on the inverse problem of statistical physics from irreversible semigroups to chaotic dynamics abstract we show that all measure preserving stationary markov processes arise as projections of kolmogorov dynamical systems which have positive entropy production and are prototypes of chaos this result not only contributes to the clarification of the relation of dynamics with stochastic processes but also shows the physical significance of the misra prigogine courbage theory of irreversibility in the more general context of the inverse problem of statistical physics because we want positivity preserving transformations our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of positive dilations

the alexander orbach conjecture holds in high dimensions we examine the incipient infinite cluster iic of critical percolation in regimes where mean field behavior has been established namely when the dimension d is large enough or when d 6 and the lattice is sufficiently spread out we find that random walk on the iic exhibits anomalous diffusion with the spectral dimension d s frac 4 3 that is p t x x t 2 3 o 1 this establishes a conjecture of alexander and orbach j phys lett paris 43 625 631 1982 en route we calculate the one arm exponent with respect to the intrinsic distance

to what class of fractals does the alexander orbach conjecture apply discussion de la decouverte numerique recente d alexander et orbach selon laquelle pour un agregat infini naissant en percolation le rapport d f d w dimension fractale de l agregat dimension fractale d une marche aleatoire dans l agregat est approximativement independant de d pour d 1

on k flows and irreversibility here we give the proof of a general theorem concerning irreversibility that was stated earlier by misra and prigogine in terms of a unitary group describing a deterministic dynamical evolution and a related markov semigroup describing an associated coarse grained probabilistic evolution it is shown that the original dynamics are necessarily those of ak flow thus a reversible dynamics which permits such an intertwining to an irreversible description must possess a high degree of instability

outer unipotent classes in automorphism groups of simple algebraic groups we study the unipotent elements of disconnected algebraic groups of the formg whereg is a simple algebraic group in characterisctic p possessing a graph automorphism of order p we classify the unipotent classes in the coset g and determine the corresponding centralizers showing that these bear a close relation to classes in a certain natural connected overgroup of g we also obtain a formula for the total number of outer unipotent elements in the finite group g where is a frobenius morphism analogous to the well known steinberg formula for the number of inner unipotent elements

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

mean field criticality for percolation on planar non amenable graphs the critical exponents and are proved to exist and to take their mean field values for independent percolation on the following classes of infinite locally finite connected transitive graphs 1 non amenable planar with one end 2 unimodular with infinitely many ends

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

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

incipient infinite percolation clusters in 2d we study several kinds of large critical percolation clusters in two dimensions we show that from the microscopic lattice scale perspective these clusters can be described by kesten s incipient infinite cluster iic as was conjectured by aizenman more specifically we establish this for incipient spanning clusters large clusters in a finite box and the inhomogeneous model of chayes chayes and durrett our results prove the equivalence of several natural definitions of the iic we also show that for any k 1 the difference in size between the kth and k 1 st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity in addition the distribution of the chayes chayes durrett cluster is shown to be singular with respect to the iic

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

intrinsical randomness of kolmogorov ℤd actions on a lebesgue space we introduce a concept of an intrinsically weak and strong randomness of a d action on a lebesgue space and we show that kolmogorov d actions are intrinsically weak random and bernoulli d actions are intrinsically strong random

asymptotic behavior of the gyration radius for long range self avoiding walk and long range oriented percolation let 0 and l 2 1 1 and suppose that the 1 step distribution d for random walk on z d decays as d x jx lj d for large jxjsuch that

scaling limit for the ant in a simple high dimensional labyrinth we prove that after suitable rescaling the simple random walk on the trace of a large critical branching random walk in mathbb z d converges to the brownian motion on the integrated super brownian excursion when d 14

on intrinsically random ℤ2 actions on a lebesgue space it is shown that a mathbb z 2 action on a lebesgue space is intrinsically random ir iff it is a kolmogorov action k action as a consequence we obtain the fact that the mathbb z 2 action defined by the lorentz gas is an ir action and the mathbb z 2 action defined by the ideal gas is not an ir action

random subgraphs of finite graphs ii the lace expansion and the triangle condition in a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph in this paper we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph the latter is readily verified for several graphs with vertex set 0 1 r 1 n including the hamming cube on an alphabet of r letters the n cube for r 2 the n dimensional torus with nearest neighbor bonds and n sufficiently large and the n dimensional torus with n 6 and sufficiently spread out long range bonds the conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples

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

the incipient infinite cluster for high dimensional unoriented percolation we consider bond percolation on mathbb z d at the critical occupation density p c for d 6 in two different models the first is the nearest neighbor model in dimension d 6 the second model is a spread out model having long range parameterized by l in dimension d 6 in the spread out case we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as x when d 6 and l is sufficiently large we also give a general criterion for this convergence to hold which is satisfied in the case d 6 in the nearest neighbor model by work of hara 12 we further give a second construction by taking p 6 the probability for invasion percolation to reach a site x is asymptotic to c x d 4 as x

products of conjugacy classes in finite and algebraic simple groups abstract we prove the arad herzog conjecture for various families of finite simple groups if a and b are nontrivial conjugacy classes then a b is not a conjugacy class we also prove that if g is a finite simple group of lie type and a and b are nontrivial conjugacy classes either both semisimple or both unipotent then a b is not a conjugacy class we also prove a strong version of the arad herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes as a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product a special case of this has been used by prasad to prove a uniqueness result for tits systems in quasi reductive groups our final result is a generalization of the baer suzuki theorem for p elements with p 5

bounds on the number and sizes of conjugacy classes in finite chevalley groups with applications to derangements we present explicit upper bounds for the number and size of conjugacy classes in finite chevalley groups and their variations these results have been used by many authors to study zeta functions associated to representations of finite simple groups random walks on chevalley groups the final solution to the ore conjecture about commutators in finite simple groups and other similar problems in this paper we solve a strong version of the boston shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups the remaining cases are settled in two other papers of the authors and applications are given in a third

construction of the incipient infinite cluster for spread out oriented percolation above 4 1 dimensions we construct the incipient infinite cluster measure iic for sufficiently spread out oriented percolation on d for d 1 4 1 we consider two different constructions for the first construction we define n e by taking the probability of the intersection of an event e with the event that the origin is connected to x n d summing this probability over x d and normalising the sum to get a probability measure we let n and prove existence of a limiting measure the iic for the second construction we condition the connected cluster of the origin in critical oriented percolation to survive to time n and let n under the assumption that the critical survival probability is asymptotic to a multiple of n 1 we prove existence of a limiting measure with in addition we study the asymptotic behaviour of the size of the level set of the cluster of the origin and the dimension of the cluster of the origin under our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super brownian motion for d 1 4 1

unfaithful minimal heilbronn characters of finite groups abstract we obtain necessary and sufficient conditions for a finite group g to possess an unfaithful minimal heilbronn character a virtual character but not a character of g whose inner product with every monomial character is nonnegative whose restriction to every proper subgroup and quotient is a character and whose restriction to some proper subgroup is unfaithful we give an application constraining hypothetical minimal counterexamples to artin s conjecture on the holomorphy of l series

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

unipotent subgroups of groups of lie type let g g be either a simple algebraic group over an algebraically closed field k or a finite chevalley group normal or twisted over k f with q 4 for the case of an algebraic group all groups discussed are taken to be closed subgroups of g in this paper we prove results that parallel those in 141 where we deterined those subgroups of g containing a maximal torus given restrictions on k here we study the overgroups of a certain type of unipotent subgroup of g which like the maximal tori are of particular importance to the understanding of the subgroup structure of g as applications we establish results about overgroups of local subgroups of g and results concerning generation by centralizers of unipotent elements a unipotent group i g is said to be full provided i contains each unipotent element of its centralizer such subgroups occur quite often for example let v be an arbitrary unipotent subgroup of g and v a maximal unipotent subgroup of either ng v or v c v then v v and v is full before stating the results we introduce the following notation and terminology let p char k a subgroup x of g is quasisimple if it is a perfect central extension of a simple group connected in case g is an algebraic group x is semisimple if it is a commuting product of quasisimple groups let r x denote the unipotent radical of x which we interpret as o x in case g is finite if k if or f4 let o x denote the group generated by all unipotent elements i e p elements of x

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

random walk on the high dimensional iic we study the asymptotic behavior of the exit times of random walk from euclidean balls around the origin of the incipient infinite cluster in a manner inspired by kumagai and misumi j theor probab 21 910 935 2008 we do this by getting bounds on the effective resistance between the origin and the boundary of these euclidean balls we show that the geometric properties of long range percolation clusters are significantly different from those of finite range clusters we also study the behavior of random walk on the backbone of the iic and we prove that the alexander orbach conjecture holds for the incipient infinite cluster in high dimensions both for long range percolation and for finite range percolation

bounds on completely convergent euclidean feynman graphs letg be a euclidean feynman graph containingl g lines we prove that ifg has massive propagators and does not contain any divergent subgraphs its value is bounded bykl g we also prove the infrared analogue of this bound

critical two point functions for long range statistical mechanical models in high dimensions we consider long range self avoiding walk percolation and the ising model on zd that are defined by power law decaying pair potentials of the form d x x d with 0 the upper critical dimension dc is 2 2 for self avoiding walk and the ising model and 3 2 for percolation let 2 and assume certain heat kernel bounds on the n step distribution of the underlying random walk we prove that for d dc and the spread out parameter sufficiently large the critical two point function gpc x for each model is asymptotically c x 2 d where the constant c 0 is expressed in terms of the model dependent lace expansion coefficients and exhibits crossover between 2 we also provide a class of random walks that satisfy those heat kernel bounds

decay of correlations in nearest neighbor self avoiding walk percolation lattice trees and animals we consider nearest neighbor self avoiding walk bond percolation lattice trees and bond lattice animals on z d the two point functions of these models are respectively the generating function for self avoiding walks from the origin to x z d the probability of a connection from the origin to x and the generating functions for lattice trees or lattice animals containing the origin and x using the lace expansion we prove that the two point function at the critical point is asymptotic to const x 2 d as x for d 5 for self avoiding walk for d 19 for percolation and for sufficiently large d for lattice trees and animals these results are complementary to those of ann probab 31 2003 349 408 where spread out models were considered in the course of the proof we also provide a sufficient and rather sharp if d 4 condition under which the two point function of a random walk on z d is asymptotic to const x 2 d as x

critical behavior and the limit distribution for long range oriented percolation ii spatial correlation we prove that the fourier transform of the properly scaled normalized two point function for sufficiently spread out long range oriented percolation with index 0 converges to e c k alpha wedge2 for some c in 0 infty above the upper critical dimension dc equiv 2 alpha wedge 2 this answers the open question remained in the previous paper chen and sakai in probab theory relat fields 142 151 188 2008 moreover we show that the constant c exhibits crossover at 2 which is a result of interactions among occupied paths the proof is based on a new method of estimating fractional moments for the spatial variable of the lace expansion coefficients

mean field behavior for long and finite range ising model percolation and self avoiding walk we consider self avoiding walk percolation and the ising model with long and flnite range by means of the lace expansion we prove mean fleld behavior for these models if d 2 fi 2 for self avoiding walk and the ising model and d 3 fi 2 for percolation where d denotes the dimension and fi the power law decay exponent of the coupling function we provide a simplifled analysis of the lace expansion based on the trigonometric approach in borgs et al 13

the temperature zero limit we prove that for a broad class of many fermion models the amplitudes of renormalized feynman diagrams converge to their temperature zero values in the limit as the temperature tends to zero

the incipient infinite cluster in two dimensional percolation letpp be the probability measure on the configurations of occupied and vacant vertices of a two dimensional graphg under which all vertices are independently occupied respectively vacant with probabilityp respectively 1 p letph be the critical probability for this system andw the occupied cluster of some fixed vertexw0 we show that for many graphsg such as mathbb z 2 or its covering graph which corresponds to bond percolation on mathbb z 2 the following two conditional probability measures converge and have the same limit v say

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

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

critical two point functions and the lace expansion for spread out high dimensional percolation and related models we consider spread out models of self avoiding walk bond percolation lattice trees and bond lattice animals on mathbb z d having long finite range connections above their upper critical dimensions d 4 self avoiding walk d 6 percolation and d 8 trees and animals the two point functions for these models are respectively the generating function for self avoiding walks from the origin to x in mathbb z d the probability of a connection from 0 to x and the generating function for lattice trees or lattice animals containing 0 and x we use the lace expansion to prove that for sufficiently spread out models above the upper critical dimension the two point function of each model decays at the critical point as a multiple of x 2 d as x to infty we use a new unified method to prove convergence of the lace expansion the method is based on x space methods rather than the fourier transform our results also yield unified and simplified proofs of the bubble condition for self avoiding walk the triangle condition for percolation and the square condition for lattice trees and lattice animals for sufficiently spread out models above the upper critical dimension

on intrinsic randomness of dynamical systems we discuss the problem of nonunitary equivalence via positivity preserving similarity transformations between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes dynamical systems admitting such nonunitary equivalence with stochastic markov processes are said to beintrinsically random in a previous work it was found that the so called bernoulli systems discrete time are intrinsically random in this sense this result is extended here by showing that a more general class of dynamical systems the so calledk systems andk flows are intrinsically random the connection of intrinsic randomness with local instability of motion is briefly discussed we also show that markov processes associated through nonunitary equivalence tononisomorphic k flows are necessarily non isomorphic

high dimensional incipient infinite clusters revisited the incipient infinite cluster iic measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite using the lace expansion we construct the iic measure for high dimensional percolation models in three different ways extending previous work by the second named author and jarai we show that each construction yields the same measure indicating that the iic is a robust object furthermore our constructions apply to spread out versions of both finite range and long range percolation models we also get estimates on structural properties of the iic such as the volume of the intersection between the iic and euclidean balls