constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

alternating multihead finite automata abstract we define alternating multihead finite automata a generalization of nondeterministic multihead finite automata based on the alternating turing machine model introduced by chandra kozen and stockmeyer 1981 we study the relationships between the classes of languages accepted by alternating multihead finite automata and the classes accepted by deterministic and nondeterministic multihead finite automata and pushdown automata we also examine basic questions about alternating multihead finite automata for example are k 1 heads better than k we conclude by placing upper bounds on the deterministic time and space complexity of the classes of languages accepted by alternating multihead finite automata as corollaries to our results about alternating multihead finite automata we obtain several facts about multihead pushdown automata indicating that the study of alternating multihead finite automata may lead to useful results about nonalternating automata

p uniform circuit complexity much complexity theoretic work on parallelism has focused on the class nc which is defined in terms of logspace uniform circuits yet p uniform circuit complexity is in some ways a more natural setting for studying feasible parallelism in this paper p uniform nc punc is characterized in terms of space bounded auxpdas and alternating turing machines with bounded access to the input the notions of general purpose and special purpose computation are considered and a general purpose parallel computer for punc is presented it is also shown that nc punc if all tally languages in p are in nc this implies that the nc punc question and the nc p question are both instances of the aspace s n aspace time s n s n o 1 question as a corollary it follows that nc punc implies pspace dtime 2 no 1

the survival probability and r point functions in high dimensions in this paper we investigate the survival probability n in high dimensional statistical physical models where n denotes the probability that the model survives up to time n we prove that if the r point functions scale to those of the canonical measure of super brownian motion and if certain self repellence and total population tail bound conditions are satised then n n 2 av where a is the asymptotic expected number of particles alive at timen andv is the vertex factor of the model our results apply to spread out lattice trees above 8 dimensions spread out oriented percolation above 4 1 dimensions and the spread out contact process above 4 1 dimensions in the case of oriented percolation this reproves a result by the rst author den hollander and slade which was proved using heavy lace expansion arguments at the cost of losing explicit error estimates we further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n our proofs are based on simple weak convergence arguments

some classes of languages in nc 1 abstract we show the containment of several classes of languages in nc 1 these include the binary encodings of semilinear sets and some subclasses of context free languages we also investigate the closure properties of nc 1

the survival probability for critical spread out oriented percolation above 4 1 dimensions i induction we consider critical spread out oriented percolation above 4 1 dimensions our main result is that the extinction probability at time n i e the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n 1 decays like 1 bn 2 as n to infty where b is a finite positive constant this in turn implies that the survival probability at time n i e the probability that the origin is connected to the hyperplane at time n decays like 1 bn as n to infty the latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster the present paper is part i in a series of two papers in part ii we derive a lace expansion for the survival probability adapted so as to deal with point to plane connections this lace expansion leads to a nonlinear recursion relation for the survival probability in part i we use this recursion relation to deduce the asymptotics via induction

typical cluster size for two dimensional percolation processes the typical cluster size for two dimensional percolation models is discussed it is shown that forw 0 x z 2 0 x lim n 1 n logp p w 0 n 1 p p c asp p c provided thate p w 0 2 e p w 0 p p c asp p c furthermore we introduce a new quantityf s p which may be thought of as the singular part of the free energy and show thatf s p p p c 2v provided that the correlation length p p c v asp p c

mean field behavior for the survival probability and the percolation point to surface connectivity we consider the critical survival probability up to time t for oriented percolation and the contact process and the point to surface of the ball of radius t connectivity for critical percolation let t denote both quantities we prove in a unifled fashion that if t exhibits a power law and both the two point function and its certain restricted version exhibit the same mean fleld behavior then t t i1 for the time oriented models with d 4 and t t i2 for percolation with d 7

evidence of dispersion relations for the nonlinear response of the lorenz 63 system along the lines of the nonlinear response theory developed by ruelle in a previous paper we have proved under rather general conditions that kramers kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of axiom a non equilibrium steady state systems to weak monochromatic forcings we present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed lorenz 63 system by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions some new theoretical results showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables are also presented our findings confirm the conceptual validity of the nonlinear response theory suggest that the theory can be extended for more general non equilibrium steady state systems and shed new light on the applicability of very general tools based only upon the principle of causality for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals in situations where the fluctuation dissipation relation is not of great help

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

non gaussian limiting behavior of the percolation threshold in a large system we study short range percolation models in a finite box we define the percolation threshold as a random variable obtained from a stochastic procedure used in actual numerical calculations and study the asymptotic behavior of these random variables as the size of the box goes to infinity we formulate very general conditions under which in two dimensions rescaled threshold variables cannot converge to a gaussian and determine the asymptotic behavior of their second moments in terms of a widely used definition of correlation length we also prove that in all dimensions the finite volume percolation thresholds converge in probability to the percolation threshold of the infinite system the convergence result is obtained by estimating the rate of decay of the limiting distribution function s tail in terms of the correlation length exponent the proofs use exponential estimates of crossing probabilities substantial parts of the proofs apply in all dimensions

hyperscaling inequalities for the contact process and oriented percolation the contact process and oriented percolation are expected to exhibit the same critical behavior in any dimension above their upper critical dimension dc they exhibit the same critical behavior as the branching process assuming existence of the critical exponents we prove a pair of hyperscaling inequalities which together with the results of refs 16 and 18 implies dc 4

stochastic climate dynamics random attractors and time dependent invariant measures a b s t r a c t this article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s the nonlinear deterministic and the linear stochastic one this unification via the theory of random dynamical systems rds allows one to consider the detailed geometric structure of the random attractors associated with nonlinear stochastically perturbed systems we report on high resolution numerical studies of two idealized models of fundamental interest for climate dynamics the first of the two is a stochastically forced version of the classical lorenz model the second one is a low dimensional nonlinear stochastic model of the el nino southern oscillation enso these studies provide a good approximation of the two models global random attractors as well as of the time dependent invariant measures supported by these attractors the latter are shown to have an intuitive physical interpretation as random versions of sinai ruelle bowen srb measures

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

symmetry breaking mixing instability and low frequency variability in a minimal lorenz like system starting from the classical saltzman two dimensional convection equations we derive via a severe spectral truncation a minimal 10 ode system which includes the thermal effect of viscous dissipation neglecting this process leads to a dynamical system which includes a decoupled generalized lorenz system the consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables with the ensuing modifications to the structural properties of the attractor and of the spectral features when the relevant nondimensional number eckert number ec is different from zero an additional time scale of o text ec ensuremath 1 is introduced in the system as shown with standard multiscale analysis and made clear by several numerical evidences moreover the system is ergodic and hyperbolic the slow variables feature long term memory with 1 f 3 2 power spectra and the fast variables feature amplitude modulation increasing the strength of the thermal viscous feedback has a stabilizing effect as both the metric entropy and the kaplan yorke attractor dimension decrease monotonically with ec the analyzed system features very rich dynamics it overcomes some of the limitations of the lorenz system and might have prototypical value in relevant processes in complex systems dynamics such as the interaction between slow and fast variables the presence of long term memory and the associated extreme value statistics this analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic in fact this leads to spurious invariances that affect essential dynamical properties ergodicity hyperbolicity and that cause the model losing ability in describing intrinsically multiscale processes

the scaling limit of self avoiding random walk in high dimensions the brydges spencer lace expansion is used to prove that the scaling limit of the finite dimensional distributions of self avoiding random walk in the d dimensional cubic lattice mathbb z d is gaussian if d is sufficiently large it is also shown that the critical exponent gamma for the number of self avoiding walks is equal to 1 if d is sufficiently large

extensional uniformity for boolean circuits imposing an extensional uniformity condition on a nonuniform circuit complexity class mathcal c means simply intersecting mathcal c with a uniform class mathcal l by contrast the usual intensional uniformity conditions require that a resource bounded machine be able to exhibit the circuits in the circuit family defining mathcal c we say that mathcal c mathcal l has the uniformity duality property if the extensionally uniform class mathcal c cap mathcal l can be captured intensionally by means of adding so called mathcal l numerical predicates to the first order descriptive complexity apparatus describing the connection language of the circuit family defining mathcal c this paper exhibits positive instances and negative instances of the uniformity duality property

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

statistics of lattice animals the authors investigate the large n behaviour of the number of lattice animals with n vertices having alpha cycles per vertex they prove concavity and continuity properties for the corresponding growth constant and in particular show that lattice trees are exponentially scarce in the set of lattice animals they also consider the corresponding generating function and prove a number of theorems which bound it and set other limits on its possible behaviour

differentiation of srb states correction and complements taking into account criticism by d dolgopyat and m jiang we present here an improved derivation of the formula for the derivative of an srb measure with respect to parameters

more inequalities for critical exponents a variety of rigorous inequalities for critical exponents is proved most notable is the low temperature josephson inequalitydv 2 2 others are 1 1 v 1 1 1 d 1 1 for d dv 3 for d 4 and 2m 2m 2 form 2 the hypotheses vary all inequalities are true for the spin 1 2 ising model with nearest neighbor ferromagnetic pair interactions

bandwidth constraints on problems complete for polynomial time abstract a graph g v e has bandwidth k under a layout l v 1 1 1 v if for all x y e l x l y k bandwidth constraints on several problems that are complete for p under long space reductions are considered in particular the solvable path system problem and the and or graph accessibility problem under various bandwidth constraints are used to prove results about subclasses of p in general restricting the bandwidth of problems complete for p results in complete problems for subclasses of p defined by simultaneous time space bounds or defined by space bounds on alternating turing machines for instance these results are used to show that the class sc of sets accepted in polynomial time and simultaneous polylog space can be characterized as the class reducible by log space transformations to sets accepted by one way log log n space bounded alternating turing machines an upper bound on the space requirements for the solvable path system problem under various bandwidth constraints is given by sps t n d space f n log n this yields as a corollary the result aspace f n k lcub0rcub d space 2 k f n for functions f that are suitably constructible and do not grow more rapidly than some logarithm function this extends the known result aspace f n k lcub0rcub d time 2 k f n which only applies to functions that grow at least as rapidly as a logarithm function

correlation length and its critical exponent for percolation processes some critical exponent inequalities are given involving the correlation length of site percolation processes on d in particular it is shown thatv 2 d which implies that the critical exponentv cannot take its mean field value for the three dimensional percolation processes

response theory for equilibrium and non equilibrium statistical mechanics causality and generalized kramers kronig relations we consider the general response theory recently proposed by ruelle for describing the impact of small perturbations to the non equilibrium steady states resulting from axiom a dynamical systems we show that the causality of the response functions entails the possibility of writing a set of kramers kronig k k relations for the corresponding susceptibilities at all orders of nonlinearity nonetheless only a special class of directly observable susceptibilities obey k k relations specific results are provided for the case of arbitrary order harmonic response which allows for a very comprehensive k k analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short time response of the perturbed system these results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models and shed light on the very general impact of considering the principle of causality for testing self consistency the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey the theory exposed in the present paper is dual to the time dependent theory of perturbations to equilibrium states and to non equilibrium steady states and has in principle similar range of applicability and limitations in order to connect the equilibrium and the non equilibrium steady state case we show how to rewrite the classical response theory by kubo so that response functions formally identical to those proposed by ruelle apart from the measure involved in the phase space integration are obtained these results taking into account the chaotic hypothesis by gallavotti and cohen might be relevant in several fields including climate research in particular whereas the fluctuation dissipation theorem does not work for non equilibrium systems because of the non equivalence between internal and external fluctuations k k relations might be robust tools for the definition of a self consistent theory of climate change

infinite canonical super brownian motion and scaling limits we construct a measure valued markov process which we call infinite canonical super brownian motion and which corresponds to the canonical measure of super brownian motion conditioned on non extinction infinite canonical super brownian motion is a natural candidate for the scaling limit of various random branching objects on mathbb z d when these objects are critical mean field and infinite we prove that icsbm is the scaling limit of the spread out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension we conjecture that it also arises as the scaling limit in various other models above the upper critical dimension such as the incipient infinite lattice tree above 8 dimensions the incipient infinite cluster for unoriented percolation above 6 dimensions uniform spanning trees above 4 dimensions and invasion percolation above 6 dimensions

some general results concerning the critical exponents of percolation processes in this paper we will give some results concerning the critical exponents of percolation processes which are valid for any model these results show that in several respects the behavior which occurs for percolation on the binary tree provides bounds on one side for what happens in general these results and their proofs are closely related to their analogues for the ising model

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

critical behavior and the limit distribution for long range oriented percolation i we consider oriented percolation on mathbb z d times mathbb z whose bond occupation probability is pd where p is the percolation parameter and d is a probability distribution on mathbb z d suppose that d x decays as x d for some 0 we prove that the two point function obeys an infrared bound which implies that various critical exponents take on their respective mean field values above the upper critical dimension d c 2 alpha wedge2 we also show that for every k the fourier transform of the normalized two point function at time n with a proper spatial scaling has a convergent subsequence to e c k alpha wedge2 for some c 0

stochastic perturbations to dynamical systems a response theory approach using the formalism of the ruelle response theory we study how the invariant measure of an axiom a dynamical system changes as a result of adding noise and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics we first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system both in the additive and in the multiplicative case we also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing this provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings using kramers kronig theory it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the green function of the system for any desired observable we then extend our results to rather general patterns of random forcing from the case of several white noise forcings to noise terms with memory up to the case of a space time random field explicit formulas are provided for each relevant case analysed as a general result we find using an argument of positive definiteness that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system we provide an example of application of our results by considering the spatially extended chaotic lorenz 96 model these results clarify the property of stochastic stability of srb measures in axiom a flows provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies and introduce new ways to study the response of a system to external perturbations taking into account the chaotic hypothesis we expect that our results have practical relevance for a more general class of system than those belonging to axiom a

on differentiability of srb states for partially hyperbolic systems consider a one parameter family of diffeomorphisms f e such that f 0 is an anosov element in a standard abelian anosov action having sufficiently strong mixing properties let e be any u gibbs state for f e we prove theorem 1 that for any c function a the map e e a is differentiable at e 0 this implies corollary 2 2 that the difference of birkhoff averages of the perturbed and unperturbed systems is proportional to e we apply this result corollary 3 3 to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique srb measure with good statistical properties

gaussian limit for critical oriented percolation in high dimensions in this paper we consider the spread out oriented bond percolation models inz d z withd 4 and the nearest neighbor oriented bond percolation model in sufficiently high dimensions let n n 1 2 be the random measures defined onr d by eta n a sum limits x in z d 1 a x sqrt n 1 0 0 to x n the mean of n denoted by bar eta n is the measure defined by bar eta n a e p eta n a we use the lace expansion method to show that the sequence of probability measures bar eta n r d 1 bar eta n converges weakly to a gaussian limit asn for everyp in the subcritical regime as well as the critical regime of these percolation models also we show that for these models the parallel correlation length xi p p c p 1 asp pc

computation times of np sets of different densities abstract in this paper we study the computational complexity of sets of different densities in np we show that the deterministic computation time for sets in np can depend on their density if and only if there is a collapse or partial collapse of the corresponding higher nondeterministic and deterministic time bounded complexity classes we show also that for np sets of different densities there exist complete sets of the corresponding density under polynomial time turing reductions finally we show that these results can be interpreted as results about the complexity of theorem proving and proof presentation in axiomatized mathematical systems this interpretation relates fundamental questions about the complexity of our intellectual tools to basic structural problems about p np conp and pspace discussed in this paper

mean field critical behavior for the contact process the contact process is a model of spread of an infectious disease combining with the result of ref 1 we prove that the critical exponents take on the mean field values for sufficiently high dimensional nearest neighbor models and for sufficiently spread out models with d 4 c as c and c 1 as c where and are the spread probability and the susceptibility of the infection respectively and c is the critical infection rate our results imply that the upper critical dimension for the contact process is at most 4

multi level dynamical systems connecting the ruelle response theory and the mori zwanzig approach we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system which are weakly coupled to the remaining variables in a previous paper we have used the ruelle response theory on such a weakly coupled system to construct a surrogate dynamics such that the expectation value of any observable agrees up to second order in the coupling strength to its expectation evaluated on the full dynamics we show here that such surrogate dynamics agree up to second order to an expansion of the mori zwanzig projected dynamics this implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related if one takes into account in addition to the widely adopted stochastic forcing the often neglected memory effects

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

application of hyperbolic dynamics to physics some problems and conjectures the long time behavior of smooth dynamical systems is in good cases given by an srb measure the measure is expected often to have very discontinuous dependence on parameters of the dynamical system this is a very unsatisfactory situation for physical applications where one would like to differentiate with respect to parameters here we propose a solution to this difficulty based on analytic continuation in a frequency variable rene thom was a philosopher and among other things but most successfully a lover of that part of philosophy which we call mathematics i use here the term philosophy with its old meaning which some contemporary philosophers and mathematicians might not like inside and outside mathematics thom s interest was turned to forms and ideas his passion was to understand geometrically the nature of things and for this he used mathematical proofs but he was not a formalist and for him proofs remained secondary to the conceptual landscape that they revealed one area where he invested a lot of effort was the study of differentiable dynamical systems this smooth dynamics is where geometric forms come closest to life i attended the regular seminar that he ran on the subject under the influence of thom the ihes was an important center for the development of differentiable dynamical systems especially hyperbolic dynamics in the golden years of the 1960 s and 1970 s steve smale and his friends were frequent visitors and the russians came when they could the big question in smooth dynamics is what do general smooth dynamical systems look like what one wants is an understanding of a large set of dynamical systems this is also a significant question for physical applications more specifically one would like to understand the long time behavior for typical time evolutions and how this changes when the parameters controlling the time evolution are changed the uniformly hyperbolic systems studied by anosov smale thom and many others form a large set and smale had conjectured that they might be dense among all dynamical systems but this is not the case and it may be too much to hope for a very large set of dynamical systems that are understood in great detail like the uniformly hyperbolic systems from a practical point of view the numerical study of dynamical systems often reveals extremely long characteristic times which received by the editors october 1 2003 2000 mathematics subject classification primary 37c40 secondary 37d45

the triangle condition for contact processes on homogeneous trees we complete work of c c wu by showing that for contact processes on homogeneous trees with degree at least 3 the triangle condition is satisfied below the second critical point in particular it holds at the first critical point and therefore at this critical point the contact process has mean field critical exponents

towards a general theory of extremes for observables of chaotic dynamical systems in this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values we show that the extremes of so called physical observables are distributed according to the classical generalised pareto distribution and derive explicit expressions for the scaling and the shape parameter in particular we derive that the shape parameter does not depend on the chosen observables but only on the partial dimensions of the invariant measure on the stable unstable and neutral manifolds the shape parameter is negative and is close to zero when high dimensional systems are considered this result agrees with what was derived recently using the generalized extreme value approach combining the results obtained using such physical observables and the properties of the extremes of distance observables it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow moreover by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments of the partial dimensions and of the kaplan yorke dimension of the attractor preliminary numerical investigations provide encouraging results on the applicability of the theory presented here the results presented here do not apply for all combinations of axiom a systems and observables but the breakdown seems to be related to very special geometrical configurations

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

complexity theory of parallel time and hardware abstract the parallel resources time and hardware and the complexity classes defined by them are studied using the aggregate model the equivalence of complexity classes defined by sequential space and uniform aggregate hardware is established aggregate time is related to bounded fanin circuit depth and similarly aggregate hardware is related to circuit width interelationships between aggregate time and hardware follow as corollaries aggregate time is related to the sequential resource reversal simultaneous relationships from aggregate hardware and time to sequential space and reversal are shown and conversely and these are used as evidence for an extended parallel computation thesis these simultaneous relationships provide new characterizations for the simultaneous parallel complexity class nc and for the complementary class sc the evaluation of monotone planar circuits is shown to be in nc in fact in logcfl

exponential decay for subcritical contact and percolation processes we study the contact process together with a version of the percolation process with one continuously varying coordinate it is proved here that the radius of the infected cluster has an exponentially decaying tail throughout the subcritical phase the same is true of the lebesgue measure in space time of this cluster certain critical exponent inequalities are derived and the critical point of the percolation process in two dimensions is determined exactly

linear response function for coupled hyperbolic attractors the srb measures of a hyperbolic system are widely accepted as the measures that are physically relevant it has been shown by ruelle that they depend smoothly on the system furthermore ruelle showed by a separate argument that the first derivative i e the linear response function admits a geometric interpretation in this paper we consider thermodynamic limits of srb measures in lattices of coupled hyperbolic attractors in a previous paper using markov partitions and thermodynamic formalism we had established the smooth dependence of thermodynamic limits of srb measures here we establish that the linear response function admits a geometric interpretation the formula is analogous to the one found by ruelle for finite dimensional systems if one term is reinterpreted appropriately we show that the limiting derivative is the thermodynamic limit of the derivatives in finite volume we also obtain similar results for the derivatives of the entropy

a generalised inductive approach to the lace expansion the lace expansion is a powerful tool for analysing the critical behaviour of self avoiding walks and percolation it gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den hollander and the authors we give conditions under which the solution to the recursion relation behaves as a gaussian both in fourier space and in terms of a local central limit theorem these conditions are shown elsewhere to hold for sufficiently spread out models of networks of self avoiding walks in dimensions d 4 and for sufficiently spread out models of critical oriented percolation in dimensions d 1 5 providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models

non commutative arithmetic circuits depth reduction and size lower bounds abstract we investigate the phenomenon of depth reduction in commutative and non commutative arithmetic circuits we prove that in the commutative setting uniform semi unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree and unrestricted depth earlier proofs did not work in the uniform setting this also provides a unified proof of the circuit characterizations of the class logcfl and its counting variant logcfl we show that ac1 has no more power than arithmetic circuits of polynomial size and degree no log log n improving the trivial bound of no log n connections are drawn between tc1 and arithmetic circuits of polynomial size and degree then we consider non commutative computation we show that over the algebra max concat arithmetic circuits of polynomial size and polynomial degree can be reduced to o log2 n depth and even to o log n depth if unbounded fanin gates are allowed this establishes that optlogcfl is in ac1 this is the first depth reduction result for arithmetic circuits over a non commutative semiring and it complements the lower bounds of kosaraju and nisan showing that depth reduction cannot be done in the general non commutative setting we define new notions called short left paths and short right paths and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible this class also can be characterized using the auxpda model finally we characterize the languages generated by efficient circuits over the semiring 2 union concat in terms of simple one way machines and we investigate and extend earlier lower bounds on non commutative circuits

geometric critical exponent inequalities for general random cluster models a set of new critical exponent inequalities d 1 1 2 dv 1 1 andd 1 is proved for a general class of random cluster models which includes independent or dependent percolations lattice animals with any interactions and various stochastic cluster growth models the inequalities imply that the critical phenomena in the models are inevitably not mean field like in the dimensions one two and three

on the susceptibility function of piecewise expanding interval maps we study the susceptibility function psi z sum n 0 infty int z n x y rho 0 y frac partial partial y varphi f n y dy associated to the perturbation f t f tx circ f of a piecewise expanding interval map f and to an observable 1 is the formal derivative at t 0 of the average mathcal r t int varphi rho t dx of with respect to the srb measure of ft our analysis is based on a spectral description of transfer operators it gives in particular sufficient conditions on f x and which guarantee that z is holomorphic in a disc of larger than one or which ensure that a number may be associated to the possibly divergent series 1 we present examples of f x and so that mathcal r t is not lipschitz at 0 and we propose a new version of ruelle s conjecture

gap exponents for percolation processes with triangle condition a study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region it is show that the gaps are given by t 2 fort 2 3 scaling theory predicts thatp p c 0 s p p c p ande p 1 c 0 c 0 s p p c p 3 wheres p is the typical cluster size it is found that p c p p p c 0 s p 1 p c p 1 2 and p c p 3 e p 1 c 0 c 0 s p 1 p c p 3 4

critical points for spread out self avoiding walk percolation and the contact process above the upper critical dimensions we consider self avoiding walk and percolation in d oriented percolation in d and the contact process in d with pd being the coupling function whose range is proportional to l for percolation for example each bond is independently occupied with probability pd y x the above models are known to exhibit a phase transition when the parameter p varies around a model dependent critical point pc we investigate the value of pc when d 6 for percolation and d 4 for the other models and l 1 we prove in a unified way that pc 1 c d o l 2d where the universal term 1 is the mean field critical value and the model dependent term c d o l d is written explicitly in terms of the random walk transition probability d we also use this result to prove that pc 1 cl d o l d 1 where c is a model dependent constant our proof is based on the lace expansion for each of these models

linear response of the lorenz system the present numerical study provides strong evidence that at standard parameters the response of the lorenz system to small perturbations of the control parameter r is linear this evidence is obtained not only directly by determining the response in the observable a x z but also indirectly by validating various implications of the assumption of a linear response like a quadratic response at twice the perturbation frequency a vanishing response in a x x the kramers kronig relations and relations between different response functions since the lorenz system is nonhyperbolic the present results indicate that in contrast to a recent speculation the large system limit thermodynamic limit need not be invoked to obtain a linear response for chaotic systems of this type

the infinite self avoiding walk in high dimensions a measure on infinite self avoiding walks is defined which is the natural limit of the uniform measure on finite self avoiding walks this limit is shown to exist in sufficiently large dimensions

linear response susceptibility and resonances in chaotic toy models abstract we consider simple examples illustrating some new features of the linear response theory developed by ruelle for dissipative and chaotic systems d ruelle smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics j stat phys 95 1999 393 468 in this theory the concepts of linear response susceptibility and resonance which are familiar to physicists have been revisited due to the dynamical contraction of the whole phase space onto attractors in particular the standard framework of the fluctuation dissipation theorem breaks down and new resonances can show up outside the power spectrum in previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by ruelle in a model of chaotic neural networks in this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the stable resonances and their consequences on the linear response of the system we consider a class of two dimensional time discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics t 1 2 t mod 2 a generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies b cessac j a sepulchre stable resonances and signal propagation in a chaotic network of coupled units phys rev e 70 2004 056111 b cessac j a sepulchre transmitting a signal by amplitude modulation in a chaotic network chaos 16 2006 013104 1 13104 12 these models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies widths and dependence on the pair of perturbed observed variables

convergence of self avoiding random walk to brownian motion in high dimensions it is proved that in sufficiently high dimensions the scaled self avoiding random walk on the hypercubic lattice converges in distribution to brownian motion convergence of the finite dimensional distributions was shown elsewhere in this paper tightness is shown

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

hyperscaling inequalities for percolation a set of critical exponent inequalities for independent percolation which saturate under the hyperscaling hypothesis is proved one of the consequences of the inequalities is the lower bounddc 6 for the upper critical dimension the proof is based on a rigorous version of the finite size scaling argument which extends easily to other systems such as ising ferromagnets

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

equivalence of non equilibrium ensembles and representation of friction in turbulent flows the lorenz 96 model we construct different equivalent non equilibrium statistical ensembles in a simple yet instructive n degrees of freedom model of atmospheric turbulence introduced by lorenz in 1996 the vector field can be decomposed into an energy conserving time reversible part plus a non time reversible part including forcing and dissipation we construct a modified version of the model where viscosity varies with time in such a way that energy is conserved and the resulting dynamics is fully time reversible for each value of the forcing the statistical properties of the irreversible and reversible model are in excellent agreement if in the latter the energy is kept constant at a value equal to the time average realized with the irreversible model in particular the average contraction rate of the phase space of the time reversible model agrees with that of the irreversible model where instead it is constant by construction we also show that the phase space contraction rate obeys the fluctuation relation and we relate its finite time corrections to the characteristic time scales of the system a local version of the fluctuation relation is explored and successfully checked the equivalence between the two non equilibrium ensembles extends to dynamical properties such as the lyapunov exponents which are shown to obey to a good degree of approximation a pairing rule these results have relevance in motivating the importance of the chaotic hypothesis in explaining that we have the freedom to model non equilibrium systems using different but equivalent approaches and in particular that using a model of a fluid where viscosity is kept constant is just one option and not necessarily the only option for describing accurately its statistical and dynamical properties

on the decomposability of nc and ac it is shown for rationale a b geq 1 that nc a nc b nc a b 1 as a consequence if for some k geq 1 and epsilon 0 nc k nc k epsilon then nc k nc a similar development can be applied to circuits with unbounded fan in it is seen that ac a ac b ac a b ac a nc b nc a b and nc a ac b ac a b 1 this shows for a geq 0 and b geq 1 that ac a ac b nc a 1 ac b and ac a nc b nc a 1 nc b an oracle a is constructed for which forall k nc k a subset ac k a and in fact ac k a nc k epsilon a neq phi for any epsilon 1 similarly there is an a so that forall k and epsilon 1 nc k 1 a ac k epsilon a neq phi and hence ac k a subset nc k 1 a combining this yields an a such that for all k and 0 epsilon l the classes ac k a and nc k epsilon a are incomparable

differentiating the absolutely continuous invariant measure of an interval map f with respect to f let the map f 1 1 1 1 have a c i m absolutely continuous f invariant measure with respect to lebesgue let be the change of corresponding to a perturbation x f f 1 of f formally we have for differentiable a

critical exponents for the contact process under the triangle condition we show that a continuous time version of the triangle condition for percolation implies mean field values for several contact process critical exponents our results support the belief that the upper critical spatial dimension for the contact process is four

parametric smoothness and self scaling of the statistical properties of a minimal climate model what beyond the mean field theories abstract a quasi geostrophic intermediate complexity model of the mid latitude atmospheric circulation is considered featuring simplified baroclinic conversion and barotropic convergence processes the model undergoes baroclinic forcing towards a given latitudinal temperature profile controlled by the forced equator to pole temperature difference t e as t e increases a transition takes place from a stationary regime hadley equilibrium to a periodic regime and eventually to a chaotic regime where evolution takes place on a strange attractor the attractor dimension metric entropy and bounding box volume in phase space have a smooth dependence on t e which results in power law scaling properties power law scalings with respect to t e are detected also for the statistical properties of global physical observables the total energy of the system and the averaged zonal wind the scaling laws which constitute the main novel result of the present work can be thought to result from the presence of a statistical process of baroclinic adjustment which tends to decrease the equator to pole temperature difference and determines the properties of the attractor of the system the self similarity could be of great help in setting up a theory for the overall statistical properties of the general circulation of the atmosphere and in guiding on a heuristic basis both data analysis and realistic simulations going beyond the unsatisfactory mean field theories and brute force approaches a leading example for this would be the possibility of estimating the sensitivity of the output of the system with respect to changes in the parameters

a first order isomorphism theorem we show that for most complexity classes of interest all sets complete under first order projections fops are isomorphic under first order isomorphisms that is a very restricted version of the berman hartmanis conjecture holds since natural complete problems seem to stay complete via fops this indicates that up to first order isomorphism there is only one natural complete problem for each nice complexity class

depth reduction for circuits of unbounded fan in we prove that constant depth circuits of size nlogo 1 n over the basis and or parity are no more powerful than circuits of this size with depth four similar techniques are used to obtain several other depth reduction theorems in particular we show every set in ac0 can be recognized by a family of depth three threshold circuits of size nlogo 1 n the size bound nlogo 1 n is optimal when considering depth reduction over and or and parity most of our results hold for both the uniform and the nonuniform case

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

lace expansion for the ising model the lace expansion has been a powerful tool for investigating mean field behavior for various stochastic geometrical models such as self avoiding walk and percolation above their respective upper critical dimension in this paper we prove the lace expansion for the ising model that is valid for any spin spin coupling for the ferromagnetic case we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace expansion coefficients for self avoiding walk as a result we obtain gaussian asymptotics of the critical two point function for the nearest neighbor model with d gg 4 and for the spread out model with d 4 and l gg 1 without assuming reflection positivity

p printable sets p printable sets arise naturally in the studies of generalized kolmogorov complexity and data compression as well as in other areas we present new characterizations of the p printable sets and present necessary and sufficient conditions for the existence of sparse sets in p that are not p printable as a corollary to one of our results we show that the class of sets of small generalized kolmogorov complexity is exactly the class of sets which are p isomorphic to a tally language

the diffusion of self avoiding random walk in high dimensions we use the brydges spencer lace expansion to prove that the mean square displacement of at step strictly self avoiding random walk in thed dimensional square lattice is asymptotically of the formdt ast approaches infinity ifd is sufficiently large the diffusion constantd is greater than one

analyticity properties and power law estimates of functions in percolation theory we consider percolation on the sites of a graphg e g a regulard dimensional lattice all sites ofg are occupied vacant with probabilityp respectively q 1 p independently of each other w denotes the cluster of occupied sites containing a fixed site which will usually be taken to be the origin andw the cardinality ofw the percolation probability is the probability that w i e p pp w some critical values ofp ph andpt are defined respectively as the smallest value ofp for which p 0 and for which the expectation of w is infinite formally ph inf p p 0 andpt inf p ep w we show for fairly general graphsgthat ifp pt thenpp w n decreases exponentially inn for the special casesg g0 the simple quadratic lattice andg1 the graph which corresponds to bond percolation on 2 we obtain upper and lower bounds for p of the formc p ph and bounds forep w of the formc p ph we also investigate smoothness properties of p ep number of clusters per site ep w 1 w 1 this function was introduced by sykes and essam who assumed that has exactly one singularity namely atp ph for the graphsg0 andg1 i e site or bond percolation on 2 we show that p is analytic atp ph and has two continuous derivatives atp ph the emphasis is on rigorous proofs

non intersection of transient branching random walks let g be a cayley graph of a nonamenable group with spectral radius rho 1 it is known that branching random walk on g with offspring distribution mu is transient i e visits the origin at most finitely often almost surely if and only if the expected number of offspring overline mu satisfies overline mu le rho 1 benjamini and muller groups geom dyn 6 231 247 2012 conjectured that throughout the transient supercritical phase 1 overline mu le rho 1 and in particular at the recurrence threshold overline mu rho 1 the trace of the branching random walk is tree like in the sense that it is infinitely ended almost surely on the event that the walk survives forever this is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely we prove this conjecture along with several other related conjectures made by the same authors a central contribution of this work is the introduction of the notion of local unimodularity which we expect to have several further applications in the future

generalized approach to the non backtracking lace expansion the lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self avoiding walk percolation and lattice trees and animals the non backtracking lace expansion noble is a modification that allows us to improve its applicability in the nearest neighbor setting on the mathbb z d lattice for percolation lattice trees and lattice animals the noble gives rise to a recursive formula that we study in this paper at a general level we state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound in two related papers we show that these conditions are satisfied for percolation in d ge 11 for lattice trees in d ge 16 and for lattice animals in d ge 18

chaotic hypothesis onsager reciprocity and fluctuation dissipation theorem it is shown that the chaoticity hypothesis recently introduced in statistical mechanics which is analogous to ruelle s principle for turbulence implies the onsager reciprocity and the fluctuation dissipation theorem in various reversible models for coexisting transport phenomena

scaling inequalities for oriented percolation we look at seven critical exponents associated with two dimensional oriented percolation scaling theory implies that these quantities satisfy four equalities we prove five related inequalitites

percolation critical exponents under the triangle condition for independent percolation models it is shown that if the diagrammatic triangle condition is satisfied then the critical exponents delta and hat beta exist and take their mean field values generalizing the criterion introduced in 1984 by aizenman and newman for the mean field value of gamma in nonoriented percolation the results apply to a broad class of nonoriented as well as oriented weakly homogeneous models in which the range of the connecting bonds need not be bounded for the nonoriented case the condition reduces to the finiteness at the critical point of nabla equiv sum x y tau 0 x tau x y tau y 0 with tau u nu the probability that the site u is connected to nu which was recently established by hara and slade for models with sufficiently spread out connections in d 6 dimensions our analysis proceeds through the derivation of complementary differential inequalities for the percolation order parameter m beta h whose value at h 0 yields the percolation density with beta parametrizing the bond or site occupation probabilities and with h h geq 0 a ghost field the conclusion is that under the triangle condition in the vicinity of the critical point beta c 0 m beta 0 approx beta beta c hat beta and m beta c h approx h 1 delta with hat beta 1 and delta 2

response formulae for n point correlations in statistical mechanical systems and application to a problem of coarse graining predicting the response of a system to perturbations is a key challenge in mathematical and natural sciences under suitable conditions on the nature of the system of the perturbation and of the observables of interest response theories allow to construct operators describing the smooth change of the invariant measure of the system of interest as a function of the small parameter controlling the intensity of the perturbation in particular response theories can be developed both for stochastic and chaotic deterministic dynamical systems where in the latter case stricter conditions imposing some degree of structural stability are required in this paper we extend previous findings and derive general response formulae describing how n point correlations are affected by perturbations to the vector flow we also show how to compute the response of the spectral properties of the system to perturbations we then apply our results to the seemingly unrelated problem of coarse graining in multiscale systems we find explicit formulae describing the change in the terms describing parameterisation of the neglected degrees of freedom resulting from applying perturbations to the full system all the terms envisioned by the mori zwanzig theory the deterministic stochastic and non markovian terms are affected at 1st order in the perturbation the obtained results provide a more comprehesive understanding of the response of statistical mechanical systems to perturbations and contribute to the goal of constructing accurate and robust parameterisations and are of potential relevance for fields like molecular dynamics condensed matter and geophysical fluid dynamics we envision possible applications of our general results to the study of the response of climate variability to anthropogenic and natural forcing and to the study of the equivalence of thermostatted statistical mechanical systems

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems