ornstein zernike theory for finite range ising models above t c we derive a precise ornstein zernike asymptotic formula for the decay of the two point function 0 x in the general context of finite range ising type models on d the proof relies in an essential way on the a priori knowledge of the strict exponential decay of the two point function and by the sharp characterization of phase transition due to aizenman barsky and fernandez goes through in the whole of the high temperature region c as a byproduct we obtain that for every c the inverse correlation length is an analytic and strictly convex function of direction

on the layering transition of an sos surface interacting with a wall i equilibrium results we consider the model of a 2d surface above a fixed wall and attracted toward it by means of a positive magnetic field h in the solid on solid sos approximation when the inverse temperature fl is very large and the external field h is exponentially small in ft we improve considerably previous results by dinaburg and mazel on the competition between the external field and the entropic repulsion with the wall leading in this case to the phenomenon of layering phase transitions in particular we show using the pirogov sinai scheme as given by zahradnik that there exists a unique critical value h fl in the interval 88 4 k 4e 4 k such that for all h h i h and fl large enough there exists a unique infinite volume gibbs state the typical configurations are small perturbations of the ground state represented by a surface at height k 1 above the wall moreover for the same choice of the thermodynamic parameters the influence of the boundary conditions of the gibbs measure in a finite cube decays exponentially fast with the distance from the boundary when h h fl we prove instead the convergence of the cluster expansion for both k and k 1 boundary conditions this fact signals the presence of a phase transition in the second paper of this series we will consider a glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external field h using the results proven in this paper we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free absent

hilbert series and degree bounds for matrix semi invariants abstract we study the ring r n m of invariants for the left right action of sl n sl n on m tuples of n n complex matrices we show that r 3 m is generated by invariants of degree 309 for all m then we use a combinatorial description of the invariants to show that r n m cannot be generated by invariants of degree n 2 for large m we also compute the hilbert series for several cases

non gaussian surface pinned by a weak potential we consider a model of a two dimensional interface of the continuous sos type with finite range strictly convex interactions we prove that under an arbitrarily weak pinning potential the interface is localized we consider the cases of both square well and potentials our results extend and generalize previous results for the case of nearest neighbours gaussian interactions in 7 and 11 we also obtain the tail behaviour of the height distribution which is not gaussian

infinite grassmannians and moduli spaces of g bundles let c be a smooth projective irreducible algebraic curve over c of any genus and g a connected simply connected simple affine algebraic group over c in this paper we elucidate the relationship between 1 the space of vacua conformal blocks defined in conformal field theory using an integrable highest weight representation of the affine kac moody algebra associated to g and 2 the space of regular sections generalised theta functions of a line bundle on the moduli space ta of semistable principal g bundles on c fix a point p in c and let p resp kp be the completion of the local ring p of c at p resp the quotient field of c p let g kp the kp rational points of the algebraic group g be the loop group of g and let g p be the standard maximal parahoric subgroup of then the generalised flag variety x n is an inductive limit of projective varieties in fact of generalised schubert varieties one has a natural r line bundle s on x cf sect 2 2 and the picard group pic x is isomorphic to z which is generated by s proposition 2 3 where f is the universal central extension of by the multiplicative group c cf sect 2 2 by an analogue of the borel weil theorem proved in the kac moody setting by kumar and also by mathieu the space h 162 s of the regular sections of the line bundle y dxo y xo for any d 0 is canonically isomorphic with the full vector space dual v dxo of the integrable highest weight irreducible module v dxo of the affine kacmoody group or of the associated affine kac moody algebra which is a certain one dimensional central extension of the loop algebra lie g with highest weight dxo cf sect 6 1

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

on the littlewood richardson polynomials we prove the equivalence of several descriptions of generators of rings of semiinvariants of quivers due to domokos and zubkov schofield and van den bergh and our earlier work we also show that the dimensions of semi invariants of weights n depend polynomially on n 2002 elsevier science usa all rights reserved

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

bounds on the mass for the high dimensional gaussian lattice field between two hard walls we consider the d 3 dimensional gaussian lattice field confined between two hard walls we show that the field becomes massive and identify the precise asymptotic behavior of the mass and the variance of the field as the height of the wall goes to infinity

universality of critical behaviour in a class of recurrent random walks let x0 0 x1 x2 be an aperiodic random walk generated by a sequence 1 2 of i i d integer valued random variables with common distribution p having zero mean and finite variance for ann step trajectory and a monotone convex functionv withv 0 0 define further let be the set of all non negative paths compatible with the boundary conditionsx0 a x n b we discuss asymptotic properties of under the probability distribution n and 0 za bn being the corresponding normalization if v grows not faster than polynomially at infinity define h to be the unique solution to the equation our main result reads that as 0 the typical height of x n scales as h and the correlations along decay exponentially on the scale h 2 using a suitable blocking argument we show that the distribution tails of the rescaled height decay exponentially with critical exponent 3 2 in the particular case of linear potential v the characteristic length h is proportional to 1 3 as 0

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

a note on the decay of correlations under δ pinning we prove that for a class of massless interface models on 2 introduction of an arbitrarily small pinning self potential leads to exponential decay of correlation or in other words to creation of mass

enhanced interface repulsion from quenched hard wall randomness we consider the harmonic crystal or massless free field that is the centered gaussian field with covariance given by the green function of the simple random walk on d our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition to be larger than is an iid field which is also independent of for every x in a large region with n a positive integer and d a bounded subset of d we are mostly motivated by results for given typical realizations of quenched set up since the conditioned harmonic crystal may be seen as a model for an equilibrium interface living in a d 1 dimensional space constrained not to go below an inhomogeneous substrate that acts as a hard wall we consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of 0 is heavier than gaussian while essentially no effect is observed if the tail is sub gaussian in the critical case that is the one of approximately gaussian tail the interplay of the two sources of randomness and leads to an enhanced repulsion effect of additive type this generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal large deviation type asymptotics as of the probability that lies above in d n

the multiplicative eigenvalue problem and deformed quantum cohomology abstract we construct deformations of the small quantum cohomology rings of homogeneous spaces g p and obtain an irredundant set of inequalities determining the multiplicative eigen polytope for the compact form k of g

universal denominators of hilbert series the denominator of the hilbert series of a finitely generated r module m does not always divide the denominator of the hilbert series of r for this reason we define the universal denominator the universal denominator of a module m is the least common multiple of the denominators of the hilbert series of all submodules of m the universal denominator behaves nicely with respect to short exact sequences and tensor products it also has interesting geometric interpretations formulas are given for the universal denominator for rings of invariants dixmier gave a conjectural formula for the denominator of the hilbert series of invariants of binary forms we show that the universal denominator is actually equal to dixmier s formula in that case

an invariance principle to ferrari spohn diffusions we prove an invariance principle for a class of tilted 1 1 dimensional sos models or equivalently for a class of tilted random walk bridges in mathbb z the limiting objects are stationary reversible ergodic diffusions with drifts given by the logarithmic derivatives of the ground states of associated singular sturm liouville operators in the case of a linear area tilt we recover the ferrari spohn diffusion with log airy drift which was derived in ferrari and spohn ann probab 33 4 1302 1325 2005 in the context of brownian motions conditioned to stay above circular and parabolic barriers

a remark on the low temperature behavior of the sos interface in half space we investigate the low temperature phase diagram of thed dimensional d 2 solid on solid sos interface constrained to lie above a rigid wall to which it is attracted by a constant force we prove uniqueness of the gibbs state and exponentially fast convergence in the diameter of the domain of the finite volume expectation of local observables for all values of parameters where uniqueness in the class of translation periodic limit gibbs states was established previously these domains of uniqueness are bordered by lines on which the system undergoes a layering transition

entropic repulsion and the maximum of the two dimensional harmonic crystal we consider the lattice version of the free field in two dimensions also called harmonic crystal the main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall the basic ingredient of the proof is the analysis of the maximum of the field which requires a multiscale analysis reducing the problem essentially to a problem on a field with a tree structure

entropic repulsion for the high dimensional gaussian lattice field between two walls the main aim of this paper is to discuss the entropic repulsion of random interfaces between two hard walls we consider the d 3 dimensional gaussian lattice field on n n n n d d and identify the repulsion of the field as n under the condition that the field lies between two hard walls at the height level 0 and l in n where l is large enough but finite we also study the same problem for two layered interfaces case

the strange duality conjecture for generic curves let sux r be the moduli space of semi stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve x of genus g 1 over c recall that a vector bundle e on x is called semi stable if for any subbundle v deg v rk v deg e rk e points of sux r correspond to isomorphism classes of semi stable rank r vector bundles with trivial determinant up to an equivalence relation for any line bundle l of degree g 1 onx define l e sux r h e l 1 this turns out be a non zero cartier divisor whose associated line bundle l o l does not depend upon l it is known that l generates the picard group of sux r for this and the precise definition of l in terms of determinant of cohomology see dn let u x k be the moduli space of semi stable rank k and degree k g 1 bundles on x recall that on u x k there is a canonical non zero theta cartier divisor k whose underlying set is f u x k h x f 0 put m o k consider the natural map k r sux r u x k u x kr given by tensor product from the theorem of the square it follows that k rm is isomorphic to l m the canonical element kr h0 u x kr m and the kunneth theorem give a map well defined up to scalars

the equations for the moduli space of n points on the line a central question in invariant theory is that of determining the relations among invariants geometric invariant theory git quotients come with a natural ample line bundle and hence often a natural projective embedding this question translates to determining the equations of the moduli space under this embedding this article deals with one of the most classical quotients the space of ordered points on the projective line we show that under any weighting of the points this quotient is cut out scheme theoretically by a particularly simple set of quadric relations with the single exception of the segre cubic threefold the space of six points with equal weight we also show that the ideal of relations is generated in degree at most four and we give an explicit description of the generators if all the weights are even e g in the case of equal weight for odd n then we show that the ideal of relations is generated by quadrics

entropic repulsion of the lattice free field consider the massless free field on thed dimensional lattice d d 3 that is the centered gaussian field on open image in new window with covariances given by the green function of the simple random walk on d we show that the probability that all the spins are positive in a box of volumen d decays exponentially at a rate of ordernd 2 logn and compute explicitly the corresponding constant in terms of the capacity of the unit cube the result is extended to a class of transient random walks with transition functions in the domain of the normal and stable law

entropic repulsion for the free field pathwise characterization in d 3 we study concentration properties of the lattice free field i e the centered gaussian field with covariance given by the green function of the discrete laplacian when constrained to be positive in a region of volume o n d hard wall condition it has been shown in 3 that as n the conditioned field is pushed to infinity more precisely the typical value of the variable to leading order is and the exact value of c was found it was moreover conjectured that the conditioned field once this diverging height is subtracted converges weakly to the lattice free field here we prove this conjecture along with other explicit bounds always in the direction of clarifying the intuitive idea that the free field with hard wall conditioning merely translates away from the hard wall we give also a proof alternative to the one presented in 3 of the lower bound on the probability that the free field is everywhere positive in a region of volume n d

dyson ferrari spohn diffusions and ordered walks under area tilts we consider families of non colliding random walks above a hard wall which are subject to a self potential of tilted area type we view such ensembles as effective models for the level lines of a class of 2 1 dimensional discrete height random surfaces in statistical mechanics we prove that under rather general assumptions on the step distribution and on the self potential such walks converge under appropriate rescaling to non intersecting ferrari spohn diffusions associated with limiting sturm liouville operators in particular the limiting invariant measures are given by the squares of the corresponding slater determinants

interface surface tension and reentrant pinning transition in the 2d ising model we develop a new way to look at the high temperature representation of the ising model up to the critical temperature and obtain a number of interesting consequences in the two dimensional case it is possible to use these tools to prove results on phase separation lines in the whole phase coexistence regime by way of a duality transformation we illustrate the power of these techniques by studying an ising model with a boundary magnetic field in which a reentrant pinning transition takes place more precisely we show that the typical configurations of the model can be described at the macroscopic level by interfaces which are solutions of the corresponding thermodynamic variational problem this variational problem is solved explicitly there exist values of the boundary magnetic field and temperatures 0 t 2 but is pinned for t 1 t 3 it is not pinned an important property of the surface tension which is used in this paper is the sharp triangle inequality about which we report some new results the techniques used in this work are robust and can be used in a variety of different situations

wetting of gradient fields pathwise estimates we consider the wetting transition in the framework of an effective interface model of gradient type in dimension 2 and higher we prove pathwise estimates showing that the interface is localized in the whole thermodynamically defined partial wetting regime considered in earlier works moreover we study how the interface delocalizes as the wetting transition is approached our main tool is reflection positivity in the form of the chessboard estimate

pinning of an interface by a weak potential we prove that in a two dimensional gaussian sos model with a small attractive potential the height of the interface remains bounded no matter how small the potential is this is in sharp contrast with the free situation in which the interface height diverges logarithmically in the thermodynamic limit

entropic repulsion of the lattice free field ii the 0 boundary case this paper is a continuation of 5 we consider the euclidean massless free field on a boxvn of volumend with o boundary condition that is the centered gaussian field with covariances given by the green function of the simple random walk on d d 3 killed as it exitsvn we show that the probability that all the spins are positive in the boxvn decays exponentially at a surface ratend 1 this is in contrast with the ratend 2 logn for the infinite field of 5

dynamics of 2 1 dimensional sos surfaces above a wall slow mixing induced by entropic repulsion we study the glauber dynamics for the 2 1 d solid on solid model above a hard wall and below a far away ceiling on an l times l box of z 2 with zero boundary conditions at large inverse temperature beta it was shown by bricmont el mellouki and frohlich 1986 that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height h asymp 1 beta log l as an essential step in understanding the effect of entropic repulsion on the glauber dynamics we determine the equilibrium height h to within an additive constant h 1 4 beta log l o 1 we then show that starting from zero initial conditions the surface rises to its final height h through a sequence of metastable transitions between consecutive levels the time for a transition from height h ah a in 0 1 to height h 1 is roughly exp c l a for some constant c 0 in particular the mixing time of the dynamics is exponentially large in l i e t mix geq e c l we also provide the matching upper bound t mix leq e c l requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques finally to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in l