asymptotics for the heat kernel in multicone domains a multicone domain rn is an open connected set that resembles a finite collection of cones far away from the origin we study the rate of decay in time of the heat kernel p t x y of a brownian motion killed upon exiting using both probabilistic and analytical techniques we find that the decay is polynomial and we characterize limt t1 p t x y in terms of the martin boundary of at infinity where 0 depends on the geometry of we next derive an analogous result for t 2px t t with 1 n 2 where t is the exit time from lastly we deduce the renormalized yaglom limit for the process conditioned on survival

conformal invariance and self avoiding walks in restricted geometries the prediction of conformal invariance for the statistics of self avoiding random walks restricted to both semi infinite and wedge shaped geometries are tested by extrapolating exact enumerations close agreement is found both for the angular distribution of the end to end vector and for the dependence of the critical exponent gamma 2 on the opening angle of the wedge

bulk and surface critical behavior of the three dimensional ising model and conformal invariance using a continuous cluster monte carlo algorithm we investigate the critical three dimensional ising model in its anisotropic limit from the ratio of the magnetic correlations in the strong and the weak coupling directions we determine the length ratio relating the isotropic ising model and the anisotropic limit on this basis we simulate the critical ising model on a spherocylinder s2 x r1 i e a curved geometry obtained from a conformal mapping of the infinite space r3 from correlation lengths along the spherocylinder combined with the prediction of conformal invariance we estimate the magnetic and thermal scaling dimensions as x h 0 5182 6 and x t 1 419 7 respectively the behavior of the binder cumulant is also determined in the limit of an infinitely long spherocylinder next free boundary conditions are imposed on the equators of the spherocylinder and thus the geometry s1 x s x r1 is obtained the surface magnetic scaling dimension is estimated as x s h 1 263 5 the consistency of the aforementioned estimations and existing results confirms that the three dimensional ising model is conformally invariant further the precision of these results reveals that as in two dimensions conformal mappings provide a powerful tool to investigate critical phenomena with the continuous cluster algorithm we also perform simulations of systems inside a conventional solid cylinder the surface magnetic correlation length differs within the estimated error margin by a factor pi 2 from that along a half spherocylinder s1 x s x r1 with the same radius

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

finite size effects at 2d ising critical points via conformal mapping using recent results by cardy 1984 based on conformal invariance of critical correlation functions the authors calculate universal results for scattering functions s k susceptibilities correlation lengths and specific heat correction terms for finite ising systems in two dimensions with circular elliptical and rectangular shapes and free boundary conditions results show the effects of critical order on these quantities for a circle s k decays as 1 k2 eta for a large range of intermediate k values with an apparent exponent eta 0 09 the probable influence of end edge and domain wall effects in the rectangular geometry is discussed application of the results to experimental systems and other theoretical models is discussed

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

scaling and local scale invariance for wetting transitions and confined interfaces the author studies the scaling behaviour of the magnetization profile m z of an ising like magnet in a parallel plate geometry with opposite h1 h2 surface fields in the limit of large plate separation l and zero bulk field m z is a scaled function of z l for temperatures tw or t tc where tw is the critical wetting temperature of each semi infinite surface the exact form of m z depends on whether t tw tc t tw or t tc the influence of the far wall on the magnetization near one surface is long ranged and is determined for t tc by wetting critical exponents the author shows that the results of the capillary wave model for a local one point function probability distribution energy density may be derived by conformally mapping the corresponding quantity defined in the semi infinite geometry at the appropriate wetting transition the author discusses the nature of local scale invariance for wetting transitions and speculates as to why conformal invariance has application to local one point functions

conformal off diagonal boundary density profiles on a semi infinite strip the off diagonal profile odb v associated with a local operator v order parameter or energy density close to the boundary of a semi infinite strip with width l is obtained at criticality using conformal methods it involves the surface exponent xs and displays a simple universal behaviour which crosses over from surface finite size scaling when v l is held constant to corner finite size scaling when v l 0

capacitary estimates for the scattering phase abstract let h 1 2 acting in l2 r n n 3 given a compact obstacle k let hy be the operator obtained from h by imposing dirichlet boundary conditions on k here y is the complement of k let y be the scattering phase for the scattering system hy h j where j denotes restriction to y we show that the variation in y modulo integer multiples of 2 as k varies may be estimated in terms of the capacity of the symmetric difference of the obstacles suppose that k has smooth boundary let h h 1k 0 and be the scattering phase for the scattering system h h we show furthermore that converges to y again modulo integer multiples of 2 roughly at the rate 1 2 as

scattering by obstacles of finite capacity abstract for very general generators of diffusion semigroups we show that the essential and absolutely continuous spectra do not change when one adds an extra dirichlet boundary condition on a small set this is done by proving that the corresponding semigroup differences are hilbert schmidt or trace class respectively our method consists in a factorization argument which is based on calculating the semigroup difference via the feynman kac formula we also derive trace class estimates for differences of resolvent powers provided the underlying semigroup has finite local dimension

semi infinite two dimensional ising model with marginally inhomogeneous couplings and conformal invariance the authors consider a semi infinite two dimensional ising model with nearest neighbour couplings that deviate from the bulk critical couplings by al 2 where l is the distance from the surface the surface critical exponents of this system are non universal under a conformal mapping onto a strip of width l the al 1 inhomogeneity transforms into a l pi sin pi l l 1 for the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit the analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance

universal order parameter profiles in confined critical systems with boundary fields the authors consider systems at the bulk critical temperature with an ordering field applied at the boundary for various boundary geometries universal expressions for the spatial dependence of the order parameter are obtained by conformal mapping of the known results for the half space analytical results for strip wedge and other two dimensional geometries and for systems with spherical boundaries in arbitrary dimension are given

conformal invariance and simulations in curved geometries abstract conformal mappings serve as useful tools for the determination of universal properties of critical models typical applications are subject to a major restriction namely that the pertinent conformal mapping should lead to a geometry that can be investigated by means of numerical methods such as monte carlo simulations since conformal mappings of 3 d systems usually lead to curved geometries which are difficult to investigate numerically most applications have thus far been restricted to 2 d systems we present a solution of this problem for discrete spin models by taking the anisotropic or hamiltonian limit which renders one of the lattice directions continuous such that the dimensionality in effect reduces by one applications to the 3 d ising and percolation models confirm the predictions obtained from the assumption of conformal invariance and lead to accurate numerical results for the scaling dimensions

positive harmonic functions vanishing on the boundary for the laplacian in unbounded horn shaped domains denote points x e rd i d 2 by x p 0 z where p o 0 e sd i and z e r let a 0 oo 0 oo be a nondecreasing c2function and define the horn shaped domain q it p 0 z lz 2 by x x z where x e rd and z e r consider a in the horn shaped domain q x x z e rd l izi a ixi where a 0 ox 0 oo is a c2 function since aq is a smooth boundary lipschitz would be enough it follows from a result of hunt and wheeden 3 that every euclidean boundary point on aq may be identified with a distinct martin boundary point thus the interesting part of the martin boundary for a in q is the martin boundary at infinity that is the part of the martin boundary arising from sequences xy c q satisfying limn 0o lxni oo or in terms of positive harmonic functions the part of the martin boundary which corresponds to positive harmonic functions on q which vanish on aq we will give a probabilistic proof of the following theorem the motivation and intuition for which is discussed below received by the editors june 26 1992 1991 mathematics subject classification primary 35j05 secondary 60j50 31c35 31b05 35j15 1994 american mathematical society 0002 9947 94 1 00 25 per page

ratio limit theorems for a brownian motion killed at the boundary of a benedicks domain we consider a brownian motion in a benedicks domain with absorption at the boundary we show ratio limit theorems for the associated heat kernel when the hole is compact therefore the martin boundary is two dimensional we obtain sharp estimates on the lifetime probabilities and we identify in probabilistic terms the various constants appearing in the theory

conformal invariance and the ising model on a spheroid we formulate conformal mappings between an infinite plane and a spheroid and one between a semi infinite plane and a half spheroid special cases of the spheroid include the surface of an infinitely long cylinder of a sphere and of a flat disc these mappings are applied to the critical ising model for the case of the sphere and the flat disc we derive analytical expressions for the second and the fourth moments of the magnetization density and thus for the binder cumulant next we investigate ising models on spheroids and half spheroids by means of a continuous cluster monte carlo method for simulations in curved geometries fixed and free boundary conditions are imposed for half spheroids the monte carlo data are analyzed by finite size scaling critical values of the binder cumulants and other ratios on the sphere and on the flat disc agree precisely with the exact calculations mentioned above at criticality we also sample two and one point correlation functions on spheroids on half spheroids respectively the magnetic and temperature scaling dimensions as determined from the monte carlo data and the theory of conformal invariance are in good agreement with exact results

sharp two sided heat kernel estimates of twisted tubes and applications we prove on diagonal bounds for the heat kernel of the dirichlet laplacian delta d omega in locally twisted three dimensional tubes in particular we show that for any fixed x the heat kernel decays for large times as rm e e 1t t 3 2 where e1 is the fundamental eigenvalue of the dirichlet laplacian on the cross section of the tube this shows that any suitably regular local twisting speeds up the decay of the heat kernel with respect to the case of straight untwisted tubes moreover the above large time decay is valid for a wide class of subcritical operators defined on a straight tube we also discuss some applications of this result such as sobolev inequalities and spectral estimates for schrodinger operators delta d omega v

subexponential behaviour of the dirichlet heat kernel abstract we obtain a formula for the asymptotic behaviour of the dirichlet heat kernel for large time in terms of the survival probability of a brownian motion under the assumption that the latter decays subexponentially for large time we also obtain a lower bound for the dirichlet heat kernel for arbitrary open and connected sets in euclidean space

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

abstract criteria for multiple solutions to nonlinear coupled equations involving magnetic schrodinger operators we consider a system of nonlinear coupled equations involving magnetic schrodinger operators and general potentials we provide the criteria for the existence of multiple solutions to these equations as special cases we get the classical results on existence of infinitely many distinct solutions within hartree and hartree fock theory of atoms and molecules subject to an external magnetic fields we also extend recent results within this theory including coulomb system with a constant magnetic field a decreasing magnetic field and a physically measurable magnetic field

solutions to quasi relativistic multi configurative hartree fock equations in quantum chemistry we establish the existence of infinitely many distinct solutions to the multi configurative hartree fock type equations for n electron coulomb systems with quasi relativistic kinetic energy sqrt a 2 d x n a 4 a 2 for the nth electron finitely many of the solutions are interpreted as excited states of the molecule moreover we prove the existence of a ground state the results are valid under the hypotheses that the total charge ztot of k nuclei is greater than n 1 and that ztot is smaller than a critical charge ztot the proofs are based on a new application of the lions fang ghoussoub critical point approach to nonminimal solutions on a complete analytic hilbert riemann manifold in combination with density operator techniques

off diagonal density profiles and conformal invariance off diagonal profiles of local densities e g order parameter or energy density are calculated at the bulk critical point by conformal methods on a strip with transverse coordinate v for different types of boundary conditions free fixed and mixed such profiles which are defined by the non vanishing matrix element of the appropriate operator between the ground state and the corresponding lowest excited state of the strip hamiltonian enter into the expression of two point correlation functions on a strip they are of interest in the finite size scaling study of bulk and surface critical behaviour since they allow the elimination of regular contributions the conformal profiles which are obtained through a conformal transformation of the correlation functions from the half plane to the strip are in agreement with the results of a direct calculation for the energy density of the two dimensional ising model

asymptotic laws for one dimensional diffusions conditioned to nonabsorption if x t is a one dimensional diffusion corresponding to the operator l 1 2 xx x starting from x 0 and t is the hitting time of we prove that under suitable conditions on the drift coefficient the following limit exists s 0 a f s lim t p x x a t 0 t we characterize this limit as the distribution of an h like process h satisfying lh h h 0 0 h 0 1 where lim t 1 t log p x t 0 t moreover we show that this parameter can only take two values 0 or where is the smallest point of increase of the spectral distribution of the operator l 1 2 xx x

percolation in restricted geometries and conformal invariance the authors determine numerically the pair connectivity for two dimensional percolation systems subject to different boundary geometries their results provide a sensitive and efficient test of the concept of conformal invariance and confirm its validity even for relatively small lattices the influence of boundaries on electrical conduction in percolative networks is also addressed briefly

existence of a solution to hartree fock equations with decreasing magnetic fields in the presence of an external magnetic field we prove existence of a ground state within the hartree fock theory of atoms and molecules the ground state exists provided the magnetic field decreases at infinity and the total charge z of k nuclei exceeds n 1 where n is the number of electrons in the opposite direction no ground state exists if n 2z k

local scale invariance for wetting and confined interfaces when a fluid or ising like magnet is confined between two parallel walls that are each completely wet by different bulk phases the interface separating the phases is subject to large scale fluctuations determined by the slit width it was noted some time ago that in two dimensions the scaling expression for the probability distribution function describing the interfacial height across the slit shows remarkable similarities with predictions of conformal invariance however this local scale invariance appears to contradict the strongly anisotropic nature of 1 1 dimensional interfacial fluctuations along and perpendicular to the interface characterized by the wandering exponent in this paper we show that similarity with conformal invariance is not coincidental and can be understood explicitly as the projection of a local scale invariance for a wandering line in 2 1 dimensions

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

large time volume of the pinned wiener sausage we calculate the form of the large time asymptotic expansion of the expected volume of the pinned wiener sausage associated to a compact set k in rd in dimensions d 3 in each case the leading coefficient is given by the newtonian capacity of k if k is a ball of radius a 0 the first three coefficients are calculated explicitly as a byproduct of our results we obtain the low energy behaviour of the krein spectral shift function for obstacle scattering systems

asymptotic behaviour of a brownian motion on exterior domains we study the asymptotic behaviour of the transition density of a brownian motion in killed at where c is a compact non polar set our main result concern dimension d 2 where we show that the transition density p t x y behaves for large t as u x u y t log t 2 1 for x y where u is the unique positive harmonic function vanishing on r such that u x log x