critical dynamics of self organizing eulerian walkers the model of self organizing eulerian walkers is numerically investigated on the square lattice the critical exponents for the distribution of a number of steps tau l and visited sites tau s characterizing the process of transformation from one recurrent configuration to another are calculated using the finite size scaling analysis two different kinds of dynamical rules are considered the results of simulations show that both the versions of the model belong to the same class of universality with the critical exponents tau l tau s 1 75 pm 0 1

conjectures and results about parabolic induction of representations of text gl _n f in 1980 zelevinsky introduced certain commuting varieties whose irreducible components classify complex irreducible representations of the general linear group over a non archimedean local field with a given supercuspidal support we formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction the conditions are in the spirit of the geiss leclerc schroer condition that occurs in the conjectural characterization of square irreducible representations we verify some special cases of the new conjecture and check that the geometric and representation theoretic conditions are compatible in various ways

structure of two dimensional sandpile i height probabilities the height probabilities of the two dimensional abelian sandpile model are the fractionial numbers of lattice sites having heights 1 2 3 4 a combinatorial method for evaluation of these quantities is proposed the method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice exact analytical expressions for all probabilities are obtained

critical exponents of the sand pile models in two dimensions abstract we study three sand pile automaton models namely the critical height critical slope and the critical laplacian models in two dimensions in which the stability criterion of the sand columns depend on the zeroth first and the second derivatives of the sand height function we carried out simulations on system sizes up to 2048 2048 and up to 108 avalanches were generated the exponents of the critical height model were calculated by taking into account the strong corrections to scaling in order to determine the exponents of the critical laplacian model accurately we introduced a height restriction in the toppling criterion that maintains universality but accelerates convergence to the steady state by orders of magnitude we see clear scaling in the critical height and the critical laplacian models and find that they belong to different universality classes however we do not find any scaling in the critical slope model

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

intrinsic construction of invariant functions on simple lie algebras abstract an algorithm for constructing primitive adjoint invariant functions on a complex simple lie algebra is presented the construction is intrinsic in the sense that it does not resort to any representation a primitive invariant function on the whole lie algebra is obtained by lifting a coordinate function on a kostant slice of the lie algebra such an intrinsic construction of invariant functions is most useful for the bigger exceptional lie algebras such as the e s the maple implementation of this algorithm is outlined at the end and will be applied to these exceptional lie algebras in a future work

dynamical and spatial aspects of sandpile cellular automata the bak tang and wiesenfeld cellular automaton is simulated in 1 2 3 4 and 5 dimensions we define a new set of scaling exponents by introducing the concept of conditional expectation values scaling relations are derived and checked numerically and the critical dimension is discussed we address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2 the scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation primarily the results of our work show that the flow of sand down the slope does not have a 1 f power spectrum in any dimension although the model does show clear critical behavior with scaling exponents depending on the dimension the power spectrum behaves as 1 f 2 in all the dimensions considered

classifying forms of simple groups via their invariant polynomials abstract let g be a simple linear algebraic group over a field f and v an absolutely irreducible representation of g we show that under some mild hypotheses there exists an invariant homogeneous polynomial f for the action of g on v defined over f such that twisted forms of f up to a scalar multiple classify twisted forms of g for which the representation v is defined over f this result extends the classical case of a quadratic form q and its orthogonal group o q

self organizing height arrow model numerical and analytical results the recently introduced self organizing height arrow ha model is studied numerically on the square lattice and analytically on the bethe lattice the concentration of occupied sites and critical exponents of distributions of avalanches are evaluated for two slightly different versions of the model the exponents found for distributions of avalanches by mass area duration and appropriate fractal dimensions are close to those for the original sandpile model which suggests that the ha model belongs to the same universality class for comparison the concentration of occupied sites in the ha model is calculated exactly on the bethe lattice with coordination number q 4

eulerian walkers as a model of self organized criticality we propose a new model of self organized criticality a particle is dropped at random on a lattice and moves along directions specified by arrows at each site as it moves it changes the direction of the arrows according to fixed rules on closed graphs these walks generate euler circuits on open graphs the particle eventually leaves the system and a new particle is then added the operators corresponding to particle addition generate an abelian group same as the group for the abelian sandpile model on the graph we determine the critical steady state and some critical exponents exactly using this equivalence

laplace operators of infinite dimensional lie algebras and theta functions abstract until recently the generalized casimir operator constructed by kac kac v g 1974 funct anal appl 8 68 70 has been the only known element of the center of a completion of the enveloping algebra of a kac moody algebra it has been conjectured deodhar v v gabber o kac v g 1982 adv math 45 92 116 however that the image of the harish chandra homomorphism contains all theta functions defined on the interior of the complexified tits cone and hence separates the orbits of the weyl group developing the ideas of feigin and fuchs feigin b l fuchs d b 1983 dokl akad nauk sssr 269 1057 1060 i prove this conjecture another application of this method is the chevalley type restriction theorem for simple finite dimensional lie superalgebras

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

sandpile models with and without an underlying spatial structure we present a simple mean field model for the sandpile model introduced by bak tang and wiesenfeld btw phys rev lett 59 381 1987 in the mean field model we are able to pinpoint the process of self organization as well as the emerging scale invariance displayed as a power law distribution of avalanche sizes we discuss the btw sandpile model on a lattice and show that the dynamical behavior can be expressed as a transport problem this implies that the average avalanche size scales with the system size and additional heuristic arguments related to the transport properties more than indicate the origin of the power law behavior we review recent work in which scaling relations and additional constraints between the various critical exponents are addressed we demonstrate that some of the proposed relations are inconsistent we present a coherent theory in which the scaling relations along with additional constraints leave only one exponent unknown

dynamics of eulerian walkers we investigate the dynamics of eulerian walkers as a model of self organized criticality the evolution of the system is divided into characteristic periods which can be seen as avalanches the structure of avalanches is described and the critical exponent in the distribution of first avalanches ensuremath tau 2 is determined we also study a mean square displacement of eulerian walkers and obtain a simple diffusion law in the critical state the evolution of an underlying medium from a random state to the critical one is described

framed m branes corners and topological invariants we uncover and highlight relations between the m branes in m theory and various topological invariants the hopf invariant over q z and z2 the kervaire invariant the f invariant and the invariant this requires either a framing or a corner structure the canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies we characterize the flux quantization condition on the c field and the topological action of the m5 brane via the hopf invariant and the dual of the c field as a refinement of an element of hopf invariant two in the signature formulation the contribution to the m brane effective action is given by the maslov index of the corner the kervaire invariant implies that the effective action of the m5 brane is quadratic our study leads to viewing the self dual string which is the boundary of the m2 brane on the m5 brane worldvolume as a string theory in the sense of cobordism of manifolds with corners we show that the dynamics of the c field and its dual are encoded in a unified way in the 4 sphere which suggests the corresponding spectrum as the generalized cohomology theory describing the fields the effective action of the corner is captured by the f invariant which is an invariant at chromatic level two finally considering m theory on manifolds with g2 holonomy we show that the canonical g2 structure minimizes the topological part of the m5 brane action this is done via the invariant and a variant that we introduce related to the one loop polynomial we uncover and highlight relations between the m branes in m theory and various topological invariants the hopf invariant over q z and z2 the kervaire invariant the f invariant and the invariant this requires either a framing or a corner structure the canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies we characterize the flux quantization condition on the c field and the topological action of the m5 brane via the hopf invariant and the dual of the c field as a refinement of an element of hopf invariant two in the signature formulation the contribution to the m brane effective action is given by the maslov index of the corner the kervaire invariant implies that the effective action of the m5 brane is quadratic our study leads to viewing the self dual string which is the boundary of the m2 brane on the m5 brane worldvolume as a string theory in the sense of cobordism of manifolds with corners

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

abelian sandpile model on the bethe lattice the authors study bak tang and wiesenfeld s abelian sandpile model 1987 of self organised criticality on the bethe lattice exact expressions for various distribution functions including the height distribution at a site and the joint distribution of heights at two sites separated by an arbitrary distance are obtained they also determine the probability distribution of the number of distinct sites that topple at least once the number of toplings at the origin and the total number of toplings in an avalanche the probability that an avalanche consists of more than n toplings varies as n 1 2 for large n the probability that its duration exceeds t decreases as 1 t for large t these exponents are the same as for the critical percolation clusters in mean field theory

exceptional lie algebras su 3 and jordan pairs part 2 zorn type representations a representation of the exceptional lie algebras reflecting a simple unifying view based on realizations in terms of zorn type matrices is presented the role of the underlying jordan pair and jordan algebra content is crucial in the development of the structure each algebra contains three jordan pairs sharing the same lie algebra of automorphisms and the same external su 3 symmetry the applications in physics are outlined

on the realizability of group actions abstract let the pair g m r denote a group g acting on an r module m r a unitary ring we ask for the existence of an r local space x such that e x k x r is equivalent in a natural way to g m r for some k 2 where e x denotes the group of homotopy classes of self homotopy equivalences of x if such an x exists we say that x realizes the group action g m r we prove that if g is finite and acts faithfully on a finitely generated q module m there exist infinitely many rational spaces realizing g m q our proof relies on providing a positive answer to kahn s realizability problem for a large class of orthogonal groups that strictly contains finite ones as a matter of fact we enlarge the class of groups that is known to be realizable in the classical kahn s sense

the octic e8 invariant we give an explicit expression for the primitive e8 invariant tensor with eight symmetric indices the result is presented in a manifestly spin 16 z2 covariant notation

introduction to the sandpile model abstract this article is based on a talk given by one of us evi at the conference statphys taipei 1997 it overviews the exact results in the theory of the sandpile model and discusses briefly the yet unsolved problem of calculation of avalanche distribution exponents the key ingredients include the analogy with the critical reaction diffusion system the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

universality classes in isotropic abelian and non abelian sandpile models universality in isotropic abelian and non abelian sandpile models is examined using extensive numerical simulations to characterize the critical behavior we employ an extended set of critical exponents geometric features of the avalanches as well as scaling functions describing the time evolution of average quantities such as the area and size during the avalanche comparing between the abelian bak tang wiesenfeld model p bak c tang and k wiensenfeld phys rev lett 59 381 1987 and the non abelian models introduced by manna s s manna j phys a 24 l363 1991 and zhang y c zhang phys rev lett 63 470 1989 we find strong indications that each one of these models belongs to a distinct universality class

logarithmic corrections of the avalanche distributions of sandpile models at the upper critical dimension we study numerically the dynamical properties of the btw model on a square lattice for various dimensions the aim of this investigation is to determine the value of the upper critical dimension where the avalanche distributions are characterized by the mean field exponents our results are consistent with the assumption that the scaling behavior of the four dimensional btw model is characterized by the mean field exponents with additional logarithmic corrections we benefit in our analysis from the exact solution of the directed btw model at the upper critical dimension which allows to derive how logarithmic corrections affect the scaling behavior at the upper critical dimension similar logarithmic corrections forms fit the numerical data for the four dimensional btw model strongly suggesting that the value of the upper critical dimension is four

how self organized criticality works a unified mean field picture we present a unified dynamical mean field theory based on the single site approximation to the master equation for stochastic self organized critical models in particular we analyze in detail the properties of sandpile and forest fire ff models in analogy with other nonequilibrium critical phenomena we identify an order parameter with the density of active sites and control parameters with the driving rates depending on the values of the control parameters the system is shown to reach a subcritical absorbing or supercritical active stationary state criticality is analyzed in terms of the singularities of the zero field susceptibility in the limit of vanishing control parameters the stationary state displays scaling characteristics of self organized criticality soc we show that this limit corresponds to the breakdown of space time locality in the dynamical rules of the models we define a complete set of critical exponents describing the scaling of order parameter response functions susceptibility and correlation length in the subcritical and supercritical states in the subcritical state the response of the system to small perturbations takes place in avalanches we analyze their scaling behavior in relation with branching processes in sandpile models because of conservation laws a critical exponents subset displays mean field values ensuremath nu frac 1 2 and ensuremath gamma 1 in any dimensions we treat bulk and boundary dissipation and introduce a critical exponent relating dissipation and finite size effects we present numerical simulations that confirm our results in the case of the forest fire model our approach can distinguish between different regimes soc ff and deterministic ff studied in the literature and determine the full spectrum of critical exponents