decomposition numbers and local properties abstract if g is a finite group and p is a prime we give evidence that the p decomposition matrix encodes properties of p sylow normalizers

hodge laplacians on graphs this is an elementary introduction to the hodge laplacian on a graph a higher order generalization of the graph laplacian we will discuss basic properties including cohomology and hodge theory t

sharp estimates for capacities and applications to symmetric diffusions we give sharp estimates for the capacity of compact setf x wherex is the state space of a strongly regular dirichlet form of diffusion type these estimates are in terms of the reference measurem and the caratheodory metric onx for instance cap 0 f leqq left mathop smallint limits 0 infty frac dr v r right 1 leqq 2 left mathop smallint limits 0 infty frac r dr v r right 1 wherev r m 0 x f r from these estimates one easily obtains lower estimates for the green function as well as sharp criteria for polarity and for recurrence the latter we apply to the question of hitting the nodal set 0 for the process associated with the operator delta 2 tfrac nabla varphi varphi nabla and to recurrence questions for divergence form operators a ond n witha a ij which degenerate at d

solving boundary value problems on networks using equilibrium measures abstract the purpose of this paper is to construct solutions of self adjoint boundary value problems on finite networks to this end we obtain explicit expressions of the green functions for all different boundary value problems the method consists of reducing each boundary value problem either to a dirichlet problem or to a poisson equation on a new network closely related with the former boundary value problem in this process we also get an explicit expression of the poisson kernel for the dirichlet problem in all cases we express the green function in terms of equilibrium measures solely which can be obtained as the unique solution of linear programming problems in particular we get analytic expressions of the green function for the following problems the poisson equation on a distance regular graph the dirichlet problem on an infinite distance regular graph and the neumann problem on a ball of an homogeneous tree

approximate bayes factors and accounting for model uncertainty in generalised linear models summary ways of obtaining approximate bayes factors for generalised linear models are described based on the laplace method for integrals we propose a new approximation which uses only the output of standard computer programs for estimating generalised linear models this appears to be quite accurate a reference set of proper priors is suggested both to represent the situation where there is not much prior information and to assess the sensitivity of the results to the prior distribution the methods can be used when the dispersion parameter is unknown when there is overdispersion to compare link functions and to compare error distributions and variance functions the methods can be used to implement the bayesian approach to accounting for model uncertainty we describe an application to inference about relative risks in the presence of control factors where model uncertainty is large and important software to implement the methods is available at no cost from statlib

random walk models for multifractals reversible random walk models for bernoulli multifractals in dimension 2 together with marginal versions of these are first studied to account for both geometrical and analytical properties of such objects using large deviation theory it is also shown how the transience recurrence of a more complex reversible random walk is related to the minimum energy flow convergence divergence property new results on the asymptotics of such a walker are then derived in both transient recurrent situations

graphical designs and extremal combinatorics abstract a graphical design is a proper subset of vertices of a graph on which many eigenfunctions of the laplacian operator have mean value zero in this paper we show that extremal independent sets make extremal graphical designs that is a design on which the maximum possible number of eigenfunctions have mean value zero we then provide examples of such graphs and sets which arise naturally in extremal combinatorics we also show that sets which realize the isoperimetric constant of a graph make extremal graphical designs and provide examples for them as well we investigate the behavior of graphical designs under the operation of weak graph product in addition we present a family of extremal graphical designs for the hypercube graph

random walks in random dirichlet environment are transient in dimension d 3 we consider random walks in random dirichlet environment rwde which is a special type of random walks in random environment where the exit probabilities at each site are i i d dirichlet random variables on z d rwde are parameterized by a 2d uplet of positive reals we prove that for all values of the parameters rwde are transient in dimension d ge 3 we also prove that the green function has some finite moments and we characterize the finite moments our result is more general and applies for example to finitely generated symmetric transient cayley graphs in terms of reinforced random walks it implies that directed edge reinforced random walks are transient for d ge 3

function theory random paths and covering spaces pour des varietes de riemann connexes m on etudie la relation entre la theorie de fonction harmonique sur m les proprietes statistiques des chemins aleatoires sur m et la structure geometrique globale de m

character tables and defect groups abstract let b be a block of a finite group g with defect group d we prove that the exponent of the center of d is determined by the character table of g in particular we show that d is cyclic if and only if b contains a large family of irreducible p conjugate characters more generally for abelian d we obtain an explicit formula for the exponent of d in terms of character values in small cases even the isomorphism type of d is determined in this situation moreover it can read off from the character table whether d d 4 where d denotes the commutator subgroup of d we also propose a new characterization of nilpotent blocks in terms of the character table

rational irreducible characters and rational conjugacy classes in finite groups we prove that a finite group g has two rational valued irreducible characters if and only if it has two rational conjugacy classes and determine the structure of any such group along the way we also prove a conjecture of gow stating that any finite group of even order has a non trivial rational valued irreducible character of odd degree

characters of relative p degree over normal subgroups let z be a normal subgroup of a finite group g let irr z be an irreducible complex character of z and let p be a prime number if p does not divide the integers 1 1 for all irr g lying over then we prove that the sylow p subgroups of g z are abelian this theorem which generalizes the gluck wolf theorem to arbitrary finite groups is one of the principal obstacles to proving the celebrated brauer height zero conjecture

every covering of a compact riemann surface of genus greater than one carries a nontrivial l2 harmonic differential this paper contains the proof of the assertion in its title my motivation for considering this problem was the following let m be a compact differentiable manifold and let d be an elliptic differential operator between the spaces of c sections of two bundles on m in 4 m f atiyah proves that if index d 0 then for every galois covering at m the operator induced by d on has nontrivial l 2 solutions u 0 atiyah goes on to ask whether the same is true for coverings which are not galois his guess is that the answer is negative in general but that the counter example will be difficult to construct the simplest situation to consider in this connection is that of a surface and the operator whose index is the euler characteristic for infinite coverings since every l 2 harmonic function would be constant and nonzero constants are not in l 2 atiyah s question reduces to the question of existence of l 2 harmonic forms of degree one it is shown here that a counter example will not be found in this simple setting from now on s will denote an oriented compact surface of genus g g s l equipped with a smooth riemannian metric s s will be an arbitrary usually infinite covering of s and a a will be the laplace operator on s with respect to the pull back metric a differential exterior form of degree one on s is harmonic i e ao 0 and in l 2 if and only if cf 15 theorem 26

the mckay conjecture and galois automorphisms the main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group g can be computed locally the simplest of these conjectures is the mckay conjecture which asserts that the number of irreducible complex characters of g of degree not divisible by p is the same if computed in a p sylow normalizer of g in this paper we propose a much stronger version of this conjecture which deals with galois automorphisms in fact the same idea can be applied to the celebrated alperin and dade conjectures

instability of the liouville property for quasi isometric riemannian manifolds and reversible markov chains on montre par un exemple que la propriete de liouville n est pas un invariant quasi isometrique des varietes de riemann et des chaines de markov reversibles

isoperimetric inequalities and transient random walks on graphs recurrent we show that any graph satisfying an isoperimetric inequality only slightly stronger than that of z2 is transient more precisely if f k denotes the smallest number of vertices in the boundary of a connected subgraph with k vertices then the graph is transient if the infinite sum e f k 2 converges this can be applied to parabolicity versus hyperbolicity of surfaces 1 introduction let g be a connected graph which is locally finite that is all vertices have finite degree we consider a random walk starting at a vertex v say such that at any vertex u the walk proceeds to a neighbour with probability 1 d u where d u is the degree i e the number of neighbours of u the graph g is recurrent if we revisit v with probability 1 otherwise g is transient it is well known that the three dimensional grid z3 is transient while z2 is recurrent more generally nash williams 13 see also 4 10 proved that any graph with smaller growth rate than z2 is recurrent lyons 10 showed that certain subgraphs of grids are transient provided they grow just a little faster than z2 other results in terms of isoperimetric inequalities supporting the statement that z2 is in a sense an extreme recurrent graph can be derived from work of fernandez 5 grigor yan 7 and varopoulos 15 varopoulos 16 used results of gromov 8 to characterize completely the recurrent cayley graphs we shall carry these results further if v is a vertex set in g then av will denote the boundary of v that is the set of vertices of v having neighbours outside of v let f be a nondecreasing positive real function defined on the natural numbers we say that g satisfies an f isoperimetric inequality if there exists a constant c 0 such that for each finite vertex set v of g

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

random walks for image segmentation a novel method is proposed for performing multilabel interactive image segmentation given a small number of pixels with user defined or predefined labels one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels by assigning each pixel to the label for which the greatest probability is calculated a high quality image segmentation may be obtained theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits this algorithm is formulated in discrete space i e on a graph using combinatorial analogues of standard operators and principles from continuous potential theory allowing it to be applied in arbitrary dimension on arbitrary graphs

the boundary of a square tiling of a graph coincides with the poisson boundary answering a question of benjamini and schramm ann probab 24 3 1219 1238 1996 we show that the poisson boundary of any planar uniquely absorbing e g one ended and transient graph with bounded degrees can be realised geometrically as a circle which arises from a discrete version of riemann s mapping theorem this implies a conjecture of northshield potential anal 2 4 299 314 1993 some of our technique apply to the non planar case and might have further applications when the graph is also hyperbolic then under mild conditions we prove the equivalence of several boundary constructions

recurrence of planar graph limits we prove that any distributional limit of nite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent as a corollary we obtain that the uniform innite planar triangulation and quadrangulation uipt and uipq are almost surely recurrent resolving a conjecture of angel benjamini and schramm we also settle another related problem of benjamini and schramm we show that in any bounded degree nite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for t steps is at most c logt

diffusions and random shadows in negatively curved manifolds abstract let m be a d dimensional complete simply connected negatively curved manifold there is a natural notion of hausdorff dimension for its boundary at infinity this is shown to provide a notion of global curvature or average rate of growth in two probabilistic senses first on surfaces d 2 it is twice the critical drift separating transience from recurrence for brownian motion with constant length radial drift equivalently it is twice the critical for the existence of a green function for the operator 2 r second for any d it is the critical intensity for almost sure coverage of the boundary by random shadows cast by balls appropriately scaled produced from a constant intensity poisson point process

some characterizations of the weil representations of the symplectic and unitary groups abstract let g be a finite symplectic or unitary group we characterize the weil representations of g via their restriction to a standard subgroup then we complete the determination of complex representations of g with specific minimal polynomials of certain elements by showing that they coincide with the weil representations

cohomology of noncompact surfaces this paper is motivated by the question of whether nonzero l2 harmonic differentials exist on coverings of a riemann surface of genus 2 our approach will be via an analogue of the de rham theorem some results concerning the invariance of l2 homology and the intersection number of lu cycles are demonstrated a growth estimate for triangulations of planar coverings of the two holed torus is derived finally the equivalence between the existence of l2 harmonic one cycles and the transience of random walks on a planar surface is shown

collapsibility of logistic regression coefficients summary in this paper we discuss collapsibility of logistic regression coefficients over a background variable and present necessary and sufficient conditions for collapsibility

eaton s markov chain its conjugate partner and mathscr p admissibility suppose that x is a random variable with density f x and that x is a proper posterior corresponding to an improper prior the prior is called p admissible if the generalized bayes estimator of every bounded function of is almost admissible under squared error loss eaton showed that recurrence of the markov chain with transition density r x f x dx is a sufficient condition for p admissibility of we show that eaton s markov chain is recurrent if and only if its conjugate partner with transition density r y x f y x d is recurrent this provides a new method of establishing p admissibility often one of these two markov chains corresponds to a standard stochastic process for which there are known results on recurrence and transience for example when x is poisson and an improper gamma prior is placed on the markov chain defined by r y x is equivalent to a branching process with immigration we use this type of argument to establish p admissibility of some priors when f is a negative binomial mass function and when f is a gamma density with known shape

a sears type self adjointness result for discrete magnetic schrodinger operators abstract in the context of a weighted graph with vertex set v and bounded vertex degree we give a sufficient condition for the essential self adjointness of the operator w where is the magnetic laplacian and w v r is a function satisfying w x q x for all x v with q v 1 the condition is expressed in terms of completeness of a metric that depends on q and the weights of the graph the main result is a discrete analogue of the results of i oleinik and m a shubin in the setting of non compact riemannian manifolds

harmonic functions on planar and almost planar graphs and manifolds via circle packings the circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant harmonic bounded dirichlet function if p is a bounded circle packing in bbb r 2 whose contacts graph is a bounded valence triangulation of a disk then with probability 1 the simple random walk on p converges to a limit point moreover in this situation any continuous function on the limit set of p extends to a continuous harmonic function on the closure of the contacts graph of p that is this dirichlet problem is solvable we define the notions of almost planar graphs and manifolds and show that under the assumptions of transience and bounded local geometry these possess non constant harmonic bounded dirichlet functions let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph

prior induction in log linear models for general contingency table analysis log linear modelling plays an important role in many statistical applications particularly in the analysis of contingency table data with the advent of powerful new computational techniques such as reversible jump mcmc bayesian analyses of these models and in particular model selection and averaging have become feasible coupled with this is the desire to construct and use suitably flexible prior structures which allow efficient computation while facilitating prior elicitation the latter is greatly improved in the case where priors can be specified on interpretable parameters about which relevant experts can express their beliefs in this paper we show how the specification of a general multivariate normal prior on the log linear parameters induces a multivariate lognormal prior on the corresponding cell counts of a contingency table we derive the parameters of this distribution in an explicit practical form and state the corresponding mean and covariances of the cell counts we discuss the importance of these results in terms of applying both uninformative and informative priors to the model parameters and provide an illustration in the context of the analysis of a 2 3 contingency table 1 introduction the analysis of general k way contingency table data is of interest in a wide variety of areas of statistical application model selection is notoriously difficult in suchsituations since th e number of models rises doubly exponentially withdimension obviously exhaustive comparisons are impossible but various computational techniques have been proposed for obtaining posterior model probabilities for problems of this sort depending upon the parameters upon which the analyst wishes to express prior opinions with the introduction of powerful new computational techniques these forms of analysis have been greatly simplified and are becoming increasingly common in the applied literature in undertaking a bayesian analysis of contingency table data it is necessary to specify priors either for the cell counts which could alternatively be expressed in terms of the cell probabilities and total cell count or equivalently the log linear parameters madigan and york 1997 choose to choose to place hyper dirichlet priors dawid and lauritzen 1993 on the cell probabilities which has the advantage that priors of this form allow a factorization of the likelihood through the identification of cliques within the corresponding model graph giudici green and tarantola 1999 illustrate how this decomposition

canonical measures on metric graphs and a kazhdan s theorem we extend the notion of canonical measures to all possibly non compact metric graphs this will allow us to introduce a notion of hyperbolic measures on universal covers of metric graphs kazhdan s theorem for riemann surfaces describes the limiting behavior of canonical arakelov measures on finite covers in relation to the hyperbolic measure we will prove a generalized version of this theorem for metric graphs allowing any infinite galois cover to replace the universal cover we will show all such limiting measures satisfy a version of gauss bonnet formula which using the theory of von neumann dimensions can be interpreted as a trace formula in the special case where the infinite cover is the universal cover we will provide explicit methods to compute the corresponding limiting hyperbolic measure our ideas are motivated by non archimedean analytic and tropical geometry

random walk on the speiser graph of a riemann surface on considere la determination du type conforme d une surface de recouvrement de la sphere de riemann a n points on montre comment definir une marche aleatoire sur les sommets du graphe de speiser de sorte que la marche aleatoire soit transitoire si et seulement si la surface est hyperbolique

incorporating prior information into the analysis of contingency tables contingency tables are often analyzed using log linear models and in some situations prior information on the value of parameters in the log linear model is available in this article we describe a prior posterior procedure that incorporates prior information directly into the analysis through a multivariate normal prior for the log linear parameters the mode and curvature of the posterior density are proposed as summary statistics

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

model determination for categorical data with factor level merging we deal with contingency table data that are used to examine the relationships between a set of categorical variables or factors we assume that such relationships can be adequately described by the conditional independence structure that is imposed by an undirected graphical model if the contingency table is large a desirable simplified interpretation can be achieved by combining some categories or levels of the factors we introduce conditions under which such an operation does not alter the markov properties of the graph implementation of these conditions leads to bayesian model uncertainty procedures based on reversible jump markov chain monte carlo methods the methodology is illustrated on a 2 x 3 x 4 and up to a 4 x 5 x 5 x 2 x 2 contingency table

disorder entropy and harmonic functions we study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on z d we prove that the vector space of harmonic functions growing at most linearly is d 1 dimensional almost surely in particular there are no non constant sublinear harmonic functions thus implying the uniqueness of the corrector the main ingredient of the proof is a quantitative annealed version of the kaimanovich vershik entropy argument this also provides bounds on the derivative of the heat kernel simplifying and generalizing existing results the argument applies to many different environments even reversibility is not necessary we also mention several open problems and conjectures on the behavior of harmonic functions on stationary random graphs

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

critical phenomena for spitzer s reversible nearest particle systems motivated by several results and open problems concerning harris basic contact process we consider the relationship between the critical behavior of the finite and infinite versions of spitzer s reversible nearest particle systems we show that the critical values for the finite and infinite systems agree but that the behavior of the two systems at the common critical value can differ the nash williams recurrence criterion for reversible markov chains is an important tool used in the proofs of the main results and we give a new treatment of that theory finally we compute several critical exponents for the nearest particle systems

markov chain monte carlo model determination for hierarchical and graphical log linear models we use reversible jump markov chain monte carlo methods green 1995 to develop strategies for calculating posterior probabilities of hierarchical graphical or decomposable log linear models for high dimensional contingency tables even for tables of moderate size these sets of models may be very large the choice of suitable prior distributions for model parameters is also discussed in detail and two examples are presented for the first example a three way table the model probabilities calculated using our reversible jump approach are compared with model probabilities calculated exactly or by using an alternative approximation the second example is a six way contingency table for which exact methods are infeasible because of the large number of possible models we identify the most probable hierarchical graphical and decomposable models and compare the results with alternatives approaches

strongly subharmonic functions graphs and their asymptotic growth let g be an infinite connected graph with uniformly bounded vertex degree and let denote the laplace operator corresponding to the simple random walk on it in this paper we obtain relations between the structure of the graph and the qualitative behaviour of the class of functions u satisfying u b 0 namely we relate the asymptotic growth of the function u and that of the cardinality of balls in g

nearest neighbor walks with low predictability profile and percolation in 2 epsilon dimensions a few years ago grimmett kesten and zhang proved that for supercritical bond percolation on z 3 simple random walk on the infinite cluster is a s transient we generalize this result to a class of wedges in z 3 including for any e 0 1 the wedge w e x y z z 3 x 0 z x e which can be thought of as representing a 2 e dimensional lattice our proof builds on recent work of benjamini pemantle and peres and involves the construction of finite energy flows using nearest neighbor walks on z with low predictability profile along the way we obtain some new results on attainable decay rates for predictability profiles of nearest neighbor walks

degrees of rational characters of finite groups abstract a classical theorem of john thompson on character degrees states that if the degree of any complex irreducible character of a finite group g is 1 or divisible by a prime p then g has a normal p complement in this paper we consider fields of values of characters and prove some improvements of this result

classification on the average of random walks during the last decade many attempts have been made to characterize absence of spontaneous breaking of continuous symmetry for the heisenberg model on graphs by using suitable classifications of random walks refs 4 and 10 we propose and study a new type problem for random walks on graphs which is particularly interesting for disordered graphs we compare this classification with the classical one and with an analogous one introduced in ref 4 various examples that are not space homogeneous are provided

2 brauer correspondent blocks with one simple module abstract one of the main problems in representation theory is to understand the exact relationship between brauer corresponding blocks of finite groups the case where the local correspondent has a unique simple module seems key we study this situation for 2 blocks

unpredictable nearest neighbor processes benjamini pemantle and peres constructed nearest neighbor processes which have predictability profiles that decay faster than that of the simple random walk haggstrom and mossel found processes with even faster decaying predictability profiles we prove that the rate of decay achieved by haggstrom and mossel is optimal

unimodular hyperbolic triangulations circle packing and random walk we show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six if and only if the triangulation is amenable in the sense of aldous and lyons 1 as a part of this we obtain an alternative proof of the benjamini schramm recurrence theorem 19 secondly in the hyperbolic case we prove that the random walk almost surely converges to a point in the unit circle that the law of this limiting point has full support and no atoms and that the unit circle is a realisation of the poisson boundary finally we show that the simple random walk has positive speed in the hyperbolic metric

asymptotic behavior of the transition probability of a random walk on an infinite graph ideas cultivated in spectral geometry are applied to obtain an asymptotic property of a reversible random walk on an infinite graph satisfying a certain periodic condition in the course of our argument we employ perturbation theory for the maximal eigenvalues of twisted transition operator as a result an asymptotic of the probabilityp n x y that a particle starting atxreachesyat timenasngoes to infinity is established

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

testing collapsibility in contingency tables on etudie en particulier le cas important des tables 2 2 k d ou il decoule les extensions aux tables plus grandes

transient random walks on graphs and metric spaces with applications to hyperbolic surfaces we introduce an r net 0 2r r of a metric space m as a maximal graph whose vertices are elements in m of pairwise distance at least r such that any two vertices of distance at most r are adjacent we show that for a large class of metric spaces including many riemannian manifolds the property of transience of a net and the property of the net carrying a non constant harmonic function of bounded energy is independent of the choice of the net we give a new necessary and sufficient condition for a graph with bounded degrees and satisfying an isoperimetric inequality to have no non constant harmonic functions for this purpose we develop equivalent analytic conditions for graphs satisfying an isoperimetric inequality some of these results have been discovered recently by others in more general settings but our treatment here is specific and self contained we use graph transience to prove that scherk s surface is hyperbolic a problem posed by osserman in 1965

recurrence and transience of the edge graph of a tiling of the euclidean plane suppose that is an infinite connected graph such that every vertex v has finite degree d v let 7b v and 7b e denote the vertex and edge set of respectively it is a standard procedure to associate with a markov chain with state space 7b v and probability d v 1 of moving from v to any neighbour v the graph is called recurrent or transient depending on whether the associated markov chain is recurrent or transient the main result of this paper states that the edge graph of a quasi normal tiling of the plane is recurrent this result can be viewed as an extension of polya s theorem on the recurrence of the square lattice of the plane

random walks and harmonic functions on infinite planar graphs using square tilings we study a wide class of transient planar graphs through a geometric model given by a square tiling of a cylinder for many graphs the geometric boundary of the tiling is a circle and is easy to describe in general the simple random walk on the graph converges with probability 1 to a point in the geometric boundary we obtain information on the harmonic measure and estimates on the rate of convergence this allows us to extend results we previously proved for triangulations of a disk

extensions of unipotent characters and the inductive mckay condition abstract we prove that unipotent characters of groups of lie type extend to their inertia groups in the full automorphism group we then apply this in order to show that certain groups of lie type are good for the prime p 3 or p 2 this deals with all the exceptions not covered by our general approach for verifying the mckay conjecture for all groups of lie type in non defining characteristic

groups whose real irreducible characters have degrees coprime to p abstract in this paper we study groups for which every real irreducible character has degree not divisible by some given odd prime p

admissible estimation dirichlet principles and recurrence of birth death chains on ℤ p we establish a connection between admissible simultaneous estimation and recurrence of reversible markov chains on specifically to each generalized bayes estimator of the mean of a vector of p independent poisson variables for a weighted quadratic loss we associate a variational problem and a reversible birth and death chain on the variational problem is closely related to the dirichlet principle for reversible chains studied recently by griffeath liggett and lyons under side conditions admissibility of the estimator is equivalent to zero infimal energy in the variational problem and to recurrence of the markov chain this yields analytic and probabilistic criteria for inadmissibility which are applied to establish a broad class of results and previous conjectures