation namely davenport s problem regarding badly approximable points on submanifolds of a euclidean space and schmidt s problem regarding the intersections of the sets of weighted badly approximable points the problems have been recently settled in dimension two but remain open in higher dimensions in this paper we develop new techniques that allow us to tackle them in full generality the techniques rest on lattice points counting and a powerful quantitative result of bernik kleinbock and margulis the main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of mathbb r n has full dimension one of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any bounded degree

the retrieval phase of the hopfield model a rigorous analysis of the overlap distribution standard large deviation estimates or the use of the hubbard stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function n on m in this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns we denote by m the modulus of the spontaneous magnetization in the curie weiss model and by the ratio between the number of the stored patterns and the system size we show that there exist strictly positive numbers 0 a c such that 1 if a m 2 then the absolute minima of are located within small balls around the points m e where e denotes the th unit vector while 2 if c m 2 at least a local minimum surrounded by extensive energy barriers exists near these points the random location of these minima is given within precise bounds these are used to prove sharp estimates on the support of the gibbs measures

inclusion of short range order in a mean field theory for the disordered magnetic lattice gas the disordered magnetic lattice gas dmlg as a unifying description of many simpler random spin systems has been investigated in an attempt to devise a mean field theory which goes beyond the infinitely long ranged model by incorporating short range order sro the authors have shown rigorously that the local thermodynamic properties of the dmlg on a cayley tree of finite coordination number z are identical to the thermodynamic properties of the dmlg in a pair approximation obtained by using the method of the distribution function further a modified pair approximation for the dmlg is presented which is exactly solvable it is formulated for general random bond distribution functions and is then examined for the special case of gaussian distributions

multidimensional continued fractions dynamical renormalization and kam theory the disadvantage of traditional multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors following ideas of dani lagarias and kleinbock margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space sl d mathbb z backslash sl d mathbb r the space of lattices of covolume one that indeed yields best possible approximations to any irrational vector the algorithm is ideally suited for a number of dynamical applications that involve small divisor problems as an example we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov

hausdorff dimension of the set of singular pairs in this paper we show that the hausdorff dimension of the set of singular pairs is 4 3 we also show that the action of diag e t e t e 2t on sl 3 r sl 3 z admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates answering a question of a n starkov as a by product of the analysis we obtain a higher dimensional generalization of the basic inequalities satisfied by convergents of continued fractions as an illustration of the technique used to compute hausdorff dimension we reprove a result of i j good asserting that the hausdorff dimension of the set of real numbers with divergent partial quotients is 1 2

ising model with competing interactions on a cayley tree an iterative scheme is developed for a renormalized effective nearest neighbor couplingk r and effective field per sitek r for spins in therth shell of a cayley tree with nearest neighborj and next nearest neighborj interactions between ising spins on the lattice in addition to the expected paramagnetic ferromagnetic and antiferromagnetic phases we find an intermediate range ofj j 0 values wherex r and kr iterate to a continuous or quasicontinuous attractor in thex k plane in this range the local magnetization is mainly chaotic with oscillatory glasslike behavior embedded in the chaos however are regions of periodic and commensurate phases

strong approximation for set indexed partial sum processes via kmt constructions ii let x i i in mathbb z d be an array of zero mean independent identically distributed random vectors with values in mathbb r k with finite variance and let mathscr l be a class of borel subsets of lbrack 0 1 rbrack d if for the usual metric mathscr l is totally bounded and has a convergent entropy integral we obtain a strong invariance principle for an appropriately smoothed version of the partial sum process sum i in nu s x i s in mathscr l with an error term depending only on mathscr l and on the tail distribution of x 1 in particular when mathscr l is the class of subsets of lbrack 0 1 rbrack d with alpha differentiable boundaries introduced by dudley we prove that our result is optimal

equivalence of ferromagnetic spin models on trees and random graphs recently we have found identical behaviour for various spin models on thin random graphs feynman diagrams and the corresponding bethe lattices in this paper we observe that the ratios of the saddle point equations in the random graph approach are identical to the fixed point s of the recursion relations which are used to solve the models on the bethe lattice the loops in the random graphs thus have no influence on the thermodynamic limit for such ferromagnetic spin models we consider the correspondence explicitly for ising and q state potts models and also note that multispin interaction models on cacti admit a similar correspondence with a randomized version of the cacti graphs which contain loops

the random bond ising model on the bethe lattice the paper studies the random bond ising model of a spin glass of a tree of coordination number q 3 the model is studied as an example of the method of ring recurrence a set of recursion relations between the moments of the probability distribution of the order parameter are produced it is shown that solutions of the recursion relations exist which form an infinite bifurcating set below a critical temperature tau c the nature of the first such solution is investigated in detail and the critical exponent analogous to beta is found

an extension of quantitative nondivergence and applications to diophantine exponents we present a sharpening of nondivergence estimates for unipotent or more gener ally polynomial like flows on homogeneous spaces applied to metric diophantine approxima tion it yields precise formulas for diophantine exponents of affine subspaces ofr n and their nondegenerate submanifolds

random walks on homogeneous spaces and diophantine approximation on fractals we extend results of y benoist and j f quint concerning random walks on homogeneous spaces of simple lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily zariski dense but satisfies some expansion properties for the adjoint action using these dynamical results we study diophantine properties of typical points on some self similar fractals in mathbb r d as examples we show that for any self similar fractal mathcal k subseteq mathbb r d satisfying the open set condition for instance any translate or dilate of cantor s middle thirds set or of a koch snowflake almost every point with respect to the natural measure on mathcal k is not badly approximable furthermore almost every point on the fractal is of generic type which means in the one dimensional case that its continued fraction expansion contains all finite words with the frequencies predicted by the gauss measure we prove analogous results for matrix approximation and for the case of fractals defined by mobius transformations

spin flip dynamics of the curie weiss model loss of gibbsianness with possibly broken symmetry we study the conditional probabilities of the curie weiss ising model in vanishing external field under a symmetric independent stochastic spin flip dynamics and discuss their set of points of discontinuity bad points we exhibit a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending the results for the corresponding lattice model where only partial answers can be obtained for initial temperature beta 1 geq 1 we prove that the time evolved measure is always gibbsian for frac 2 3 leq beta 1 1 the time evolved measure loses its gibbsian character at a sharp transition time for beta 1 frac 2 3 we observe the new phenomenon of symmetry breaking in the set of points of discontinuity bad points corresponding to non zero spin average appear at a sharp transition time and give rise to biased non gibbsianness of the time evolved measure these bad points become neutral at a later transition time while the measure stays non gibbs in our proof we give a detailed description of the phase diagram of a curie weiss random field ising model with possibly non symmetric random field distribution based on bifurcation analysis

modulated structures of an ising model with competing nearest neighbour interactions the authors use a two dimensional mapping to analyse an ising model with competing ferromagnetic and antiferromagnetic nearest neighbour interactions on a cayley tree the phase diagram which had some resemblance to the case of an ising spin glass displays some tricritical points and many modulated phases

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

the weak limit of ising models on locally tree like graphs we consider the ising model with inverse temperature and without external field on sequences of graphs g n which converge locally to the k regular tree we show that for such graphs the ising measure locally weakly converges to the symmetric mixture of the ising model with boundary conditions and the boundary conditions on the k regular tree with inverse temperature in the case where the graphs g n are expanders we derive a more detailed understanding by showing convergence of the ising measure conditional on positive magnetization sum of spins to the measure on the tree

flows on homogeneous spaces and diophantine properties of matrices we generalize the notions of badly approximable resp singular systems of m linear forms in n variables and relate these generalizations to certain bounded resp divergent trajectories in the space of lattices in rm n

expanding translates of curves and dirichlet minkowski theorem on linear forms we show that a multiplicative form of dirichlet s theorem on simulta neous diophantine approximation as formulated by minkowski cannot be improved for almost all points on any analytic curve on r k which is not contained in a proper affine subspace such an investigation was initiated by davenport and schmidt in the late sixties the diophantine problem is then settled via showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in r k 1 gets equidistributed in the limit we use ratner s theorem on unipotent flows linearization techniques and a new observation about intertwined linear dynamics of various sl m r s contained in sl k 1 r

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

on the ising model with random boundary condition the infinite volume limit behavior of the 2d ising model under possibly strong random boundary conditions is studied the model exhibits chaotic size dependence at low temperatures and we prove that the and phases are the only almost sure limit gibbs measures assuming that the limit is taken along a sparse enough sequence of squares in particular we provide an argument to show that in a sufficiently large volume a typical spin configuration under a typical boundary condition contains no interfaces in order to exclude mixtures as possible limit points a detailed multi scale contour analysis is performed

moderate deviations for longest increasing subsequences the upper tail we derive the upper tail moderate deviations for the length of a longest increasing subsequence in a random permutation this concerns the regime between the upper tail large deviation regime and the central limit regime our proof uses a formula to describe the relevant probabilities in terms of the solution of the rank 2 riemann hilbert problem rhp this formula was invented by baik deift and johansson 3 to find the central limit asymptotics of the same quantities in contrast to the work of these authors who apply a third order nonstandard steepest descent approximation at an inflection point of the transition matrix elements of the rhp our approach is based on a more classical second order gaussian saddle point approximation at the stationary points of the transition function matrix elements 2001 john wiley sons inc