on the eisenstein cohomology of arithmetic groups the cohomology h gamma e of an arithmetic subgroup gamma of a connected reductive algebraic group g defined over some algebraic number field f can be interpreted in terms of the automorphic spectrum of gamma with this framework in place there is a sum decomposition of the cohomology into the cuspidal cohomology i e classes represented by cuspidal automorphic forms for g and the so called eisenstein cohomology constructed as the span of appropriate residues or derivatives of eisenstein series attached to cuspidal automorphic forms on the levi components of proper parabolic f subgroups of g the main objective of this paper is to isolate a specific structural part in the eisenstein cohomology this pertains to regular eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered it is shown that the cohomological degree of these classes is bounded from below by the constant q 0 g mathbb r 1 2 dim x g mathbb r rk g mathbb r rk k mathbb r where k mathbb r denotes a maximal compact subgroup of the real lie group g mathbb r where x g mathbb r is the associated symmetric space this investigation has various applications one of these is a vanishing result for the cohomology in the generic case i e where the representation determining the coefficient system e has regular highest weight in degrees below q 0 g mathbb r this is a sharp bound depending only on the underlying real lie group g mathbb r corollary 5 6 proposition 5 8 this result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular eisenstein cohomology classes

the μ ordinary hasse invariant of unitary shimura varieties we construct a generalization of the hasse invariant for any shimura variety of pel type a over a prime of good reduction whose vanishing locus is the open and dense mu ordinary locus

hierarchical pinning models quadratic maps and quenched disorder we consider a hierarchical model of polymer pinning in presence of quenched disorder introduced by derrida et al j stat phys 66 1189 1213 1992 which can be re interpreted as an infinite dimensional dynamical system with random initial condition the disorder it is defined through a recurrence relation for the law of a random variable r n n 1 2 which in absence of disorder i e when the initial condition is degenerate reduces to a particular case of the well known logistic map the large n limit of the sequence of random variables 2 n log r n a non random quantity which is naturally interpreted as a free energy plays a central role in our analysis the model depends on a parameter 0 1 related to the geometry of the hierarchical lattice and has a phase transition in the sense that the free energy is positive if the expectation of r 0 is larger than a certain threshold value and it is zero otherwise it was conjectured in derrida et al j stat phys 66 1189 1213 1992 that disorder is relevant respectively irrelevant or marginally relevant if 1 2 1 2 we find the correct scaling form for weak disorder of the critical point shift

on the cuspidal cohomology of arithmetic groups it is the main objective of this paper to show a vanishing result for cuspidal cohomology of arithmetic groups in classical groups g defined over some number field k our approach relies on the fact that cuspidal automorphic forms are nonsingular in the sense of howe this result puts a strong constraint on the archimedean components of irreducible cuspidal automorphic representations of g that can possibly contribute to the cuspidal cohomology combining this with the classification of vogan zuckerman of unitary representations with nonzero cohomology provides a constant r 0 g k only depending on g k below which the cuspidal cohomology vanishes we will give a formula for this constant for each classical group of type i in the classification scheme due to weil we conclude with making this result explicit for some split classical groups over a totally real algebraic number field

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

sub gaussian estimates of heat kernels on infinite graphs we prove that a two sided sub gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the green kernel admits a uniform polynomial decay

cubic structures equivariant euler characteristics and lattices of modular forms we use the theory of cubic structures to give a fixed point riemann roch formula for the equivariant euler characteristics of coherent sheaves on projective at schemes over z with a tame action of a finite abelian group this formula supports a conjecture concerning the extent to which such equivariant euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus our results are applied to study the module structure of modular forms having fourier coeficients in a ring of algebraic integers as well as the action of diamond hecke operators on the mordell weil groups and tate shafarevich groups of jacobians of modular curves

a strong law of large numbers for iterated functions of independent random variables we study sequences of random variables obtained by iterative procedures which can be thought of as nonlinear generalizations of the arithmetic mean we prove a strong law of large numbers for a class of such iterations this gives rise to the concept of generalized expected value of a random variable for which we prove an analog of the classical jensen inequality we give several applications to models arising in mathematical physics and other areas

partial regularity and amplitude the author continues the study of frobenius amplitude introduced in an earlier paper and compares this to various other positivity notions these are applied to deduce vanishing theorems a refinement of castelnuovo mumford regularity is also given

estimates of transition densities for brownian motion on nested fractals we obtain upper and lower bounds for the transition densities of brownian motion on nested fractals compared with the estimate on the sierpinski gasket the results require the introduction of a new exponent dj related to the shortest path metric and chemical exponent on nested fractals further holder order of the resolvent densities sample paths and local times are obtained the results are obtained using the theory of multi type branching processes

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

frobenius amplitude and strong vanishing theorems for vector bundles the primary goal of this paper is to systematically exploit the method of deligne illusie to obtain kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties the key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the frobenius or f amplitude the f amplitude enters into the statement of the basic vanishing theorem and this leads to the problem of calculating or at least estimating this number most of the work in this paper is devoted to doing this various situations

vanishing theorems for torsion automorphic sheaves on general pel type shimura varieties abstract we consider the interior cohomology and the hodge graded pieces in the case of the de rham realization of general not necessarily compact pel type shimura varieties with coefficients in the local systems corresponding to sufficiently regular algebraic representations of the associated reductive group for primes p bigger than an effective bound we prove that the f p and z p cohomology groups are concentrated in the middle degree that the z p cohomology groups are free of p torsion and that every f p cohomology class lifts to a z p cohomology class

polymers on disordered hierarchical lattices a nonlinear combination of random variables the problem of directed polymers on disordered hierarchical and hypercubic lattices is considered for the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way we present the results of numerical simulations of two hierarchical lattices finding evidence of a phase transition in one case for a limiting case we extend the perturbation theory developed by derrida and griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties overlaps of the model in this limit we obtain an interpolation formula allowing one to obtain the noninteger moments of the partition function from the integer moments we obtain bounds for the transition temperature for hierarchical and hypercubic lattices and some similarities between the problem on the two different types of lattice are discussed

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

vibration modes of 3n gaskets and other fractals we rigorously study eigenvalues and eigenfunctions vibration modes on the class of self similar symmetric finitely ramified fractals which include the sierpinski gasket and other 3n gaskets we consider the classical laplacian on fractals which generalizes the usual one dimensional second derivative is the generator of the self similar diffusion process and has possible applications as the quantum hamiltonian we develop a theoretical matrix analysis including analysis of singularities which allows us to compute eigenvalues eigenfunctions and their multiplicities exactly we support our theoretical analysis by symbolic and numerical computations our analysis in particular allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues i e integrated density of states we consider such examples as the level 3 sierpinski gasket a fractal 3 tree and the diamond fractal

on the generic part of the cohomology of compact unitary shimura varieties the goal of this paper is to show that the cohomology of compact unitary shimura varieties is concentrated in the middle degree and torsion free after localizing at a maximal ideal of the hecke algebra satisfying a suitable genericity assumption along the way we establish various foundational results on the geometry of the hodge tate period map in particular we compare the fibres of the hodge tate period map with igusa varieties

heat kernel fluctuations for a resistance form with non uniform volume growth in this article we consider the problem of estimating the heat kernel on measure metric spaces equipped with a resistance form such spaces admit a corresponding resistance metric that reflects the conductivity properties of the set in this situation it has been proved that when there is uniform polynomial volume growth with respect to the resistance metric the behaviour of the on diagonal part of the heat kernel is completely determined by this rate of volume growth however recent results have shown that for certain random fractal sets there are global and local point wise fluctuations in the volume as r 0 and so these uniform results do not apply motivated by these examples we present global and local on diagonal heat kernel estimates when the volume growth is not uniform and demonstrate that when the volume fluctuations are non trivial there will be non trivial fluctuations of the same order up to exponents in the short time heat kernel asymptotics we also provide bounds for the off diagonal part of the heat kernel these results apply to deterministic and random self similar fractals and metric space dendrites the topological analogues of graph trees

potential automorphy and the leopoldt conjecture in the period during which this research was conducted jack thorne served as a clay research fellow chandrashekhar khare was supported by nsf grant dms 1161671 and by a humboldt research award and thanks the tata instititute of fundamental research mumbai for hospitality during the period in which most of the work on this paper was done

semiregularity and obstructions of complete intersections abstract we prove that on a smooth projective variety over an algebraically closed field of characteristic 0 the semiregularity map annihilates every obstruction to embedded deformations of a local complete intersection subvariety with extendable normal bundle the proof is based on the theory of l algebras and tamarkin tsygan calculus on the de rham complex of dg schemes

universal eigenvarieties trianguline galois representations and p adic langlands functoriality using the overconvergent cohomology modules introduced by ash and stevens we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces building on work of ash stevens urban and others we also formulate a precise modularity conjecture linking trianguline galois representations with overconvergent cohomology classes in the course of giving evidence for this conjecture we establish several new instances of p adic langlands functoriality our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of chenevier s interpolation theorem

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

volume growth and heat kernel estimates for the continuum random tree in this article we prove global and local point wise volume and heat kernel bounds for the continuum random tree we demonstrate that there are almost surely logarithmic global fluctuations and log logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r 0 we also show that the on diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t 0 almost surely finally we prove that this quenched almost sure behaviour contrasts with the local annealed averaged over all realisations of the tree volume and heat kernel behaviour which is smooth

stationary random metrics on hierarchical graphs via min type recursive distributional equations this paper is inspired by the problem of understanding in a mathematical sense the liouville quantum gravity on surfaces here we show how to define a stationary random metric on self similar spaces which are the limit of nice finite graphs these are the so called hierarchical graphs they possess a well defined level structure and any level is built using a simple recursion stopping the construction at any finite level we have a discrete random metric space when we set the edges to have random length using a multiplicative cascade with fixed law m we introduce a tool the cut off process by means of which one finds that renormalizing the sequence of metrics by an exponential factor they converge in law to a non trivial metric on the limit space such limit law is stationary in the sense that glueing together a certain number of copies of the random limit space according to the combinatorics of the brick graph the obtained random metric has the same law when rescaled by a random factor of law m in other words the stationary random metric is the solution of a distributional equation when the measure m has continuous positive density on mathbf r the stationary law is unique up to rescaling and any other distribution tends to a rescaled stationary law under the iterations of the hierarchical transformation we also investigate topological and geometric properties of the random space when m is log normal detecting a phase transition influenced by the branching random walk associated to the multiplicative cascade

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

a quotient of the lubin tate tower ii in this article we construct the quotient mathcal m mathbf 1 p k of the infinite level lubin tate space mathcal m mathbf 1 by the parabolic subgroup p k subset mathrm gl n k of block form n 1 1 as a perfectoid space generalizing the results of ludwig forum math sigma 5 e17 41 2017 to arbitrary n and k mathbb q p finite for this we prove some perfectoidness results for certain harris taylor shimura varieties at infinite level as an application of the quotient construction we show a vanishing theorem for scholze s candidate for the mod p jacquet langlands and mod p local langlands correspondence an appendix by david hansen gives a local proof of perfectoidness of mathcal m mathbf 1 p k when n 2 and shows that mathcal m mathbf 1 q k is not perfectoid for maximal parabolics q not conjugate to p

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

dirac and magnetic schrodinger operators on fractals abstract in this paper we define local dirac operators and magnetic schrodinger hamiltonians on fractals and prove their essential self adjointness to do so we use the concept of 1 forms and derivations associated with dirichlet forms as introduced by cipriani and sauvageot and further studied by the authors jointly with rockner ionescu and rogers for simplicity our definitions and results are formulated for the sierpinski gasket with its standard self similar energy form we point out how they may be generalized to other spaces such as the classical sierpinski carpet

brownian motion on a random recursive sierpinski gasket we introduce a random recursive fractal based on the sierpinski gasket and construct a diffusion upon the fractal via a dirichlet form this form and its symmetrizing measure are determined by the electrical resistance of the fractal the effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion the main result is to obtain uniform upper and lower bounds for the transition density of the brownian motion on the fractal in terms of this metric the bounds are not tight as there are logarithmic corrections due to the randomness in the environment and the behavior of the shortest paths in the effective resistance metric is not well understood the results are deduced from the study of a suitable general branching process

a relation between the parabolic chern characters of the de rham bundles in this paper we consider the weight i de rham gauss manin bundles on a smooth variety arising from a smooth projective morphism f x u longrightarrow u for i geq 0 we associate to each weight i de rham bundle a certain parabolic bundle on s and consider their parabolic chern characters in the rational chow groups for a good compactification s of u we show the triviality of the alternating sum of these parabolic bundles in the positive degree rational chow groups this removes the hypothesis of semistable reduction in the original result of this kind due to esnault and viehweg

high temperature scaling limit for directed polymers on a hierarchical lattice with bond disorder diamond lattices are sequences of recursively defined graphs that provide a network of directed pathways between two fixed root nodes a and b the construction recipe for diamond graphs depends on a branching number b in mathbb n and a segmenting number s in mathbb n for which a larger value of the ratio s b intuitively corresponds to more opportunities for intersections between two randomly chosen paths by attaching i i d random variables to the bonds of the graphs we construct a random gibbs measure on the set of directed paths by assigning each path an energy through summing the random variables along the path for the case b s we propose a scaling regime in which the temperature grows along with the number of hierarchical layers of the graphs and the partition function the normalization factor of the gibbs measure appears to converge in law we prove that all of the positive integer moments of the partition function converge in this limiting regime the motivation of this work is to prove a functional limit theorem that is analogous to a previous result obtained in the b s case

the residual eisenstein cohomology of sp_ 4 over a totally real number field let g sp4 k be the k split symplectic group of k rank 2 where k is a totally real numbereld in this paper we compute the eisenstein cohomology of g with respect to anynite dimensional irreducible k rational representation e of g1 rk qg r where rk q denotes the restriction of scalars from k to q the approach is based on the work of schwermer regarding the eisenstein cohomology for sp4 q kim s description of the residual spectrum of sp4 and the frankeltration of the space of automorphic forms in fact taking the representation theoretic point of view we write for the group g the frankeltration with respect to the cuspidal support and give a precise description of theltration quotients in terms of induced representations this is then used as a prerequisite for the explicit computation of the eisenstein cohomology the special focus is on the residual eisenstein cohomology under a certain compatibility condition for the coefficient system e and the cuspidal support we prove the existence of non trivial residual eisenstein cohomology classes which are not square integrabl e that is represented by a non square integrabl residue of an eisenstein series

eigenvarieties for reductive groups we develop the theory of overconvergent cohomology introduced by g stevens and we use it to give a construction of eigenvarieties associated to any reductive group g over q such that g r has discrete series we prove that the so called eigenvarieties are equidimensional and generically at over the weight space

on the cohomology groups of local systems over hilbert modular varieties via higgs bundles let x be a hilbert modular variety and v a non trivial local system over x in this paper we study deligne saito s mixed hodge structure mhs on the cohomology groups h k x v using the method of higgs bundles among other results we give a dimension formula for the hodge numbers and show that the mixed hodge structure is split over r these results are analogous to 18 in the cocompact case and complement the results in 8 for constant coefficients

on the galois structure of equivariant line bundles on curves let it be a finite field and let x be a smooth protective curve over k with structure sheaf o let g be a finite group and write cl 0 g for the reduced grothendieck group of the category of g vector bundles in this paper we describe explicitly the subgroup of cl 0 g which is generated by the classes arising from g stable invertible sheaves on tame galois covers of x which have galois group g introduction let k be a finite field of characteristic p9 and let g be a finite abelian group suppose that y x is a tamely ramified galois covering of smooth projective curves over k with galois group g we shall refer to such coverings as tame g covers of x in this paper we study the structure of g stable line bundles on such curves y in order to be more precise we let cl ox g 9 respectively cl k g 9 denote the reduced grothendieck group of x g vector bundles respectively of finitely generated fc g modules which are cohomologically trivial for g suppose that a is a g stable line bundle on y then 4 is an ox g vector bundle and it gives rise to a class f a g c ox g we shall say that an element of c1 0x g is realizable if it may be obtained in this manner for some choice of y and a we shall combine techniques of c m and bu to obtain an explicit description of the subgroup 9 x g of cl ox g which is generated by the subset of realizable classes in conjunction with a riemann roch theorem these results are sufficient to describe explicitly those elements of c k g which arise as the image of the vector bundles a under the refined euler poincare characteristic

explicit formulas for heat kernels on diamond fractals this paper provides explicit pointwise formulas for the heat kernel on compact metric measure spaces that belong to a mathbb n times mathbb n parameter family of fractals which are regarded as projective limits of metric measure graphs and do not satisfy the volume doubling property the formulas are applied to obtain uniform continuity estimates of the heat kernel and to derive an expression of the fundamental solution of the free schrodinger equation the results also open up the possibility to approach infinite dimensional spaces based on this model

diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice we construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal a dirichlet form can be constructed on the limit set and we consider the properties of the associated laplace operator and diffusion process in particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on diagonal heat kernel for the percolation clusters and for the underlying lattice in this setting a number of features of the lattice are inherited by the critical cluster

rational structures on automorphic representations this paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups we identify cases where the constructed rational structures are optimal which includes the case of mathrm mathrm gl n as an application we deduce the existence of a natural set of periods attached to cuspidal automorphic representations of mathrm mathrm gl n this has consequences for the arithmetic of special values of l functions that we discuss in januszewski on period relations for automorphic l functions i pp 1 46 arxiv 1504 06973 2015 on period relations for automorphic l functions ii pp 1 65 arxiv 1604 04253 2015 in the course of proving our results we lay the foundations for a general theory of harish chandra modules over arbitrary fields of characteristic 0 in this context a rational character theory translation functors and an equivariant theory of cohomological induction are developed we also study descent problems for harish chandra modules in quadratic extensions where we obtain a complete theory over number fields

eisenstein cohomology for congruence subgroups of so n 2 the automorphic cohomology of a connected reductive algebraic group defined over mathbb q decomposes as a direct algebraic sum of cuspidal and eisenstein cohomology the present paper investigates the eisenstein cohomology for congruence subgroups of a rational form g of mathbb q rank 2 of so n 2 in the generic case the main results provide a description of the internal structure of the summands in the eisenstein cohomology corresponding to the associate classes of maximal parabolic mathbb q subgroups

classicality for small slope overconvergent automorphic forms on some compact pel shimura varieties of type c we study the rigid cohomology of the ordinary locus in some compact pel shimura varieties of type c with values in automorphic local systems and use it to prove a small slope criterion for classicality of overconvergent hecke eigenforms this generalises work of coleman and is a first step in an ongoing project to extend the cohomological approach to classicality to higher dimensional shimura varieties

harmonic calculus on limits of networks and its application to dendrites abstract the main purpose of this paper is to give a general framework to analysis on fractals including finitely ramified self similar sets random fractals canter sets dendrites and the recurrent case of infinitely ramified self similar sets we study a limit of a sequence of electrical networks satisfying certain compatibility which is a generalization of the decimation invariance of random walks playing an important role in constructing diffusion processes on self similar sets as a limit we propose notions of finite resistance forms and effective resistance metrics also it is shown that a finite resistance form induces an effective resistance metric and vice versa a finite resistance form becomes a regular dirichlet form if the domain is dense in l 2 space in such a case the effective resistance metric is thought of as the intrinsic metric for the dirichlet form in the latter half of the paper we apply the general framework to dendrites which are topological spaces containing no loops a shortest path metric where all arcs are geodesics on a dendrite is shown to be an effective resistance metric and the corresponding finite resistance form turns out to be a local regular dirichlet form furthermore we establish a simple relation between the hausdorff dimensions and the spectral dimensions of dendrites with shortest path metrics

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

oscillatory critical amplitudes in hierarchical models and the harris function of branching processes oscillatory critical amplitudes have been repeatedly observed in hierarchical models and in the cases that have been taken into consideration these oscillations are so small to be hardly detectable hierarchical models are tightly related to iteration of maps and in fact very similar phenomena have been repeatedly reported in many fields of mathematics like combinatorial evaluations and discrete branching processes it is precisely in the context of branching processes with bounded off spring that t harris in 1948 first set forth the possibility that the logarithm of the moment generating function of the rescaled population size in the super critical regime does not grow near infinity as a power but it has an oscillatory prefactor the harris function these oscillations have been observed numerically only much later and while the origin is clearly tied to the discrete character of the iteration the amplitude size is not so well understood the purpose of this note is to reconsider the issue for hierarchical models and in what is arguably the most elementary setting the pinning model that actually just boils down to iteration of polynomial maps and notably quadratic maps in this note we show that the oscillatory critical amplitude for pinning models and the harris function coincide moreover we make explicit the link between these oscillatory functions and the geometry of the julia set of the map making thus rigorous and quantitative some ideas set forth in derrida et al commun math phys 94 115 132 1984

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

comparison between analytic and algebraic constructions of toroidal compactifications of pel type shimura varieties using explicit identications between algebraic and analytic theta functions we compare the algebraic constructions of toroidal compactications by faltings chai and the author with the analytic constructions of toroidal compactications following ash mumford rapport tai as one of the appli cations we obtain the corresponding comparison for fourier jacobi expansions of holomorphic automorphic forms

classicite en theorie de hida in this article we prove an effective classicity theorem in hida s theory we focus more precisely on the case of the symplectic group and siegel modular forms we also prove that certain mod p siegel modular forms lift to characteristic zero and study carefully the divisibility by p of certain hecke operators

on the asymptotics of the eigenvalue counting function for random recursive sierpinski gaskets we consider natural laplace operators on random recursive affine nested fractals based on the sierpinski gasket and prove an analogue of weyl s classical result on their eigenvalue asymptotics the eigenvalue counting function n is shown to be of order ds 2 as where we can explicitly compute the spectral dimension d s moreover the limit n ds 2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable this random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal