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

isoperimetric phase transitions of two dimensional droplets we consider two dimensional assemblies of particles governed by hamiltonians depending on the area and the perimeter of their convex hull provided the hamiltonian is quadratically homogeneous in the coordinates we find an exact formula for the free energy phase transitions resulting from the competition between area and perimeter can easily be produced and explicitly dealt with we illustrate those features by a simple example undergoing a second order transition

schauder estimates for degenerate monge ampere equations and smoothness of the eigenfunctions we obtain c 2 beta estimates up to the boundary for solutions to degenerate monge ampere equations of the type begin aligned det d 2 u f text in omega quad f sim text dist alpha cdot partial omega text near partial omega alpha 0 end aligned as a consequence we obtain global c infty estimates up to the boundary for the eigenfunctions of the monge ampere operator det d 2 u 1 n on smooth bounded uniformly convex domains in mathbb r n

diffusion equation techniques in stochastic monotonicity and positive correlations a diffusion equation approach is investigated for the study of stochastic monotonicity positive correlations and the preservation of lipschitz functions necessary and sufficient conditions are given for diffusion semigroups to be stochastically monotonic and to preserve the class of positively correlated measures applications are given which discuss the shape of the ground state for schrodinger operators v with fkg potentialsv

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

global calderon zygmund theory for nonlinear elliptic obstacle problems with asymptotically regular nonlinearities abstract we study a nonlinear elliptic problem with an irregular obstacle in a bounded nonsmooth domain when the nonlinearity is merely asymptotically regular we find an optimal regularity requirement on the associated nonlinearity and a minimal geometric condition on the boundary to ensure a global calderon zygmund estimate for such an asymptotically regular obstacle problem we assume that the associated nonlinearity has a small bmo and the boundary is sufficiently flat in the reifenberg sense

asymptotically regular problems i higher integrability abstract we consider weak solutions u of non linear systems of partial differential equations assuming that the system exhibits a certain kind of elliptic behavior near infinity we prove higher integrability results for the gradient du in particular we establish holder continuity of u in low dimensions moreover we obtain analogous results for vectorial minimizers of multi dimensional variational integrals finally we discuss an extension to minimizing sequences and applications to generalized minimizers

radially symmetric boundary value problems for real and complex elliptic monge ampere equations abstract radially symmetric dirichlet and neumann problems for real and complex monge ampere equations are considered existence of radially symmetric solutions is proved by transforming the differential equations into integral ones solvable by means of fixed point arguments then taking advantage of integral formulae regularity and convexity of the radial solutions are checked fairly weak assumptions are required in that process in the real case a priori radial symmetry is also discussed

hydrodynamic limit for φ interface model on a wall we consider random evolution of an interface on a hard wall under periodic boundary conditions the dynamics are governed by a system of stochastic differential equations of skorohod type which is langevin equation associated with massless hamiltonian added a strong repelling force for the interface to stay over the wall we study its macroscopic behavior under a suitable large scale space time limit and derive a nonlinear partial differential equation which describes the mean curvature motion except for some anisotropy effects with reflection at the wall such equation is characterized by an evolutionary variational inequality

on path correlation and pert bias most studies of project time estimation assume that a activity times are mutually independent random variables many also assume that b path completion times are mutually independent in this paper we subject the impact of both these assumptions to close scrutiny using tools from multivariate analysis we make a theoretical study of the direction of the error in the classical pert method of estimating mean project completion time when correlation is ignored we also investigate the effect of activity dependence on the normality of path length via simulation

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 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

thick points of the gaussian free field let u c be a bounded domain with smooth boundary and let f be an instance of the continuum gaussian free field on u with respect to the dirichlet inner product u f x g x dx the set t a u of a thick points of f consists of those z u such that the average of f on a disk of radius r centered at z has growth a log 1 r as r 0 we show that for each 0 a 2 the hausdorff dimension of t a u is almost surely 2 a that 2 a t a u when 0 2 furthermore we prove that t a u is invariant under conformal transformations in an appropriate sense the notion of a thick point is connected to the liouville quantum gravity measure with parameter y given formally by dz e 2 f z dz considered by duplantier and sheffield

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

sos approximants for potts crystal shapes abstract an approximation made when replacing the orientation dependent potts model surface tension by the sos one is rigorously estimated as a result an sos model may be used for an explicit and precise construction of crystal and meniscus shapes for potts models at low temperatures

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

weighted orlicz estimates for general nonlinear parabolic equations over nonsmooth domains we introduce a parabolic analogue of muckenhoupt weights to study optimal weighted regularity in orlicz spaces for a general nonlinear parabolic problem of p laplacian type in divergence form over a nonsmooth domain assuming that the nonlinearity is measurable with respect to the time variable and has a small bounded mean oscillation bmo with respect to the spatial variables that the lateral boundary of the parabolic cylinder is reifenberg flat and that the associated weight belongs to a suitable parabolic muckenhoupt class we obtain a global gradient estimate for such a nonlinear parabolic problem by essentially proving that the gradient of the weak solution is as globally integrable as the nonhomogeneous term in the weighted orlicz space our results extend the existing regularity estimates in lebesgue spaces to weighted orlicz spaces

on the microscopic validity of the wulff construction and of the generalized young equation for a large class of 1 1 dimensional interfaces of the solid on solid type we prove on a microscopic basis the validity of the wulff construction and of the generalized young equation which gives the contact angle of a sessile drop on a wall our proof relies on a new method to treat random walks with a finite number of global constraints

boundary behavior of large solutions to the monge ampere equations with weights abstract this paper is concerned with the monge ampere equation det d 2 u x b x f u x x where is a strictly convex bounded smooth domain in r n with n 2 and b c which is positive in but may be vanishing on the boundary we find a new structure condition on f which plays a crucial role in the boundary behavior of strictly convex large solutions our results are obtained in a more general setting than those in cirstea and trombetti 2008 12 where f is regularly varying at infinity with index p n

liouville metric of star scale invariant fields tails and weyl scaling we study the liouville metric associated to an approximation of a log correlated gaussian field with short range correlation we show that below a parameter gamma c 0 the left right length of rectangles for the riemannian metric e gamma phi 0 n ds 2 with various aspect ratio is concentrated with quasi lognormal tails that the renormalized metric is tight when gamma min gamma c 0 4 and that subsequential limits are consistent with the weyl scaling

lorentz estimate for nonlinear parabolic obstacle problems with asymptotically regular nonlinearities abstract in this paper we mainly prove an interior lorentz estimate of the spatial gradient of weak solutions to a class of nonlinear parabolic problems with an irregular obstacle in a bounded cylindrical domain when the nonlinearity is merely asymptotically regular we assume that the associated nonlinearity satisfies certain ellipticity and growth conditions and the r vanishing condition here the approach of constructing a regular obstacle problem via poisson s formula is used to establish lorentz estimate for such an asymptotically regular obstacle problem

existence and estimates of solutions to a singular dirichlet problem for the monge ampere equation given a strictly convex smooth and bounded domain in rn we establish the existence of a negative convex solution in c c with zero boundary value to the singular monge ampere equation det d2u p x g u an associated dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem estimates of solutions will also be given and regularity of solutions will be deduced from the estimates

on singular boundary value problems for the monge ampere operator abstract we consider different types of singular boundary value problems for the monge ampere operator the approach is based on existing regularity theory and a subsolution supersolution method nonexistence and uniqueness results are also given

optimal uniqueness theorems and exact blow up rates of large solutions in this paper we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non linearity and combine these results with the localization method introduced in lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 to prove that any large solution l of u a x up p 1 a 0 must satisfy limx x0l x fx0 dist x i0 pp 1p 1p 1p 1p 1foreachx0 where fx0 t t 0sfx01p 1 p 1p 1ds i0 limt 0fx0 t fx0 t fx0 t 2 and fx0 is any smooth extension of the boundary normal section of a at x0 i e fx0 t a x0 tnx0 t 0 t 0 subsequently nx0 stands for the outward unit normal at x0 therefore the theory can be extended to cover the general case when fx0 l1p 1

resistance of the edwards walk an analytical and numerical study of the scaling of edwards random walks defined as the zero component limit n to 0 of an n component g0 mod phi 2 mod 2 theory is done in three dimensions the hausdorf dimension 1 nu and the resistance exponent x is calculated using renormalisation group methods in 4 in dimensions the numerical study is done by defining a monte carlo algorithm on the cubic lattice to generate ensembles of edwards walks

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

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

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

deconvolving multivariate kernel density estimates from contaminated associated observations we consider the estimation of the multivariate probability density function f x sub 1 x sub p of x sub 1 x sub p of a stationary positively or negatively associated pa or na random process x sub i sub i 1 sup spl infin from noisy observations both ordinary smooth and super smooth noise are considered quadratic mean and asymptotic normality results are established

lorentz estimates for degenerate and singular evolutionary systems we prove estimates of calderon zygmund type for evolutionary p laplacian systems in the setting of lorentz spaces we suppose the coefficients of the system to satisfy only a vmo condition with respect to the space variable our results hold true mutatis mutandis also for stationary p laplacian systems

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

critical region for droplet formation in the two dimensional ising model we study the formation dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence specifically we consider the 2d ising model in volumes of size l2 inverse temperature c and overall magnetization conditioned to take the value m l2 2m vl where c 1 is the critical temperature m m is the spontaneous magnetization and vl is a sequence of positive numbers we find that the critical scaling for droplet formation dissolution is when vl3 2l 2 tends to a definite limit specifically we identify a dimensionless parameter proportional to this limit a non trivial critical value c and a function such that the following holds for c there is a single wulff shaped droplet containing a fraction c 2 3 of the magnetization deficit and there are no other droplets beyond the scale of log l moreover and are related via a universal equation that apparently is independent of the details of the system

the empirical process of a short range dependent stationary sequence under gaussian subordination consider the stationary sequencex 1 g z 1 x 2 g z 2 whereg is an arbitrary borel function andz 1 z 2 is a mean zero stationary gaussian sequence with covariance functionr k e z 1 z k 1 satisfyingr 0 1 and 1 r k m where withi denoting the indicator function andf the continuous marginal distribution function of the sequence x n the integerm is the hermite rank of the family i g x f x x r letf n be the empirical distribution function ofx 1 x n we prove that asn the empirical processn 1 2 f n f converges in distribution to a gaussian process in the spaced

blow up rates of large solutions for elliptic equations in this paper we mainly study the boundary behavior of solutions to boundary blow up elliptic problems for more general nonlinearities f which may be rapidly varying at infinity u b x f u x u where is a bounded domain with smooth boundary in rn and b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary further when f s sp f1 s for s sufficiently large where p 1 and f1 is normalized regularly varying at infinity with index p1 0 p we show the influence of the geometry of on the boundary behavior for solutions to the problem we also give the existence and uniqueness of solutions

partial to complete wetting a microscopic derivation of the young relation this paper is devoted to the study of the young equation which gives a connection between surface tensions and contact angle we derive the generalized form of this equation for anisotropic models using thermodynamic considerations in two dimensions with sos like approximations of the interface we prove that the surface tension may be computed explicitly as a simple integral which of course depends upon the orientation of the interface this allows a complete study of the wetting transition when a constant wall attraction is taken into account within the sos and gaussian models we therefore give a complete analysis of the variation of the contact angle with the temperature for those models it is found that for certain values of the parameters two wetting transitions may successively appear one at low temperature and one at high temperature giving the following states film droplet film this study rests upon the generalized young equation the validity of which is proved for the gaussian model with a constant wall attraction using microscopic considerations

hard rods statistics of parking configurations we compute the correlation function in the equilibrium version of renyi s parking problem the correlation length is found to diverge as 2 1 2 1 2 when 1 maximum density and as 2 2 1 2 when 1 2 minimum density

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

extremes of the discrete two dimensional gaussian free field we consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high or low we then extend some of our computations to the case of the free field conditioned on being everywhere nonnegative for example we compute the width of the largest downward spike of a given length through the prism of these results we find that the extrema of the free field under entropic repulsion minus its mean and those of the unconditioned free field are identical finally when compared to previous results these findings reveal a suggestive analogy between the square of the free field and the two dimensional simple random walk on the discrete torus

optimal boundary regularity for a singular monge ampere equation abstract in this paper we study the optimal global regularity for a singular monge ampere type equation which arises from a few geometric problems we find that the global regularity does not depend on the smoothness of domain but it does depend on the convexity of the domain we introduce a type to describe the convexity as a result we show that the more convex is the domain the better is the regularity of the solution in particular the regularity is the best near angular points

a characterization of monotone unidimensional latent variable models recently the problem of characterizing monotone unidimensional latent variable models for binary repeated measures was studied by ellis and van den wollenberg and by junker we generalize their work with a de finetti like characterization of the distribution of repeated measures x x 1 x 2 that can be represented with mixtures of likelihoods of independent but not identically distributed random variables where the data satisfy a stochastic ordering property with respect to the mixing variable the random variables x j may be arbitrary real valued random variables we show that the distribution of x can be given a monotone unidimensional latent variable representation that is useful in the sense of junker if and only if this distribution satisfies conditional association ca and a vanishing conditional dependence vcd condition which asserts that finite subsets of the variables in x become independent as we condition on a larger and larger segment of the remaining variables in x it is also interesting that the mixture representation is in a certain ordinal sense unique when ca and vcd hold the characterization theorem extends and simplifies the main result of junker and generalizes methods of ellis and van den wollenberg to a much broader class of models exchangeable sequences of binary random variables also satisfy both ca and vcd as do exchangeable sequences arising as location mixtures in the same way that de finetti s theorem provides a path toward justifying standard i i d mixture components in hierarchical models on the basis of our intuitions about the exchangeability of observations this theorem justifies one dimensional latent variable components in hierarchical models in terms of our intuitions about positive association and redundancy between observations because these conditions are on the joint distribution of the observable data x they may also be used to construct asymptotically power 1 tests for unidimensional latent variable models

m matrices as covariance matrices of multinormal distributions abstract the class of multivariate normal densities n 0 whose inverse covariance matrix 1 is an m matrix is equivalent to this normal density being multivariate totally positive of order 2 mtp 2 equivalent characterizations are given in terms of certain partial correlation coefficients being positive it is further shown that related partial and multiple regression coefficients and canonical correlation are positive when is an m matrix the corresponding normal random vector components are negatively associated this concept and some extensions are discussed

weighted lorentz estimate for asymptotically regular parabolic equations of p x t laplacian type abstract we prove a global weighted lorentz estimate of the spatial gradients of weak solution to the cauchy dirichlet problem for asymptotically regular parabolic equations of p x t laplacian type in a reifenberg flat domain it is mainly assumed that the associated regular nonlinearity is measurable with respect to the time variable and satisfies the r vanishing condition with respect to the spatial variables

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

lorentz estimates for obstacle parabolic problems we prove that the spatial gradient of variational solutions to parabolic obstacle problems of p laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of lorentz spaces

large solutions to the monge ampere equations with nonlinear gradient terms existence and boundary behavior abstract in this paper we obtain conditions about the existence and boundary behavior of strictly convex solutions to the monge ampere equations with boundary blow up det d 2 u x b x f u x u q x u and det d 2 u x b x f u x 1 u q x u where is a strictly convex bounded smooth domain in r n with n 2 q 0 n or q 0 n b c which is positive in but may vanish or blow up on the boundary f c 0 f 0 0 and f is strictly increasing on 0 or f c r f s 0 s r and f is strictly increasing on r

large deviations and concentration properties for ϕ interface models we consider the massless field with zero boundary conditions outside d n d d n n d a suitable subset of d i e the continuous spin gibbs measure n on d n with hamiltonian given by h x y x y 1 v x y and x 0 for x d n c the interaction v is taken to be strictly convex and with bounded second derivative this is a standard effective model for a d 1 dimensional interface represents the height of the interface over the base d n due to the choice of scaling of the base we scale the height with the same factor by setting n n we study various concentration and relaxation properties of the family of random surfaces n and of the induced family of gradient fields n n as the discretization step 1 n tends to zero n in particular we prove a large deviation principle for n and show that the corresponding rate function is given by d u x dx where is the surface tension of the model this is a multidimensional version of the sample path large deviation principle we use this result to study the concentration properties of n under the volume constraint i e the constraint that 1 n d x dn n x stays in a neighborhood of a fixed volume v 0 and the hard wall constraint i e n x 0 for all x this is therefore a model for a droplet of volume v lying above a hard wall we prove that under these constraints the field n of rescaled heights concentrates around the solution of a variational problem involving the surface tension as it would be predicted by the phenomenological theory of phase boundaries our principal result however asserts local relaxation properties of the gradient field n n to the corresponding extremal gibbs states thus our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments the proofs have both probabilistic and analytic aspects essential analytic tools are p estimates for elliptic equations and the theory of young measures on the side of probability tools a central role is played by the helffer sjostrand 31 pde representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of t funaki and h spohn 25 on the structure of gradient fields

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

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

phase separation in random cluster models iii circuit regularity we study the droplet that results from conditioning the subcritical fortuin kasteleyn random cluster model on the presence of an open circuit encircling the origin and enclosing an area of at least or exactly n 2 in this paper we prove that the resulting circuit is highly regular we define a notion of a regeneration site in such a way that for any such site v the circuit cuts through the radial line segment through v only at v we show that provided that the conditioned circuit is centred at the origin in a natural sense the set of regeneration sites reaches into all parts of the circuit with maximal distance from one such site to the next being at most logarithmic in n with high probability the result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations and as such it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in hammond arxiv 1001 1527 2009 arxiv 1001 1528 2009

ferromagneticity of simplicial fields on two dimensional compact manifolds smooth triangulations of a compact smooth connected two dimensional riemannian manifold m are considered the q simplicial fields are defined with values in the space of q cochains and a natural gaussian measure is defined giving their distribution with covariance defined essentially in terms of the combinatorial laplacian c in the continuum limit this measure for q 0 is the free quantum field measure over m in this case it is shown that for a certain collection of triangulations there exists a sequence of subdivisions of each triangulation such that the corresponding measure is ferromagnetic it is also shown that for sufficiently fine subdivisions m2i m 0 has nonpositive off diagonal elements the proofs are obtained by a result on triangulations by simplexes with acute angles it is also proven that the probability measures describing quantum fields on m with polynomial trigonometric or exponential interactions satisfy fkg inequalities

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

the first passage sets of the 2d gaussian free field convergence and isomorphisms in a previous article we introduced the first passage set fps of constant level a of the two dimensional continuum gaussian free field gff on finitely connected domains informally it is the set of points in the domain that can be connected to the boundary by a path along which the gff is greater than or equal to a this description can be taken as a definition of the fps for the metric graph gff and in the current article we prove that the metric graph fps converges towards the continuum fps in the hausdorff distance we also draw numerous consequences in particular we obtain a relatively simple proof of the fact that certain natural interfaces of the metric graph gff converge to hbox sle 4 level lines these results improve our understanding of the continuum gff by strengthening its relationship with the critical brownian loop soup indeed a new construction of the fps using clusters of brownian loops and excursions helps to strengthen the known gff isomorphism theorems and allows us to use brownian loop soup techniques to prove technical results on the geometry of the gff we also obtain a new representation of brownian loop soup clusters and as a consequence we prove that the clusters of a critical brownian loop soup admit a non trivial minkowski content in the gauge r mapsto log r 1 2 r 2

a berry esseen theorem for associated random variables uniform rates of convergence in the central limit theorem for associated random variables are given applications to the ising model and diffusion and gaussian processes are discussed

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

exponential and double exponential tails for maximum of two dimensional discrete gaussian free field we study the tail behavior for the maximum of discrete gaussian free field on a 2d box with dirichlet boundary condition after centering by its expectation we show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail in particular our result implies that the variance of the maximum is of order 1 improving an o log n bound by chatterjee chaos concentration and multiple valleys 2008 and confirming a folklore conjecture an important ingredient for our proof is a result of bramson and zeitouni commun pure appl math 2010 who proved the tightness of the centered maximum together with an evaluation of the expectation up to an additive constant

boundary behavior for large solutions to elliptic equations with singular weights abstract in this paper we analyze the boundary behavior of large positive solutions to some semilinear elliptic equations which include a singular weight the most important point is that the growth of the solutions can be determined in terms of the solution to a one dimensional first order equation we also consider the questions of existence and uniqueness of positive solutions

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

optimal global and boundary asymptotic behavior of large solutions to the monge ampere equation abstract this paper is mainly concerned with optimal global and boundary asymptotic behavior of strict convex large solutions to the monge ampere equation det d 2 u b x f u x where is a strict convex and bounded smooth domain in r n with n 2 f c 1 0 or f c 1 r which is increasing in 0 or r and satisfies the keller osserman type condition b c is positive in but may vanish or blow up on the boundary properly we find new structure conditions on f which play a crucial role in both global and boundary behavior of such solutions moreover we reveal asymptotic behavior of such solutions when the parameters on b tend to the corresponding critical values in addition when f does not satisfy the keller osserman type condition and is a ball we supply a necessary and sufficient condition on b for the existence of an infinitude of strict convex radially symmetric positive solutions to such problem

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

elliptically contoured random fields in space and time elliptically contoured random fields are natural extensions of the gaussian random field and may or may not have second order moments a second order elliptically contoured random field is determined by its mean and covariance functions just like the gaussian one this paper proposes a class of covariance functions for second order elliptically contoured random fields by using the nonnegative mixture method which possess the following properties nonseparable nonstationary but with the stationary case as a special case and allowing for both positive and negative correlation some long range dependent models are also derived

the existence and asymptotic behavior of boundary blow up solutions to the k hessian equation abstract in this paper we consider the existence and asymptotic behavior of k convex solution to the boundary blow up k hessian problem s k d 2 u h x f u for x u x as dist x 0 where k 1 2 n s k d 2 u is the k hessian operator is a smooth bounded strictly convex domain in r n n 2 h c is positive in but is not necessarily bounded on and f is a smooth positive function that satisfies the so called keller osserman condition further results are obtained for the special case that is a ball our approach to show the existence and asymptotic behavior exploits the method of sub and super solutions and karamata regular variation theory

fluctuations of shapes of large areas under paths of random walks we discuss statistical properties of random walks conditioned by fixing a large area under their paths we prove the functional central limit theorem invariance principle for these conditional distributions the limiting gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous gaussian random process obtained by an integral transformation of the white noise from the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one dimensional sos model

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

existence and boundary asymptotic behavior of large solutions of hessian equations abstract in this paper we establish the existence of large solutions of hessian equations and obtain a new boundary asymptotic behavior of solutions

fluctuations of interfaces and anisotropy the mean square vertical displacement w2 is considered for 1 1 dimensional interfaces of the sos type which make an angle theta with respect to the horizontal axis it is proved that w2 per unit horizontal length is proportional to the inverse of beta gamma theta beta gamma theta where beta is the inverse temperature gamma theta is the surface tension of the interface at angle theta and gamma theta is the second derivative of gamma with respect to its argument this result establishes on a microscopic basis the validity of the corresponding formula proposed by akutsu and akutsu 1986 using thermodynamical arguments

limit theorems in financial market models abstract invariance principle states that a scaled simple random walk converges to the standard brownian motion in this article we present a discrete time stochastic process which reflects a microstructure of market dynamics and prove a convergence to a scaling limit process with a drift term and a jump term these terms are derived from a macroscopic condition on volumes traded in some time intervals the mathematical tools for obtaining our results are dobrushin hryniv theory and the method of cluster expansion developed in mathematical studies of statistical mechanics

fluctuations of the phase boundary in the 2d ising ferromagnet we discuss some statistical properties of the phase boundary in the 2d low temperature ising ferromagnet in a box with the two component boundary conditions we prove the weak convergence in c 0 1 of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them the limiting gaussian measure coincides with the conditional distribution of certain gaussian process obtained by the integral transformation of the white noise

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