second order accurate projective integrators for multiscale problems we introduce new projective versions of second order accurate runge kutta and adams bashforth methods and demonstrate their use as outer integrators in solving stiff differential systems an important outcome is that the new outer integrators when combined with an inner telescopic projective integrator can result in fully explicit methods with adaptive outer step size selection and solution accuracy comparable to those obtained by implicit integrators if the stiff differential equations are not directly available our formulations and stability analysis are general enough to allow the combined outer inner projective integrators to be applied to legacy codes or perform a coarse grained time integration of microscopic systems to evolve macroscopic behavior for example

telescopic projective methods for parabolic differential equations projective methods were introduced in an earlier paper c w gear i g kevrekidis projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum nec research institute report 2001 029 available from http www neci nj nec com homepages cwg projective pdf abbreviated version to appear in sisc as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active in this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi discretization of pdes with parabolic components

percolation of hypersurfaces and finite size scaling the authors consider the percolation of a surface without holes in a system of randomly occupied plaquettes they show that duality can relate not only the critical points but also the correlation length exponents of this problem to the usual random bond percolation the generalisation to hypersurfaces in higher dimensions is discussed

when is an interacting particle system ergodic we consider stochastic flip dynamics for an infinite number of ising spins on the lattice d we find a sequence of constructive criteria for the system to be exponentially ergodic the main idea is to approximate the continuous time process with discrete time processes itseuler polygon and to use an improved version of previous results ms about constructive ergodicity of discrete time processes

percolation in half spaces equality of critical densities and continuity of the percolation probability renormalization arguments are developed and applied to independent nearest neighbor percolation on various subsets of d d 2 yielding equality of the critical densities p c for a half space quarter space etc and ford 2 equality with the limit of slab critical densities continuity of the phase transition for the half space quarter space etc i e vanishing of the percolation probability p atp p c

on the euler characteristics of random surfaces in a monte carlo computer experiment we simulate the gibbs distribution of nonconnected two dimensional surfaces isometrically embedded in three dimensional euclidean space with fixed boundary and the action given by the area the simulation involves surfaces built out of plaquettes in a cubical lattice the foam structure is analyzed in terms of correlations of the local fluctuations in the euler characteristic and the area the scaling behavior of the area and the euler characteristic is discussed by varying the boundary we show evidence of a phase transition point which is independent of the choice of the boundary an existence proof is given of the thermodynamic limit for the models considered

the monotone iterative technique for a parabolic boundary value problem with discontinuous nonlinearity let be a bounded lebesgue measurable domain of r n with a lipschitz boundary the following initial boundary value problem is considered u t lu f u h in q bu o on u 0 where q o t o t and 0 0 l is a uniformly elliptic operator b a boundary operator defined by bu u v bu in which v denotes the outer normal derivative on and f represents a discontinuous non linearity

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

slab percolation for the ising model for the fk representation of the ising model we prove that the slab percolation threshold coincides with the critical temperature in any dimension d 3

a high order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws we present a general high order fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions the scheme consists of two steps in a first relaxation step the nonlinear hyperbolic conservation law is approximated by a kinetic equation with stiff bgk source term then this kinetic equation is integrated in time using a projective integration method after taking a few small inner steps with a simple explicit method such as direct forward euler to damp out the stiff components of the solution the time derivative is estimated and used in an outer rungekutta method of arbitrary order we show that with an appropriate choice of inner step size the time step restriction on the outer time step is similar to the cfl condition for the hyperbolic conservation law moreover the number of inner time steps is also independent of the stiffness of the bgk source term we discuss stability and consistency and illustrate with numerical results linear advection burgers equation and the shallow water and euler equations in one and two spatial dimensions

monotone solutions of a class of second order nonlinear differential equations on etudie le comportement asymptotique des solutions de l equation p t f x t x t g t g x t on donne des conditions necessaires et suffisantes pour que ces solutions soient bornees

crabgrass measles and gypsy moths an introduction to modern probability on etudie cinq modeles qui permettent d expliquer des processus lies aux systemes de particules en interaction ainsi que d autres processus

the large n limit of the threshold values in mandelbrot s fractal percolation process the authors consider mandelbrot s fractal percolation process characterised by a density parameter p in 0 1 and an integer subdivision index n 1 for each n the process is known to have a percolation transition at density pc n in 0 1 they prove that lim pc n pc where pc is the threshold value of the ordinary square lattice site percolation model

an upper bound on the critical percolation probability for the three dimensional cubic lattice we prove that the critical probability for site percolation on the three dimensional cubic lattice satisfies the inequality p 3 c 1 2 an application to the three dimensional ising model is given

phase diagram and correlation length bounds for mandelbrot aerogels the authors study a variant of the mandelbrot percolation process which is of current use as a model of aerogels the model has two parameters one of them q is the usual multiscale parameter of the mandelbrot percolation process and the other p is a bernoulli percolation parameter that is reserved for the last step of the construction they investigate the phase diagram of this model in the q p plane there are two phases a sol phase and a gel phase classified according to whether a limit of crossing probabilities vanishes or is nonzero in the sol phase they define a correlation length via the rate of decay of a rescaled connectivity function they show that this length scale diverges at the phase boundary furthermore they demonstrate that if the phase boundary is approached with q fixed and p tending up to its critical value pg q then up to logarithmic corrections the divergence is at least as fast as mod pg p mod 2dh where dh 2 mod log q log n mod can be identified as the hausdorff dimension of the background medium

explicit time stepping for stiff odes we present a new strategy for solving stiff odes with explicit methods by adaptively taking a small number of stabilizing small explicit time steps when necessary a stiff ode system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis for many stiff problems the cost of the stabilizing small time steps is small so the improvement is large we illustrate the technique on a number of well known stiff test problems

uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation for independent translation invariant irreducible percolation models it is proved that the infinite cluster when it exists must be unique the proof is based on the convexity or almost convexity and differentiability of the mean number of clusters per site which is the percolation analogue of the free energy the analysis applies to both site and bond models in arbitrary dimension including long range bond percolation in particular uniqueness is valid at the critical point of one dimensional 1 x y 2 models in spite of the discontinuity of the percolation density there corollaries of uniqueness and its proof are continuity of the connectivity functions and except possibly at the critical point of the percolation density related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent e g 1 2 1 1

filtering techniques for systems of stiff ordinary differential equations i consider linear systems of ordinary differential equations frac dy dt a t y f t quad 0 leqq t leqq t quad y 0 text given which contain two different time scales a slow one and a fast one in many applications such systems possess special smooth solutions and one would like to compute these smooth solutions using a step size which only resolves the slow time scales in the equation there are several different strategies to use in order to compute these smooth solutions one of these strategies is to filter the numerical solution in this paper we introduce a large class of filtering techniques which combines a digital filter interactively with a linear multistep method we also introduce various stability concepts and make a thorough investigation of the stability properties of the filtering methods

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

filtering techniques for systems of stiff ordinary differential equations ii error estimates consider the initial value problem for systems of linear ordinary differential equations which contain two different time scales a slow one and a fast one under very general conditions the solution of this initial value problem can be decomposed into the sum of a slowly varying solution and a rapidly varying solution of the given system of equations the decomposition is unique in a manner which is stated in this paper in many applications one would like to compute only the slowly varying solution in the stated decomposition using a step size which resolves the slow time scale in the system of equations and not the fast one in this paper we approximate these equations by a large class of filtering techniques which combines a digital filter interactively with a linear multistep method for the purpose of computing slowly varying solutions automatically these methods are called r q text dflmm s and were introduced in a previous paper 13 we show that if the r q text dflmm is suitably stab

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

strict monotonicity for critical points in percolation and ferromagnetic models when is the numerical value of the critical point changed by an enhancement of the process or of the interaction ferromagnetic spin models independent percolation and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction e g the addition of ferromagnetic couplings can only increase the transition temperature the question explored here is whether enhancements do indeed change the value of the critical point we present a generally applicable approach to this issue for ferromagnetic ising spin systems with pair interactions of finite range ind 2 dimensions it is shown that the critical temperaturetc is strictly monotone increasing in each coupling with the first order derivatives bounded by positive functions which are continuous on the set of fullyd dimensional interactions for independent percolation with 0 pc 1 we prove that any essential enhancement of the process has an effect on the critical probability a result with applications to the question of the existence of entanglements and to invasion percolation with trapping

entanglement in percolation we study the existence of finite and infinite entangled graphs in the bond percolation process in three dimensions with density p after a discussion of the relevant definitions the entanglement critical probabilities are defined the size of the maximal entangled graph at the origin is studied for small p and it is shown that this graph has radius whose tail decays at least as fast as exp n log n indeed the logarithm may be replaced by any iterate of logarithm for an appropriate positive constant we explore the question of almost sure uniqueness of the infinite maximal open entangled graph when p is large and we establish two relevant theorems we make several conjectures concerning the properties of entangled graphs in percolation

second order chebyshev methods based on orthogonal polynomials stabilized methods also called chebyshev methods are explicit runge kutta methods with extended stability domains along the negative real axis these methods are intended for large mildly stiff problems originating mainly from parabolic pdes the aim of this paper is to show that with the use of orthogonal polynomials we can construct nearly optimal stability polynomials of second order with a three term recurrence relation these polynomials can be used to construct a new numerical method which is implemented in a code called rock2 this new numerical method can be seen as a combination of van der houwen sommeijer type methods and lebedev type methods

on convergence to sle6 i conformal invariance for certain models of the bond triangular type we establish convergence to sle6 of the exploration processes for the correlated bond triangular type models studied by two of us in an earlier work chayes and lei in rev math phys 19 511 565 2007 this rigorously establishes that these models are in the same universality class as the standard site percolation model on the triangular lattice in the context of these models the result is proven for all domains with boundary minkowski dimension less than two moreover the proof of convergence applies in the context of general critical 2d percolation models and for general domains under the stipulation that cardy s formula can be established for domains in this generality

invasion percolation and the incipient infinite cluster in 2d we establish two links between two dimensional invasion percolation and kesten s incipient infinite cluster iic we first prove that the kth moment of the number of invaded sites within the box n n n n is of order n2 n k for k 1 where n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n this improves a result of y zhang we show that the size of the invaded region when scaled by n2 n is tight

rigorous studies of critical behavior the task addressed in the talk is the analysis of the critical behavior in statistical mechanical systems and related problems in the theory of random fields while it would have been very useful to have mathematical means of approaching this subject at the level of generality presented here in professor dobrushin s analysis of the high temperature phases we find that the richness of the behavior exhibited at low temperatures calls for more restricted treatments of special models nevertheless it was found that the somewhat delicate issues of the phase structure and the critical behavior may be approached by methods which apply to a number of systems in particular the picture offered by random walk representations led to very useful insights for the study of ferromagnetic systems 4 d field theory and percolation models it is hoped that random surface expansions might play a somewhat similar role for gauge fields and some models of spin glasses

the q state potts model in the standard pirogov sinai theory surface tensions and wilson loops theq state potts model both scalar and gauge versions is rewritten with the help of the duality transformation into a form of the pirogov sinai theory with noninteracting contours that can be controlled by cluster expansions onceq is large enough this is then used in a new proof of the existence of a unique transition inverse temperature t where the mean internal energy is discontinuous moreover we prove for the scalar model again forq large enough that there are discontinuities at t of the magnetization and of the mass gap with the magnetization vanishing below t and the mass gap vanishing above t we also show that the surface tensions between ordered stable phases are strictly positive up to t and the surface tension between an ordered phase and the disordered one is strictly positive at t for the three dimensional gauge model the wilson parameter exhibits a direct transition from an area law decay quark confinement to a perimeter law decay deconfinement

connectivity properties of mandelbrot s percolation process in 1974 mandelbrot introduced a process in 0 1 2 which he called canonical curdling and later used in this book s on fractals to generate self similar random sets with hausdorff dimension d 0 2 in this paper we will study the connectivity or percolation properties of these sets proving all of the claims he made in sect 23 of the fractal geometry of nature and a new one that he did not anticipate there is a probability pc 0 1 so that if p pc then the set is duslike i e the largest connected component is a point whereas if p pc notice the opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles more succinctly put the system has a first order phase transition

interface motion in random media we study the pinning phase transition for discrete surface dynamics in random environments a renormalization procedure is devised to prove that the interface moves with positive velocity under a finite size condition this condition is then checked for different examples of microscopic dynamics to illustrate the flexibility of the method we show in our examples the existence of a phase transition for various models including high dimensional interfaces dependent environments and environments with arbitrarily deep obstacles finally our ballisticity criterion is proved to be valid up to the critical threshold for a lipschitz interface model

limit theorems for maximum flows on a lattice we independently assign a non negative value as a capacity for the quantity of flows per unit time with a distribution f to each edge on the mathbf z d lattice we consider the maximum flows through the edges from a source to a sink in a large cube in this paper we show that the ratio of the maximum flow and the size of the source is asymptotic to a constant this constant is denoted by the flow constant by the max flow and min cut theorem this is equivalent to a statement about the asymptotic behavior of the minimal value assigned to any surface on the large cube we can also show that there exists such a surface that is proportional to the size of the faces of the cube

one dimensional 1 j i s percolation models the existence of a transition for s 2 consider a one dimensional independent bond percolation model withpj denoting the probability of an occupied bond between integer sitesi andi j j 1 ifpj is fixed forj 2 and mathop lim limits j to infty j2pj 1 then unoriented percolation occurs forp1 sufficiently close to 1 this result analogous to the existence of spontaneous magnetization in long range one dimensional ising models is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size oriented percolation is shown to occur forp1 close to 1 if mathop lim limits j to infty jspj 0 for somes 2 analogous results are valid for one dimensional site bond percolation models

equation free implementation of statistical moment closures we present a general numerical scheme for the practical implementation of statistical moment closures suitable for modeling complex large scale nonlinear systems building on recently developed equation free methods this approach numerically integrates the closure dynamics the equations of which may not even be available in closed form although closure dynamics introduce statistical assumptions of unknown validity they can have significant computational advantages as they typically have fewer degrees of freedom and may be much less stiff than the original detailed model the numerical closure approach can in principle be applied to a wide class of nonlinear problems including strongly coupled systems either deterministic or stochastic for which there may be no scale separation we demonstrate the equation free approach for implementing entropy based eyink levermore closures on a nonlinear stochastic partial differential equation

a space time formulation and improved spatial reconstruction for the divide and conquer multiscale method abstract a time discontinuous space time formulation for the divide and conquer multiscale method is presented this multiscale method aims at stable and accurate solutions of transient convection diffusion reaction problems the method is constituted by a time discontinuous space time finite element method on the fine scale level an explicit procedure for advancing the coarse scale solution on the respective level and inter scale operators providing the data transfer between the coarse and fine scale level in both directions the results from three numerical test cases show that for both problematic flow regimes that is the regime of dominant convection and the regime of dominant convection and absorption the proposed method provides completely stable solutions which are not achieved by standard stabilized methods particularly for the later regime for remedying a still to be noted shortcoming of the proposed method that is a too smooth resolution of sharp gradient regions in the solution field considerable improvement can be provided by the use of cubic spline interpolation as the spatial reconstruction i e the spatial coarse to fine inter scale operator

the accurate numerical solution of highly oscillatory ordinary differential equations we consider systems of ordinary differential equations with rapidly oscillating solutions conventional numerical methods require an excessively small time step t 0 eh where h is the step size necessary for the resolution of a smooth function of t and 1 e measures the size of the large eigenvalues of the system s jacobian for the linear problem with well separated large eigenvalues we introduce smooth transformations which lead to the separation of the time scales and computation with a large time step t 0 h for more general problems including systems with weak polynomial nonlinearities we develop an asymptotic theory which leads to expansions whose terms are suitable for numerical approximation resonances can be detected and resolved often with a large time step t 0 h passage through resonance in nonautonomous systems can be resolved by a moderate time step t 0 eh

a note on uniqueness for parabolic problems with discontinuous nonlinearities parabolic problems with discontinuous nonlinearities arise as a natural simplification of various problems of mathematical physics in a model of porous medium combustion e g see norbury and stuart 4 a chemical reaction and the corresponding heat supply is started by reaching a threshold temperature and on the other hand it is expected to end when the concentration drops below a critical level there are of course many other situations where the use of discontinuous nonlinearities may be fully justified and leads to a considerable simplification of the problem fleishman and mahar 3 carl l and the literature quoted therein the resulting equations are easier to handle and the solutions are very often in good agreement with observed behavior of the system see however the remarks in fleishman and mahar 3 consequently the question of uniqueness for this kind of equation seems to be of great importance both from the analytical and numerical point of view as a model problem we consider the equation

a note on implementations of the boosting algorithm and heterogeneous multiscale methods we present improved convergence results for the boosting algorithm ba and demonstrate that an existing formulation of the heterogeneous multiscale methods hmm is accurate to first order only in the macro time step regardless of the order of the numerical solvers employed these results are obtained by considering the ba and two other formulations of hmm as special cases of a general formulation of hmm applied to dissipative stiff ordinary differential equations

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

deciding the nature of the coarse equation through microscopic simulations the baby bathwater scheme recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations in closed form the closure is obtained on demand through appropriately initialized bursts of microscopic simulation the effective coupling of microscopic simulators with macroscopic behavior requires certain decisions about the nature of the unavailable coarse equation such decisions include a the highest spatial derivative active in the coarse equation b whether the equation satisfies certain conservation laws or c whether the coarse dynamics is hamiltonian or dissipative these decisions affect the number and type of boundary conditions as well as the algorithms employed in the absence of an explicit formula for the temporal derivative we propose implement and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation in the same spirit we show how to determine whether a certain coarse conservation law exists or not and we discuss plausibility tests for the existence of a coarse hamiltonian or integrability we believe that such schemes constitute an important part of the equation free approach to multiscale computation

on convergence of higher order schemes for the projective integration method for stiff ordinary differential equations we present a convergence proof for higher order implementations of the projective integration method pi for a class of deterministic multi scale systems in which fast variables quickly settle on a slow manifold the error is shown to contain contributions associated with the length of the microsolver the numerical accuracy of the macrosolver and the distance from the slow manifold caused by the combined effect of micro and macrosolvers respectively we also provide stability conditions for the pi methods under which the fast variables will not diverge from the slow manifold we corroborate our results by numerical simulations

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

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

on convergence to sle6 ii discrete approximations and extraction of cardy s formula for general domains we show how to extract cardy s formula for a general class of domains given the requisite interior analyticity statement this is accomplished by a careful study of the interplay between discretization schemes and extraction of limiting boundary values of particular importance to the companion work binder et al in j stat phys 2010 we establish these results for slit domains and for the critical percolation models introduced in chayes and lei rev math phys 19 511 565 2007

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

critical points and intermediate phases on wedges of zd the authors examine the phase structure of self avoiding walks the ising magnet and bond percolation defined on subsets of zd d or 2 which have the geometry of wedges they prove that if the cross sectional area of the wedge diverges with its width then the high temperature critical point of any of these models defined on the wedge coincides with that of the corresponding model on the full lattice for the ising magnet and bond percolation they show that there is a non trivial low temperature critical point if the cross sectional area of the wedge diverges logarithmically with its width moreover for bond percolation they show that the low temperature critical point of a logarithmic wedge may be made arbitrarily close to that of the full lattice by taking the coefficient of the logarithm large enough corollaries to these theorems include the existence of an intermediate phase for the ising magnet and bond percolation on logarithmic wedges and the existence of a first order transition for percolation models on a subclass of these wedges

a high order asymptotic preserving scheme for kinetic equations using projective integration we investigate a high order fully explicit asymptotic preserving scheme for a kinetic equation with linear relaxation both in the hydrodynamic and diffusive scalings in which a hyperbolic resp parabolic limiting equation exists the scheme first takes a few small inner steps with a simple explicit method such as direct forward euler to damp out the stiff components of the solution and estimate the time derivative of the slow components these estimated time derivatives are then used in an outer runge kutta method of arbitrary order we show that with an appropriate choice of inner step size the time step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation moreover the number of inner time steps is also independent of the scaling parameter we analyze stability and consistency and illustrate with numerical results

discontinuity of the wilson string tension in the 4 dimensional lattice pure gauge potts model we consider the 4 dimensionalq state pure gauge potts model forq large enough we give a new proof of the existence of a unique coupling constant t where a first order phase transition occurs moreover we prove the following new results the string tension is discontinuous at t the wilson parameter exhibits at t a direct transition from an area law decay quark confinement to a perimeter law decay quark deconfinement

bond percolation in frustrated systems conwe study occurrence and properties of percolation of occupied bonds in systems with random interactions and hence frustration we develop a general argument somewhat like peierls argument by which we show that in z d d 2 percolation occurs for all possible interactions provided they are bounded away from zero if the parameter p e 0 1 regulating the density of occupied bonds is high enough if the interactions are i i d random variables then we determine bounds on the values of p for which percolation occurs for all almost all but not all almost none but some or none of the interactions motivations of this work come from the rigorous analysis of phase transitions in frustrated statistical mechanics systems

gaussian fluctuations of connectivities in the subcritical regime of percolation we consider thed dimensional bernoulli bond percolation model and prove the following results for allp pc the finite volume conditional measures defined by requiring the existence of a dual path between opposing faces of the boundary converge in the infinite volume limit to the standard bernoulli measure

decay of correlations in random cluster models we prove exponential decay for the tail of the radius r of the cluster at the origin for subcritical random cluster models under an assumption slightly weaker than that here d is the number of dimensions specifically if throughout the subcritical phase then for some 0 this implies the exponential decay of the two point correlation function of subcritical potts models subject to a hypothesis of at least polynomial decay of this function similar results are known already for percolation and ising models and for potts models when the number q of available states is sufficiently large indeed the hypothesis of polynomial decay has been proved rigorously for these cases in two dimensions the hypothesis that is weaker than requiring that the susceptibility be finite i e that the two point function be summable the principal new technique is a form of russo s formula for random cluster models reported by bezuidenhout grimmett and kesten for the current application this leads to an analysis of a first passage problem for random cluster models and a proof that the associated time constant is strictly positive if and only if the tail of r decays exponentially

rigidity of the interface in percolation and random cluster models we study conditioned random cluster measures with edge parameter p and cluster weighting factor q satisfying q 1 the conditioning corresponds to mixed boundary conditions for a spin model interfaces may be defined in the sense of dobrushin and these are proved to be rigid in the thermodynamic limit in three dimensions and for sufficiently large values of p this implies the existence of non translation invariant conditioned random cluster measures in three dimensions the results are valid in the special case q 1 thus indicating a property of three dimensional percolation not previously noted

projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum we show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters one containing the stiff or fast components and one containing the slow components these methods have large average step sizes relative to the fast components conventional implicit methods involve the solution of nonlinear equations at each step which for large problems requires significant communication between processors on a multiprocessor machine for such problems the methods proposed here have significant potential for speed improvement

the probability of a large finite cluster in supercritical bernoulli percolation we consider standard bernoulli site percolation on zd with probability p for each site to be occupied c denotes the occupied cluster of the origin and ici its cardinality we show that for p critical probability of the halfspacezd l x 7 one has pp ici n 0 this improves a recent result of chayes chayes and newman the proof is based on a peierls argument which shows exponential decay of the distribution of the size of an exterior boundary of c

inhomogeneous percolation problems and incipient infinite clusters the authors consider inhomogeneous percolation models with density pc f x and examine the forms of f x which produce incipient structures taking f x approximately mod x mod lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model they prove that in d 2 the borderline value of lambda is lambda b 1 v if lambda 1 v then with probability one there is no infinite cluster while if lambda 2 the models studied suggest what sort of incipient objects should be examined in random surface models

random surface correlation functions truncated pair functions for free random surface models and bernoulli ensembles are examined in both cases the pair function is shown to obey ornstein zernike scaling whenever various correlation lengths of the system satisfy a nonperturbative criterion under the same conditions the transverse displacement of surfaces contributing to the pair function is shown to be normally distributed a new type of transition which concerns the width of typical surfaces is introduced and studied whenever the system is below the melting transition temperature of a related lower dimensional model the width of typical surfaces is shown to be finite a thermodynamic formalism for free random surface models is developed the formalism is used to obtain sharp estimates of the entropy of surfaces contributing to the pair function

invariant manifolds and the initialization problem for some atmospheric equations abstract an invariant manifold of quasi geostrophic states is shown to exist for a finite dimensional system of atmospheric equations there is another set of scalings for the parameters with respect to the rossby number which leads to the same limiting geostrophic equations but such that there need not be an invariant manifold also discussed are computational methods

percolation phenomena in low and high density systems we consider the 2d quenched disordered q state potts ferromagnets and show that in the translation invariant measure averaged over the disorder at self dual points any amalgamation of q 1 species will fail to percolate despite an overall high density of 1 q 1 further in the dilute bond version of these systems if the system is just above threshold then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density finally we demonstrate both phenomena in a single model by considering a perturbation of the dilute model that has a self dual point we also demonstrate that these phenomena occur by a similar mechanism in a simple coloring model invented by o haggstrom

asymptotic preserving projective integration schemes for kinetic equations in the diffusion limit we investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path in which the kinetic description approaches a diffusion phenomenon the scheme first takes a few small steps with a simple explicit method such as a spatial centered flux forward euler time integration and subsequently projects the results forward in time over a large time step on the diffusion time scale we show that with an appropriate choice of the inner step size the time step restriction on the outer time step is similar to the stability condition for the diffusion equation whereas the required number of inner steps does not depend on the mean free path we also provide a consistency result the presented method is asymptotic preserving in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path the analysis is illustrated with numerical results and we present an application to the su olson test

projective and coarse projective integration for problems with continuous symmetries temporal integration of equations possessing continuous symmetries e g systems with translational invariance associated with traveling solutions and scale invariance associated with self similar solutions in a co evolving frame i e a frame which is co traveling co collapsing or co exploding with the evolving solution leads to improved accuracy because of the smaller time derivative in the new spatial frame the slower time behavior permits the use of projective and coarse projective integration with longer projective steps in the computation of the time evolution of partial differential equations and multiscale systems respectively these methods are also demonstrated to be effective for systems which only approximately or asymptotically possess continuous symmetries the ideas of projective integration in a co evolving frame are illustrated on the one dimensional translationally invariant nagumo partial differential equation pde a corresponding kinetic monte carlo model motivated from the nagumo kinetics is used to illustrate the coarse grained method a simple one dimensional diffusion problem is used to illustrate the scale invariant case the efficiency of projective integration in the co evolving frame for both the macroscopic diffusion pde and for a random walker particle based model is again demonstrated

sharpness of the phase transition for continuum percolation in mathbb r 2 r 2 we study the phase transition of random radii poisson boolean percolation around each point of a planar poisson point process we draw a disc of random radius independently for each point the behavior of this process is well understood when the radii are uniformly bounded from above in this article we investigate this process for unbounded and possibly heavy tailed radii distributions under mild assumptions on the radius distribution we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter lambda c moreover the techniques we develop in this article can be applied to other models such as the poisson voronoi and confetti percolation

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

bulk transport properties and exponent inequalities for random resistor and flow networks the properties of random resistor and flow networks are studied as a function of the density p of bonds which permit transport it is shown that percolation is sufficient for bulk transport in the sense that the conductivity and flow capacity are bounded away from zero wheneverp exceeds an appropriately defined percolation threshold relations between the transport coefficients and quantities in ordinary percolation are also derived assuming critical scaling these relations imply upper and lower bounds on the conductivity and flow exponents in terms of percolation exponents the conductivity exponent upper bound so derived saturates in mean field theory

analyticity properties and power law estimates of functions in percolation theory we consider percolation on the sites of a graphg e g a regulard dimensional lattice all sites ofg are occupied vacant with probabilityp respectively q 1 p independently of each other w denotes the cluster of occupied sites containing a fixed site which will usually be taken to be the origin andw the cardinality ofw the percolation probability is the probability that w i e p pp w some critical values ofp ph andpt are defined respectively as the smallest value ofp for which p 0 and for which the expectation of w is infinite formally ph inf p p 0 andpt inf p ep w we show for fairly general graphsgthat ifp pt thenpp w n decreases exponentially inn for the special casesg g0 the simple quadratic lattice andg1 the graph which corresponds to bond percolation on 2 we obtain upper and lower bounds for p of the formc p ph and bounds forep w of the formc p ph we also investigate smoothness properties of p ep number of clusters per site ep w 1 w 1 this function was introduced by sykes and essam who assumed that has exactly one singularity namely atp ph for the graphsg0 andg1 i e site or bond percolation on 2 we show that p is analytic atp ph and has two continuous derivatives atp ph the emphasis is on rigorous proofs

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

the probability distribution of the percolation threshold in a large system we show that the distribution of the percolation threshold in a large finite system does not converge to a gaussian when the size of the system goes to infinity provided that the two widely accepted definitions of correlation length are equivalent the shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length the result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v all results are rigorously proven in the 2d case generalizations for three dimensions are also discussed

fourth order chebyshev methods with recurrence relation in this paper a new family of fourth order chebyshev methods also called stabilized methods is constructed these methods possess nearly optimal stability regions along the negative real axis and a three term recurrence relation the stability properties and the high order make them suitable for large stiff problems often space discretization of parabolic pdes a new code rock4 is proposed illustrated at several examples and compared to existing programs

heterogeneous multiscale methods for mechanical systems with vibrations we are concerned with heterogeneous multiscale methods for the numerical integration of the equations of motion of mechanical systems subjected to fast vibrations we suggest easily formulated asynchronous algorithms that bypass the need for explicitly determining the relations between macro and microstates the new algorithms have good geometric properties and in some cases may incorporate a simplified filtering technique that leads to large savings in computational effort the problems considered may be modeled either by ordinary differential equations state space description or by differential algebraic equations descriptor form a shake shake and a rattle shake algorithm are presented but higher order versions exist

correlation length bounds for disordered ising ferromagnets thed dimensional nearest neighbor disordered ising ferromagnet h sum j ij sigma i sigma j is studied as a function of both temperature t and a disorder parameter which measures the size of fluctuations of couplingsjij 0 a finite size scaling correlation length f t is defined in terms of the magnetic response of finite samples this correlation length is shown to be equivalent in the scaling sense to the quenched average correlation length t defined as the asymptotic decay rate of the quenched average two point function furthermore the magnetic response criterion which defines f is shown to have a scale invariant property at the critical point the above results enable us to prove that the quenched correlation length satisfies c left log xi t right xi t geqq left t t c right 2 mathord left vphantom 2 d right kern nulldelimiterspace d which implies the boundv 2 d for the quenched correlation length exponent

the wulff construction and asymptotics of the finite cluster distribution for two dimensional bernoulli percolation we consider two dimensional bernoulli percolation at densityp p c and establish the following results

heterogeneous multiscale methods for stiff ordinary differential equations the heterogeneous multiscale methods hmm is a general framework for the numerical approximation of multiscale problems it is here developed for ordinary differential equations containing differe

rkc an explicit solver for parabolic pdes the fortran program rkc is intended for the time integration of parabolic partial differential equations discretized by the method of lines it is based on a family of runge kutta chebyshev formulas with a stability bound that is quadratic in the number of stages remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size moreover they make it possible to evaluate the explicit formulas in just a few vectors of storage these characteristics of the program make it especially attractive for problems in several spatial variables rkc is compared to the bdf solver vodpk on two test problems in three spatial variables