an effective lie kolchin theorem for quasi unipotent matrices abstract we establish an effective version of the classical lie kolchin theorem namely let a b gl m c be quasi unipotent matrices such that the jordan canonical form of b consists of a single block and suppose that for all k 0 the matrix a b k is also quasi unipotent then a and b have a common eigenvector in particular a b gl m c is a solvable subgroup we give applications of this result to the representation theory of mapping class groups of orientable surfaces

structure of chevalley groups over rings via universal localization abstract in the current article we study the structure of a chevalley group g r over a commutative ring r we generalize and improve the following results on the standard relative and multi relative commutator formulas the nilpotent structure of relative k 1 the bounded word length of commutators to this end we enlarge the elementary group construct a generic element for the extended elementary group and use localization in the universal ring the key step is a construction of a generic element for the principal congruence subgroup corresponding to a principal ideal

a martingale approach to homogenization of unbounded random flows we study the asymptotic behavior of brownian motion in steady unbounded incompressible random flows we prove an invariance principle for almost all realizations of random flows the key compactness result is obtained by moser s iterative scheme in pde theory

popular matchings in the stable marriage problem we consider the problem of computing a maximum cardinality popular matching in a bipartite graph g a b e where each vertex u a b ranks its neighbors in a strict order of preference such a graph is called an instance of the stable marriage problem with strict preferences and incomplete lists a matching m is popular if for every matching m in g the number of vertices that prefer m to m is at most the number of vertices that prefer m to m every stable matching of g is popular however a stable matching is a minimum cardinality popular matching the complexity of computing a maximum cardinality popular matching was unknown in this paper we show a simple characterization of popular matchings in g a b e we also show a sufficient condition for a popular matching to be a maximum cardinality popular matching we construct a matching that satisfies our characterization and sufficient condition in o mn 0 time where m e and n 0 min a b thus the maximum cardinality popular matching problem in g a b e can be solved in o mn 0 time

popular edges and dominant matchings given a bipartite graph g a cup b e with strict preference lists and given an edge e in e we ask if there exists a popular matching in g that contains e we call this the popular edge problem a matching m is popular if there is no matching m such that the vertices that prefer m to m outnumber those that prefer m to m it is known that every stable matching is popular however g may have no stable matching with the edge e in this paper we identify another natural subclass of popular matchings called dominant matchings and show that if there is a popular matching that contains the edge e then there is either a stable matching that contains e or a dominant matching that contains e this allows us to design a linear time algorithm for identifying the set of popular edges when preference lists are complete we show an o n 3 algorithm to find a popular matching containing a given set of edges or report that none exists where n a b

non linear residually finite groups abstract we give the first example of a non linear residually finite 1 related group a t a t 2 a 2

application of pair approximation method to modeling and analysis of a marriage network numerous studies on marital issues have frequently resorted to statistical analysis or mathematical modeling on a well mixed homogeneous population however such mathematical models have largely ignored the effects of heterogeneous contact pattern among individuals of the population in this paper we propose a pair model to describe the evolution of a marriage network where an unmarried male and an unmarried female become friends with a certain probability through the introduction of their common friends for convenience this probability is termed as the introduction probability a threshold is obtained to determine whether there is a married couple or not we verify the model by good agreement between the numerical solution of the model computational simulations and the real marriage data in beijing the results show that the positive equilibrium of the model is steady and increases with the threshold moreover the introduction probability contributes to increase both the threshold and the married population size furthermore it is found that increasing the number of same sex friends and balancing the sex ratio can also enlarge the married population size all these results can provide some insights into the evolution and the structure of the marriage network

palindromic automorphisms of free groups abstract let f n be the free group of rank n with free basis x x 1 x n a palindrome is a word in x 1 that reads the same backwards as forwards the palindromic automorphism group a n of f n consists of those automorphisms that map each x i to a palindrome in this paper we investigate linear representations of a n and prove that a 2 is linear we obtain conjugacy classes of involutions in a 2 and investigate residual nilpotency of a n and some of its subgroups let ia n be the group of those automorphisms of f n that act trivially on the abelianisation pi n be the palindromic torelli group of f n and let e a n be the elementary palindromic automorphism group of f n we prove that pi n ia n e a n this result strengthens a recent result of fullarton 2

harnack inequalities and sub gaussian estimates for random walks we show that the beta parabolic harnack inequality for random walks on graphs is equivalent on one hand to the sub gaussian estimate for the transition probability and on the other hand to the conjunction of the elliptic harnack inequality the doubling volume property and the fact that the mean exit time in any ball of radius r is of the order r beta the latter condition can be replaced by a certain estimate of the resistance of annuli

sur les groupes de transformations birationnelles des surfaces ni agissant par transformations bi rationnelles sur les surfaces complexes compactes kahl eriennes nous mon trons a que le groupe des transformations birationnelles d une surface satisfait l alternative de tits b que les actions birationnelles de groupes de kazhdan sur les surfaces sont toutes birationnellement conjugu eesa des actions homographiques sur le plan projectif et c que si f et g sont deux transformations birationnelles de surfaces qui commutent alors ou bien f pr eserve un pinceau de courbes ou bien l un des it er es g m de g m 0 concide avec un it er e f n de f n 2 z let s be any compact complex kahler surface and bir s the group of birational transformations of s we study the structure of nitely genera ted subgroups of bir s and prove three main results a bir s satises the tits alternative b if a group with kazhdan property t acts on a compact kahler surface by birational transformations then the action is conjugate to an action by linear projective transformations on the projec tive plane and c if f is an element of bir s which does not preserve any pencil of curves and if g commutes with f then a positive iterate of g coincides with an iterate of f the second statement provides a positive answer to zimmer s conjecture for birational actions on surfaces

quenched invariance principles for random walks on percolation clusters we consider a supercritical bernoulli percolation model in d2 and study the simple symmetric random walk on the infinite percolation cluster the aim of this paper is to prove the almost sure q

harnack inequalities on weighted graphs and some applications to the random conductance model we establish elliptic and parabolic harnack inequalities on graphs with unbounded weights as an application we prove a local limit theorem for a continuous time random walk x in an environment of ergodic random conductances taking values in 0 infty satisfying some moment conditions

random walks on supercritical percolation clusters we obtain gaussian upper and lower bounds on the transition density qt x y of the continuous time simple random walk on a supercritical percolation cluster c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions hold with constants ci depending only on p the percolation probability and d the irregular nature of the medium means that the bound for qt x only holds for t sx where the constant sx depends on the percolation congura tion

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

on the first cohomology of automorphism groups of graph groups abstract we study the virtual indicability of the automorphism group aut a of the right angled artin group a associated to a simplicial graph first we identify two conditions denoted b1 and b2 on which together imply that h 1 g z 0 for certain finite index subgroups g aut a on the other hand we will show that b2 is equivalent to the matrix group h im aut a aut h 1 a gl n z not being virtually indicable and also to h having kazhdan s property t as a consequence aut a virtually surjects onto z whenever does not satisfy b2 in addition we give an extra property of ensuring that aut a and out a virtually surject onto z finally in the appendix we offer some remarks on the linearity problem for aut a

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

random walk on the infinite cluster of the percolation model we consider random walk on the infinite cluster of bond percolation on d we show that in the supercritical regime whend 3 this random walk is a s transient this conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance it is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite

products of elementary and idempotent matrices over integral domains abstract a ring r such that invertible matrices over r are products of elementary matrices is called after cohn generalized euclidean extending results proved by ruitenburg for bezout domains characterizations of generalized euclidean commutative domains are obtained that relate them with the property that singular matrices are products of idempotent matrices this latter property is investigated focusing on 2 2 matrices which is not restrictive in the context of bezout domains it is proved that domains r that satisfy a suitable property of ideals called princ are necessarily bezout domains if 2 2 singular matrices over r are products of idempotent matrices the class of rings satisfying princ includes factorial and projective free domains the connection with the existence of a weak euclidean algorithm a notion introduced by o meara for dedekind domains is also investigated

invariance principle for the random conductance model we study a continuous time random walk x in an environment of i i d random conductances mu e in 0 infty in mathbb z d we assume that mathbb p mu e 0 p c so that the bonds with strictly positive conductances percolate but make no other assumptions on the law of the e we prove a quenched invariance principle for x and obtain green s functions bounds and an elliptic harnack inequality

limit theory for random walks in degenerate time dependent random environments abstract we study continuous time variable speed random walks in random environments on z d d 2 where at time t the walk at x jumps across edge x y at time dependent rate a t x y the rates which we assume stationary and ergodic with respect to space time shifts are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time we formulate conditions on the environment under which the law of diffusively scaled random walk path tends to brownian motion for almost every sample of the rates the proofs invoke moser iteration to prove sublinearity of the corrector in pointwise sense a key additional input is a conversion of certain weighted energy norms to ordinary ones our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the helffer sjostrand representation of gradient models with certain non strictly convex potentials

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

universal lattices and property tau we prove that the universal lattices the groups g sl d r where r x 1 x k have property for d 3 this provides the first example of linear groups with which do not come from arithmetic groups we also give a lower bound for the constant with respect to the natural generating set of g our methods are based on bounded elementary generation of the finite congruence images of g a generalization of a result by dennis and stein on k 2 of some finite commutative rings and a relative property t of mathrm sl 2 r ltimes r 2 r 2

the automorphism group of a free group is not linear a linear group has ascending chain condition on centralizers malcev s p 511 finitely generated linear groups are residually finite malcev s p 453 a solvable linear group is nilpotent by abelian byfinite tits 5 pp 14551461 a linear group either contains a free group of rank two or is solvable by locally finite lubotzky 3 has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions these conditions appear difficult to check although they have been used to show linearity for certain groups however we do not use his theorem the main result of this paper theorem 5 is that for n 2 3 the automorphism group of a free group of rank n is not a linear group the proof uses the representation theory of algebraic groups to show that a kind of diophantine equation between the irreducible representations of

the braid group b4 is linear we study a linear representation b n gl m z q 1 t 1 with m n n 1 2 we will show that for n 4 this representation is faithful we prove a relation with the new charney length function we formulate a conjecture implying that is faithful for all n

characterizing a free group in its automorphism group for bea and geg if the center of g is trivial then i g a g is one to one and g can be identified with a normal subgroup of a g with the action on g given by conjugation a group g is said to be complete if i g a g is an isomorphism a theorem of burnside ru p 951 says that if g is a group with trivial center then a g is complete if and only if g is a characteristic subgroup of a g let f be a free group of finite rank n 2 in d f joan dyer and 1 used burnside s criterion to show that a f is a complete group more precisely wc showed that if q4 a f a f is an automorphism of a f then d p f

comparison of quenched and annealed invariance principles for random conductance model we show that there exists an ergodic conductance environment such that the weak annealed invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle does not hold

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

bertrand s postulate and subgroup growth abstract in this article we investigate the l 1 norm of certain functions on groups called divisibility functions using these functions their connection to residual finiteness and integration theory on profinite groups we define the residual average of a finitely generated group one of the main results in this article is the finiteness of residual averages on finitely generated linear groups whether or not the residual average is finite depends on growth rates of indices of finite index subgroups our results on index growth rates are analogous to results on gaps between primes and provide a variant of the subgroup growth function which may be of independent interest

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

on the mixing time of a simple random walk on the super critical percolation cluster we study the robustness under perturbations of mixing times by studying mixing times of random walks in percolation clusters inside boxes in z d we show that for d 2 and p p c z d the mixing time of simple random walk on the largest cluster inside is n 2 thus the mixing time is robust up to a constant factor the mixing time bound utilizes the lovasz kannan average conductance method this is the first non trivial application of this method which yields a tight result

abelian covers of graphs and maps between outer automorphism groups of free groups we explore the existence of homomorphisms between outer automorphism groups of free groups out f n out f m we prove that if n 8 is even and n m 2n or n is odd and n m 2n 2 then all such homomorphisms have finite image in fact they factor through det rm out f n to mathbb z 2 in contrast if m r n n 1 1 with r coprime to n 1 then there exists an embedding rm out f n hookrightarrow rm out f m in order to prove this last statement we determine when the action of out f n by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian galois group

stability of parabolic harnack inequalities let g e be a graph with weights a xy for which a parabolic harnack inequality holds with space time scaling exponent 2 suppose xy is another set of weights that are comparable to a xy we prove that this parabolic harnack inequality also holds for g e with the weights a xy we also give stable necessary and sufficient conditions for this parabolic harnack inequality to hold

popular matchings with two sided preference lists and matroid constraints abstract in this paper we consider the popular matching problem with two sided preference lists and matroid constraints which is based on the variants of the popular matching problem proposed by brandl and kavitha and nasre and rawat we prove that there always exists a popular matching in our model and a popular matching can be found in polynomial time furthermore we prove that if every matroid is weakly base orderable then we can find a maximum size popular matching in polynomial time

quenched invariance principle for random walks with time dependent ergodic degenerate weights we study a continuous time random walk xx on zdzd in an environment of dynamic random conductances taking values in 0 0 we assume that the law of the conductances is ergodic with respect to space time shifts we prove a quenched invariance principle for the markov process xx under some moment conditions on the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

symmetric random walks in random environments we consider a random walk on thed dimensional lattice d where the transition probabilitiesp x y are symmetric p x y p y x different from zero only ify x belongs to a finite symmetric set including the origin and are random we prove the convergence of the finite dimensional probability distributions of normalized random paths to the finite dimensional probability distributions of a wiener process and find our an explicit expression for the diffusion matrix

unramified subextensions of ray class field towers abstract fix a prime l in this paper we explore various senses in which the ramification in an infinite l extension of number fields can be small in particular 1 we construct infinitely many pairs of primes p q distinct from l such that q admits an infinite l extension unramified outside p q and 2 we explore the possibility of finding infinite unramified subextensions inside infinite ramified l extensions of number fields

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

word length in elementary matrices let r be an associative ring with identity recall that the group e n r is generated by those n x n matrices over r differing from the identity matrix z in at most one entry which must be off the main diagonal defining v a to be the minimal number of nontrivial generators necessary to obtain the product a one may ask whether there is an absolute bound on v a as a varies in e n r if such a bound exists denote by e r the maximum attained by v on e n r it is then of interest to estimate e r as a by product of their results on commutator products r k dennis and l n vaserstein 2 give the following relative bound which holds for n m1 sr r where w r is the stable rank in s

universal lattices and unbounded rank expanders we study the representations of non commutative universal lattices and use them to compute lower bounds of the constant for the commutative universal lattices g d k sl d x 1 x k for d 3 with respect to several generating sets as an application we show that the cayley graphs of the finite groups text sl 3k mathbb f p can be made expanders with a suitable choice of generators this provides the first example of expander families of groups of lie type where the rank is not bounded and provides counter examples to two conjectures of a lubotzky and b weiss

isoperimetry and heat kernel decay on percolation clusters we prove that the heat kernel on the infinite bernoulli percolation cluster in zd almost surely decays faster than t d 2 we also derive estimates on the mixing time for the random walk confined to a finite box our approach is based on local isoperimetric inequalities some of the results of this paper were previously announced in the note of mathieu and remy c r acad sci paris ser i math 332 2001 927 931

invariance principle for the random conductance model in a degenerate ergodic environment we study a continuous time random walk x on zd in an environment of random conductances taking values in 0 we assume that the law of the conductances is ergodic with respect to space shifts we prove a quenched invariance principle for x under some moment conditions of the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

symmetric groups and expander graphs we construct explicit generating sets sn and tilde s n of the alternating and the symmetric groups which turn the cayley graphs mathcal c text alt n s n and mathcal c text sym n tilde s n into a family of bounded degree expanders for all n

polynomial identities in graded group rings restricted lie algebras and p adic analytic groups let g be any finitely generated group and let k be a field of characteristic p 0 it is shown that the graded group ring gr kg satisfies a nontrivial polynomial identity if and only if the pro p completion of g is p adic analytic i e can be given the structure of a lie group over the p adic field q p the proof applies theorems of lazard quillen and passman as well as results on engel lie algebras and on dimension subgroups in positive characteristic

low dimensional representations of matrix groups and group actions on cat 0 spaces and manifolds abstract we study low dimensional representations rigidity problem of matrix groups over general rings by considering group actions on cat 0 spaces spheres and acyclic manifolds

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

the diffusion limit for reversible jump processes on z d with ergodic random bond conductivities we consider a reversible jump process on d whose jump rates themselves are random we show mean square convergence of this process under diffusion scaling to a limiting brownian motion with a certain diffusion matrix characterizing effective conductivity

a group theoretic characterization of linear groups let r be a group when is f linear this is an old problem the first to study this question systematically was malcev in 1940 m who essentially reduced the problem to finitely generated groups note that a finitely generated group is linear over some field of characteristic zero if and only if it can be embedded in cl c for some n very little progress was made since that paper of malcev although as linear groups are a quite special type of group many necessary conditions were obtained e g r should be residually finite and even virtually residually p for almost all primes p f should be virtually torsion free and if not solvable by finite it has a free non abelian subgroup etc cf z of course none of these properties characterizes the finitely generated linear groups over in this paper we give such a characterization using the congruence structure of r first some definitions for a group h d h denotes the minimal number of generators for h

products of involutions in the stable general linear group abstract in the stable general linear group over an arbitrary field we prove that every element with determinant 1 is the product of three involutions and of no less in general we also obtain several results of the same flavor with applications to decompositions of automorphisms of an infinite dimensional vector space that are scalar multiples of finite rank perturbations of the identity

an invariance principle for diffusion in turbulence we prove an almost sure invariance principle for diffusion driven by velocities with unbounded stationary vector potentials the result generalizes to multiple particles motion driven by a common velocity field and independent molecular brownian motions

random walks in quenched i i d space time random environment are always a s diffusive we consider a general model of discrete time random walk x t on the lattice 1 in a random environment t x t x 1 with i i d components t x previous results on the a s validity of the central limit theorem for the quenched model required a small stochasticity condition in this paper we show that the result holds provided only that an obvious non degeneracy condition is met the proof is based on the analysis of a suitable generating function which allows to estimate l 2 norms by contour integrals

on triangular representations of aut fr abstract if f r gl n k is a representation of a finitely generated free group f r and a m i for each basic element i e element which occurs in some basis a then we show that if f r is triangularisable it is finite this can be thought of as a generalisation of the burnside problem for these linear groups

on groups of polynomial subgroup growth let be a finitely generated group andan the number of its subgroups of indexn we prove that assuming is residually nilpotent e g linear thenan grows polynomially if and only if is solvable of finite rank this answers a question of segal the proof uses a new characterization ofp adic analytic groups the theory of algebraic groups and the prime number theorem the method can be applied also to groups of polynomial word growth

on representations and k theory of the braid groups let be the fundamental group of the complement of a k 1 hyperplane arrangement such as artin s pure braid group or more generally a homologically toroidal group as defined below the triviality of bundles arising from orthogonal representations of is characterized completely as follows an orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups furthermore the subgroup of elements in the complex k theory of b which arises from complex unitary representations of is shown to be trivial in the case of real k theory the subgroup of elements which arises from real orthogonal representations of is an elementary abelian 2 group which is characterized completely in terms of the first two stiefel whitney classes of the representation in addition quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the jacobi identity for certain choices of poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized heisenberg groups these cohomology relations correspond to non trivial spin representations of the pure braid groups which give rise to trivial bundles

anchored nash inequalities and heat kernel bounds for static and dynamic degenerate environments abstract we introduce anchored versions of the nash inequality they allow to control the l 2 norm of a function by dirichlet forms that are not uniformly elliptic we then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments as an example we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium

groups of triangular automorphisms of a free associative algebra and a polynomial algebra abstract we study the structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semidirect product of abelian groups using this decomposition we describe the structure of the lower central series and the derived series for the group of unitriangular automorphisms and prove that every element of the derived subgroup is a commutator in addition we prove that the group of unitriangular automorphisms of a free associative algebra of rank greater than 2 is not linear and describe some two generated subgroups of this group also we give a more simple system of generators for the group of tame automorphisms in comparison to that system in umirbaev s paper

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

on a question of m newman on the number of commutators abstract m newman asked whether there is an absolute constant c such that every matrix in slnr is the product of at most c commutators where r ranges over euclidean commutative rings and n 3 we give here a negative answer however if for the ring r every matrix in slmr is the product of a bounded number of commutators for some fixed m 3 then for all sufficiently large n every matrix in slnr is the product of six commutators

uniform kazhdan constant for some families of linear groups abstract let r be a ring generated by l elements with stable range r assume that the group el d r has kazhdan constant 0 0 for some d r 1 we prove that there exist 0 l 0 and k n s t for every n d el n r has a generating set of order k and a kazhdan constant larger than as a consequence we obtain for sl n z where n 3 a kazhdan constant which is independent of n w r t generating set of a fixed size

mixing time for random walk on supercritical dynamical percolation we consider dynamical percolation on the d dimensional discrete torus znd of side length n where each edge refreshes its status at rate n 1 2 to be open with probability p we study random walk on the torus where the walker moves at rate 1 2d along each open edge in earlier work of two of the authors with a stauffer it was shown that in the subcritical case p pc zd the mixing time is n2 1 here the implied constants depend only on d and p we prove a quenched and hence annealed version of this conjecture up to a poly logarithmic factor under the assumption p 1 2 when p 0 we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times the latter implies sharp up to poly logarithmic factors upper bounds on exit times of large balls throughout the supercritical regime our proofs are based on percolation results e g the grimmett marstrand theorem and an analysis of the volume biased evolving set process the key point is that typically the evolving set has a substantial intersection with the giant percolation cluster at many times this allows us to use precise isoperimetric properties of the cluster due to g pete to infer rapid growth of the evolving set which in turn yields the upper bound on the mixing time

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models