the γ limit of the two dimensional ohta kawasaki energy droplet arrangement via the renormalized energy this is the second in a series of papers in which we derive a expansion for the two dimensional non local ginzburg landau energy with coulomb repulsion known as the ohta kawasaki model in connection with diblock copolymer systems in this model two phases appear which interact via a nonlocal coulomb type energy here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction thus creating small droplets of the minority phase in a sea of the majority phase in our previous paper we computed the limit of the leading order energy which yields the averaged behavior for almost minimizers namely that the density of droplets should be uniform here we go to the next order and derive a next order limit energy which is exactly the coulombian renormalized energy obtained by sandier and serfaty as a limiting interaction energy for vortices in the magnetic ginzburg landau model the derivation is based on the abstract scheme of sandier serfaty that serves to obtain lower bounds for 2 scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem thus without appealing to the euler lagrange equation we establish for all configurations which have almost minimal energy the asymptotic roundness and radius of the droplets and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense via a kind of equivalence the obtained results also yield an expansion of the minimal energy and a characterization of the zero super level sets of the minimizers for the original ohta kawasaki energy this leads to the expectation of seeing triangular lattices of droplets as energy minimizers

the sharp quantitative sobolev inequality for functions of bounded variation abstract the classical sobolev embedding theorem of the space of functions of bounded variation bv r n into l n r n is proved in a sharp quantitative form

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

large deviations for occupation time profiles of random interlacements we derive a large deviation principle for the density profile of occupation times of random interlacements at a fixed level in a large box of mathbb z d d ge 3 as an application we analyze the asymptotic behavior of the probability that atypically high values of the density profile insulate a macroscopic body in a large box as a step in this program we obtain a similar large deviation principle for the occupation time measure of brownian interlacements at a fixed level in a large box of mathbb r d and we derive a new identity for the laplace transform of the occupation time measure which is based on the analysis of certain schrodinger semi groups

on the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems it is the purpose of this paper to discuss the equality sign in d and in related inequalities in particular we shall give an answer to the following question does the equality in d imply that u u this question has been raised by g polya g szego in their book on isoperimetric inequalities l l p 186 they state that their method of proving d unfortunately leaves little hope for discussing the sign of equality another comment on this question can be found in 8 where e lieu conjectures that probably a strict version of d is true as it turns out the doubts of g polya g szeg were indeed justified because there are functions u for which equality holds in d while u does not coincide with u in section 1 of this paper i give an example of such a function on the other hand the conjecture of e lieb is also reasonable since generically i can give a positive answer to the question above there is a dense subset of nonnegative functions in wol 2 1 1 for which the equality in d implies

entropic repulsion for a gaussian lattice field with certain finite range interaction consider the centered gaussian field on z d d 2l 1 with covariance matrix given by j l k q j j 1 where is the discrete laplacian on z d 1 l k and q j r l j k are constants satisfying j l k q j r j 0 for r 0 2 and a certain additional condition we show the probability that all spins are positive in a box of volume n d decays exponentially at a rate of order n d 2l log n and under this hard wall condition the local sample mean of the field is repelled to a height of order log n this extends the previously known result for the case that the covariance is given by the green function of simple random walk on z d i e k l 1 q 1 1

critical large deviations in harmonic crystals with long range interactions we continue our study of large deviations of the empirical measures of a massless gaussian field on z d whose covariance is given by the green function of a long range random walk in this paper we extend techniques and results of bolthausen and deuschel to the nonlocal case of a random walk in the domain of attraction of the symmetric stable law with a 0 2 a d in particular we show that critical large deviations occur at the capacity scale n d with a rate function given by the dirichlet form of the embedded stable process we also prove that if we impose zero boundary conditions the rate function is given by the dirichlet form of the killed stable process

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

generalizations of fixed point theorems and computation as a powerful mechanism fixed point theorems have many applications in mathematical and economic analysis in this paper the well known brouwer fixed point theorem and kakutani fixed point theorem are generalized to a class of nonconvex sets and a globally convergent homotopy method is developed for computing fixed points on this class of nonconvex sets

the perimeter inequality under steiner symmetrization cases of equality steiner symmetrization is known not to increase perimeter of sets in rn the sets whose perimeter is preserved under this symmetrization are characterized in the present paper

robustness of the non gibbsian property some examples we discuss some examples of measures on lattice systems which lack the property of being a gibbs measure in a rather strong sense

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

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

large deviations and variational principle for harmonic crystals we consider massless gaussian fields with covariance related to the green function of a long range random walk on d these are viewed as gibbs measures for a linear quadratic interaction we establish thermodynamic identities and prove a version of gibbs variational principle showing that translation invariant gibbs measures are characterized as minimizers of the relative entropy density we then study the large deviations of the empirical field of a gibbs measure we show that a weak large deviation principle holds at the volume order with rate given by the relative entropy density

the construction of the d 1 dimensional gaussian droplet the aim of this note is to study the asymptotic behavior of a gaussian random field under the condition that the variables are positive and the total volume under the variables converges to some fixed numberv 0 in the context of statistical mechanics this corresponds to the problem of constructing a droplet on a hard wall with a given volume we show that properly rescaled the profile of a gaussian configuration converges to a smooth hypersurface which solves a quadratic variational problem our main tool is a scaling dependent large deviation principle for random hypersurfaces

hole probability for nodal sets of the cut off gaussian free field abstract let g be a closed connected surface equipped with a riemannian metric let n n n and n n n be the increasing sequence of eigenvalues and the sequence of corresponding l 2 normalized eigenfunctions of the laplacian on for each l 0 we consider l 0 n l n n n where the n are i i d centered gaussians with variance 1 as l l converges a s to the gaussian free field on in the sense of distributions we first compute the asymptotic behavior of the covariance function for this family of fields as l we then use this result to obtain the asymptotics of the probability that l is positive on a given open proper subset with smooth boundary in doing so we also prove the concentration of the supremum of l around 1 2 ln l

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

entropic repulsion for the gaussian free field conditioned on disconnection by level sets we investigate level set percolation of the discrete gaussian free field on mathbb z d d ge 3 in the strongly percolative regime we consider the event that the level set of the gaussian free field below a level alpha disconnects the discrete blow up of a compact set a subseteq mathbb r d from the boundary of an enclosing box we derive asymptotic large deviation upper bounds on the probability that the local averages of the gaussian free field deviate from a specific multiple of the harmonic potential of a when disconnection occurs these bounds combined with the findings of the recent work by duminil copin goswami rodriguez and severo show that conditionally on disconnection the gaussian free field experiences an entropic push down proportional to the harmonic potential of a in particular due to the slow decay of correlations the disconnection event affects the field on the whole lattice furthermore we provide a certain profile description for the field in the presence of disconnection we show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of a it locally retains the structure of a gaussian free field shifted by a constant value our proofs rely crucially on the solidification estimates developed in nitzschner and sznitman to appear in j eur math soc arxiv 1706 07229

stability in the isoperimetric problem for convex or nearly spherical domains in ⁿ for convex bodies d in rn the deviation d from spherical shape is estimated from above in terms of the dimensionless isoperimetric deficiency a of d as follows d f a for a sufficiently small here f is an explicit elementary function vanishing continuously at 0 the estimate is sharp as regards the order of magnitude of f the dimensions n 2 and 3 present anomalies as to the form of f in the planar case n 2 the result is contained in an inequality due to t bonnesen a qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies d in rn in the sense that as d varies d p 0 for a a 0 the proof of the estimate d f a is based on a related estimate in the case of domains not necessarily convex that are supposed a priori to be nearly spherical in a certain sense

vacant set of random interlacements and percolation we introduce a model of random interlacements made of a countable collection of doubly infinite paths on d d 3 a nonnegative parameter u measures how many trajectories enter the picture this model describes in the large n limit the microscopic structure in the bulk which arises when considering the disconnection time of a discrete cylinder n d 1 x by simple random walk or the set of points visited by simple random walk on the discrete torus n d at times of order u n d in particular we study the percolative properties of the vacant set left by the interlacement at level u which is an infinite connected translation invariant random subset of d we introduce a critical value u such that the vacant set percolates for u u our main results show that u is finite when d 3 and strictly positive when d 7

critical large deviations for gaussian fields in the phase transition regime i we investigate large deviations for the empirical distribution functional of a gaussian random field on mathbb r mathbb z d d geq 3 in the phase transition regime we first prove that the specific entropy governs an n d volume order large deviation principle outside the gibbsian class within the gibbsian class we derive an n d 2 capacity order large deviation principle with exact rate function and we apply this result to the asymptotics of microcanonical ensembles we also give a spins profile description of the field and show that smooth profiles obey n d 2 order large deviations whereas discontinuous profiles obey n d 1 surface order large deviations

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