exact large deviation bounds up to t c for the ising model in two dimensions we prove an upper large deviation bound for the block spin magnetization in the 2d ising model in the phase coexistence region the precise rate given by the wulff construction is shown to hold true for all c combined with the lower bounds derived in i those results yield an exact second order large deviation theory up to the critical temperature

the droplet in the tube a case of phase transition in the canonical ensemble we consider the 2 dimensional ising model with ferromagnetic nearest neighbour interaction at inverse temperature lets n t be the total magnetization inside ann n square box per be the gibbs state in with periodic b c andm be the spontaneous magnetization we show the existence of the limit psi varrho mathop lim limits n to infty left frac 1 beta n right ln mu lambda per s n n varrho for m provided is large enough it turns out that the quantity is closely related to the wulf construction and the dependence of the function on is singular

harmonic crystal on the wall a microscopic approach a three dimensional winterbottom type construction in the regime of partial wetting is derived in a scaling limit of a gas of microscopic gaussian sos droplets under the fixed volume constraint the proof is based on a coarse graining of the random microscopic region wetted by the crystal random walk representations of various quantities related to free massless fields and a stability analysis of the torsional rigidity problem

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

large deviations and continuum limit in the 2d ising model we study the 2d ising model in a rectangular box l of linear size o l we determine the exact asymptotic behaviour of the large deviations of the magnetization p t2 l ae t when l 1 for values of the parameters of the model corresponding to the phase coexistence region where the order parameter m is strictly positive we study in particular boundary eects due to an arbitrary real valued boundary magnetic field using the self duality of the model a large part of the analysis consists in deriving properties of the covariance function hae 0 ae t i as t 1 at dual values of the parameters of the model to do this analysis we establish new results about the high temperature representation of the model these results are valid for dimensions d 2 and up to the critical temperature they give a complete non perturbative exposition of the high temperature representation we then study the gibbs measure conditioned by p t2 l ae t im l l l ic with 0 c 1 4 and im m m we construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperi metric type

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

equivalences of the large deviation principle for gibbs measures and critical balance in the ising model in this paper we obtain the equivalence of the large deviation principle for gibbs measures with and without an external field for the ising model the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube b h ash 0 for variousb we find a critical balance at a valueb0 ofb

constrained variational problem with applications to the ising model we continue our study of the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh initiated in our earlier work we strengthen further a result previously proven by martirosyan at low enough temperature which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the system if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the system in our earlier work this result was extended to every subcritical value of the temperature here for every subcritical value of the temperature we show the existence of a critical valueb0 t which separates the two regimes specified above we also find the asymptotic shape of the region occupied by the phase in the second regime which turns out to be a squeezed wulff shape the main step in our study is the solution of the variational problem of finding the curve minimizing the wulff functional which curve is constrained to the unit square other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field extended to all temperatures below the critical one

entropic repulsion of an interface in an external field we consider an interface above an attractive hard wall in the complete wetting regime and submitted to the action of an external increasing convex potential and study its delocalization as the intensity of this potential vanishes our main motivation is the analysis of critical prewetting which corresponds to the choice of a linear external potential we also present partial results on critical prewetting in the two dimensional ising model as well as a few weak results on pathwise estimates for the pure wetting problem for effective interface models

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

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

slow droplet driven relaxation of stochastic ising models in the vicinity of the phase coexistence region we consider the stochastic ising models glauber dynamics corresponding to the infinite volume basic ising model in arbitrary dimensiond 2 with nearest neighbor interaction and under a positive external magnetic fieldh under minimal assumptions on the rates of flip so that all the common choices are included we obtain results which state that when the system is at low temperaturet the relaxation time when the evolution is started with all the spins down blows up whenh 0 as exp t hd 1 the precise results are lower and upper bounds of this form moreover after a time which does not scale withh and before a time which also grows as an exponential of a multiple of 1 hd 1 ash 0 the law of the state of the process stays whenh is small close to the minus phase of the same ising model without an external field these results may be considered as a partial vindication of a conjecture raised by aizenman and lebowitz in connection to the metastable behavior of these stochastic ising models

application of adaptive quadrature to axi symmetric vortex sheet motion studies of the formation of fine structures on free surfaces in liquids such as curvature singularities or interface pinching demand that the motion of the interface must be computed very accurately boundary integral techniques are a popular choice in such studies because they reduce the dimension of the problem by one on the other hand the boundary integrals are singular and their accurate evaluation can prove quite challenging in two dimensional motion the interface is just a curve when this curve is closed or periodic the singularity in the integrand may be removed and the trapezoidal rule may be applied with spectral accuracy unfortunately the nature of the singularity in the integrand for three dimensional motion is much more difficult to treat in this paper we present an accurate adaptive quadrature to compute the motion of a vortex sheet in axi symmetric flow the technique is based on a vector potential formulation which offers some computational advantages over other methods based on the biot savart integral direct numerical computations show that our technique is much more accurate and efficient than existing techniques we apply our technique to study the evolution of an initially spherical vortex sheet we present evidence of the formation of a 3 2 power singularity in the curvature of the vortex sheet

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

wulff droplets and the metastable relaxation of kinetic ising models we consider the kinetic ising models glauber dynamics corresponding to the infinite volume ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h minimal conditions on the flip rates are assumed so that all the common choices are being considered we study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature t and the evolution is started from a distribution which is stochastically lower than the phase we show that as h 0 the relaxation time blows up as exp c t h with lgr c t w t 2 12 t m t here m t is the spontaneous magnetization and w t is the integrated surface tension of the wulff body of unit volume moreover for 0 c the state of the process at time exp h is shown to be close when h is small to the phase the difference between this state and the phase can be described in terms of an asymptotic expansion in powers of the external field this expansion can be interpreted as describing a set of continuations in h of the family of gibbs distributions with the negative magnetic fields into the region of positive fields

equilibrium shapes for planar crystals in an external field the equilibrium shape of a two dimensional crystal in a convex background potential g x is analyzed for g 0 the shape of minimum energy may be deduced from surface tension via the wulff construction but if g is not constant little is known beyond the case of a crystal sitting in a uniform field only an unpublished result of okikiolu shows each connected component of the equilibrium crystal to be convex here it will be shown that any such component minimizes energy uniquely among convex sets of its area if the wulff shape and g x are symmetric under x x it follows that the equilibrium crystal is unique convex and connected this last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature driven flow subsequent work with felix otto shows without assuming symmetry that no equilibrium crystal has more than two convex components

on the two dimensional stochastic ising model in the phase coexistence region near the critical point we consider the two dimensional stochastic ising model in finite square with free boundary conditions at inverse temperature 0 and zero external field using duality and recent results of ioffe on the wulff construction close to the critical temperature we extend some of the results obtained by martinelli in the low temperature regime to any temperature below the critical one in particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of with a rate constant determined by the surface tension along one of the coordinate axes we also extend to the same range of temperatures the result due to shlosman on the equilibrium large deviations of the magnetization with free boundary conditions

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

on criticality for competing influences of boundary and external field in the ising model we continue a study of schonmann 1994 schonmann and shlosman 1996 and greenwood and sun 1997 regarding the competing influences of boundary conditions and external field for the ising model we find a critical point b0 in the competing influences for low temperature in dimension d 2a7e 2

interpretation and estimation of summary ratios under heterogeneity of the several types of summary odds ratios in use in epidemiology only those proposed by mantel and haenszel and miettinen do not assume homogeneity constancy of stratum specific parameters under heterogeneity miettinen s estimators consistently estimate meaningful epidemiologic parameters whereas the asymptotic expectation of the mantel haenszel odds ratio varies according to irrelevant features of the sampling design although one can modify the mantel haenszel summary to overcome the latter problem one can also extend the miettinen estimators to sparse data and thus limit the use of mantel haenszel summaries to situations in which the homogeneity hypothesis is reasonable this paper provides formulae for the extended miettinen estimators and large sample variances and confidence limits for summary ratios under heterogeneity