uniqueness of gradient gibbs measures with disorder we consider in a uniformly strictly convex potential regime two versions of random gradient models with disorder in model a the interface feels a bulk term of random fields while in model b the disorder enters through the potential acting on the gradients we assume a general distribution on the disorder with uniformly bounded finite second moments it is well known that for gradient models without disorder there are no gibbs measures in infinite volume in dimension d 2 while there are shift invariant gradient gibbs measures describing an infinite volume distribution for the gradients of the field as was shown by funaki and spohn commun math phys 185 1 36 1997 van enter and kulske proved in ann appl probab 18 1 109 119 2008 that adding a disorder term as in model a prohibits the existence of such gradient gibbs measures for general interaction potentials in d 2 in cotar and kulske ann appl probab 22 5 1650 1692 2012 we proved the existence of shift covariant random gradient gibbs measures for model a when d ge 3 the disorder is i i d and has mean zero and for model b when d ge 1 and the disorder has a stationary distribution in the present paper we prove existence and uniqueness of shift covariant random gradient gibbs measures with a given expected tilt u in mathbb r d and with the corresponding annealed measure being ergodic for model a when d ge 3 and the disordered random fields are i i d and symmetrically distributed and for model b when d ge 1 and for any stationary disorder dependence structure we also compute for both models for any gradient gibbs measure constructed as in cotar and kulske ann appl probab 22 5 1650 1692 2012 when the disorder is i i d and its distribution satisfies a poincare inequality assumption the optimal decay of covariances with respect to the averaged over the disorder gradient gibbs measure

on the u invariant of a real function field we obtain a new upper bound on the dimensions of anisotropic quadratic torsion forms over a field that is an extension of finite transcendence degree of a real closed field

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

lattice gas models in contact with stochastic reservoirs local equilibrium and relaxation to the steady state extending the results of a previous work we consider a class of discrete lattice gas models in a finite interval whose bulk dynamics consists of stochastic exchanges which conserve the particle number and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials we establish here the local equilibrium structure of the stationary measures for these models further we prove as a law of large numbers that the time dependent empirical density field converges to a deterministic limit process which is the solution of the initial boundary value problem for a nonlinear diffusion equation

on the probability of staying above a wall for the 2 1 dimensional sos model at low temperature we obtain sharp asymptotics for the probability that the 2 1 dimensional discrete sos interface at low temperature is positive in a large region for a square region lambda both under the infinite volume measure and under the measure with zero boundary conditions around lambda this probability turns out to behave like exp tau beta 0 l log l with tau beta 0 the surface tension at zero tilt also called step free energy and l the box side this behavior is qualitatively different from the one found for continuous height massless gradient interface models bolthausen et al commun math phys 170 2 417 443 1995 deuschel et al stochastic process appl 89 2 333 354 2000

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

on homogenization and scaling limit of some gradient perturbations of a massless free field we study the continuum scaling limit of some statistical mechanical models defined by convex hamiltonians which are gradient perturbations of a massless free field by proving a central limit theorem for these models we show that their long distance behavior is identical to a new homogenized continuum massless free field we shall also obtain some new bounds on the 2 point correlation functions of these models

the u invariant of the function fields of p adic curves the u invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field it is an open question whether the u invariant of function fields of p aidc curves is 8 in this paper we answer this question in the affirmative for function fields of nondyadic p adic curves

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

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

tail approximations of integrals of gaussian random fields this paper develops asymptotic approximations of p tef t dt b as b for a homogeneous smooth gaussian random field f living on a compact d dimensional jordan measurable set t the integral of an exponent of a gaussian random field is an important random variable for many generic models in spatial point processes portfolio risk analysis asset pricing and so forth the analysis technique consists of two steps 1 evaluate the tail probability p ef t dt b over a small domain depending on b where mes 0 as b and mes is the lebesgue measure 2 with appropriately chosen we show that p tef t dt b 1 o 1 mes t mes 1 p ef t dt b

fluctuations of one dimensional ginzburg landau models in nonequilibrium we study the fluctuation of one dimensional ginzburg landau models in nonequilibrium along the hydrodynamic diffusion limit the hydrodynamic limit has been proved to be a nonlinear diffusion equation by fritz guo papanicolaou varadhan etc we proved that if the potential is uniformly convex then the fluctuation process is governed by an ornstein uhlenbeck process whose drift term is obtained by formally linearizing the hydrodynamic equation

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

moderate deviations for the overlap parameter in the hopfield model we derive moderate deviation principles for the overlap parameter in the hopfield model of spin glasses and neural networks if the inverse temperature is different from the critical inverse temperature c 1 and the number of patterns m n satisfies m n n 0 the overlap parameter multiplied by n 1 2 1 obeys a moderate deviation principle with speed n1 2 and a quadratic rate function i e the gaussian limit for 1 2 remains visible on the moderate deviation scale at the critical temperature we need to multiply the overlap parameter by n 1 4 1 if then m n satisfies m n 6 log n m n 2n4 log n n 0 the rescaled overlap parameter obeys a moderate deviation principle with speed n1 4 and a rate function that is basically a fourth power the random term occurring in the central limit theorem for the overlap at c 1 is no longer present on a moderate deviation scale if the scaling is even closer to n1 4 e g if we multiply the overlap parameter by n1 4 log log n the moderate deviation principle breaks down the case of variable temperature converging to one is also considered if n converges to c fast enough i e faster than open image in new window the non gaussian rate function persists whereas for n converging to one slower than open image in new window the moderate deviations principle is given by the gaussian rate at the borderline the moderate deviation rate function is the one at criticality plus an additional gaussian term

hitting times of rare events in markov chains a general condition for a markov process with finite states is given for the convergence of the normalized hitting time of a rare event to a mean one exponential law it is then applied to the boundary driven symmetric simple exclusion process for an open set of density profiles which does not contain the stationary density profile

precise large deviations for dependent regularly varying sequences we study a precise large deviation principle for a stationary regularly varying sequence of random variables this principle extends the classical results of nagaev theory probab appl 14 51 64 193 208 1969 and nagaev ann probab 7 745 789 1979 for iid regularly varying sequences the proof uses an idea of jakubowski stoch proc appl 44 291 327 1993 68 1 20 1997 in the context of central limit theorems with infinite variance stable limits we illustrate the principle for stochastic volatility models real valued functions of a markov chain satisfying a polynomial drift condition and solutions of linear and non linear stochastic recurrence equations

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

convexity properties of the surface tension and equilibrium crystals we study the thermodynamic limit of the orientation dependent surface tension under general conditions which we show to hold true for a large class of lattice systems we prove that the limit exists and that it satisfies some convexity properties related to the pyramidal inequality introduced by r l dobrushin and s b shlosman 1 we discuss some consequences of these results for the equilibrium crystal shape

one sided local large deviation and renewal theorems in the case of infinite mean if s n n 0 is an integer valued random walk such that s n a n converges in distribution to a stable law of index 0 1 as n then gnedenko s local limit theorem provides a useful estimate for p s n r for values of r such that r a n is bounded the main point of this paper is to show that under certain circumstances there is another estimate which is valid when r a n in other words to establish a large deviation local limit theorem we also give an asymptotic bound for p s n r which is valid under weaker assumptions this last result is then used in establishing some local versions of generalized renewal theorems

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

hydrodynamics of stationary non equilibrium states for some stochastic lattice gas models we consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior which conserve the particle number and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials the unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non reversible we study the structure of the stationary measure in the hydrodynamic limit as the microscopic lattice size goes to infinity in particular we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by fick s law

fibers of flat morphisms and weierstrass preparation theorem abstract we characterize flat extensions of commutative rings satisfying the weierstrass preparation theorem using this characterization we prove a variant of the weierstrass preparation theorem for rings of functions on a normal curve over a complete local domain of dimension one this generalizes recent works of harbater hartmann and krashen with a different method of proof

large deviations for solutions to stochastic recurrence equations under kesten s condition in this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation we assume kesten s condition acta math 131 1973 207 248 under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails while the noise sequence of the equations can have light tails the results of the paper are analogs to those obtained by a v nagaev theory probab appl 14 1969 51 64 193 208 and s v nagaev ann probab 7 1979 745 789 in the case of partial sums of i i d random variables in the latter case the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum for the solution to a stochastic recurrence equation the magnitude of the large deviation probabilities is again given by the tail of the maximum summand but the exact asymptotic tail behavior is also influenced by clusters of extreme values due to dependencies in the sequence we apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

surface tension in ising systems with kac potentials we consider an ising spin system with kac potentials in a torus of d d 2 and fix the temperature below its lebowitz penrose critical value we prove that when the kac scaling parameter vanishes the log of the probability of an interface becomes proportional to its area and the surface tension related to the proportionality constant converges to the van der waals surface tension the results are based on the analysis of the rate functionals for gibbsian large deviations and on the proof that they converge to the perimeter functional of geometric measure theory which extends the notion of area our considerations include nonsmooth interfaces proving that the gibbsian probability of an interface depends only on its area and not on its regularity

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

nonequilibrium measures which exhibit a temperature gradient study of a model we give some rules to define measures which could describe heat flow in homogeneous crystals we then study a particular model which is explicitly solvable the one dimensional nearest neighborhood ising model we analyze two cases in the first one the spins at the two boundaries interact with reservoirs at different temperatures in the thermodynamical limit the measure we introduce converges locally to gibbs measures and a temperature profile is so derived we obtain an explicit expression for the thermal conductivity coefficient which depends on the temperature in the second case we study the asymptotic behavior starting from an initial state in which each half of the space is at a different temperature we find again a temperature profile which asymptotically obeys the heat equation with the thermal conductivity coefficient previously derived from a mathematical point of view the analysis of the invariant measure is made possible by studying a time reversed process related to a graphical representation of an associated process this provides us with an explicit formula for then fold correlation function and we study the limiting behavior using both this representation for proving an exchangeability result and a donsker type spacetime renormalization procedure

large deviations of square root insensitive random sums we provide a large deviation result for a random sum s nxn 0 x n wheren xis a renewal counting process and x n n 0are i i d random variables independent ofn x with a common distribution that belongs to a class of square root insensitive distributions asymptotically the tails of these distributions are heavier thane vxand have zero relative decrease in intervals of length v x hence square root insensitive using this result we derive the asymptotic characterization of the busy period distribution in the stable gi g 1 queue with square root insensitive service times this characterization further implies that the tail behavior of the busy period exhibits a functional change for distributions that are lighter thane vx

strict convexity of the free energy for a class of non convex gradient models we consider a gradient interface model on the lattice with interaction potential which is a non convex perturbation of a convex potential we show using a one step multiple scale analysis the strict convexity of the surface tension at high temperature this is an extension of funaki and spohn s result 8 where the strict convexity of potential was crucial in their proof

motion by mean curvature from the ginzburg landau interface model we consider the scalar field t with a reversible stochastic dynamics which is defined by the standard dirichlet form relative to the gibbs measure with formal energy the potential v is even and strictly convex we prove that under a suitable large scale limit the t field becomes deterministic such that locally its normal velocity is proportional to its mean curvature except for some anisotropy effects as an essential input we prove that for every tilt there is a unique shift invariant ergodic gibbs measure for the field

applications of patching to quadratic forms and central simple algebras this paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields in particular we use a patching approach to reprove and generalize a recent result of parimala and suresh in preprint arxiv 0708 3128 2007 on the u invariant of p adic function fields p 2 the strategy relies on a local global principle for homogeneous spaces for rational algebraic groups combined with local computations

dobrushin kotecký shlosman theorem up to the critical temperature we develop a non perturbative version of the dobrushin koteck shlosman theory of phase separation in the canonical 2d ising ensemble the results are valid for all temperatures below critical

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

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

local behaviour of first passage probabilities suppose that s is an asymptotically stable random walk with norming sequence cn and that tx is the time that s first enters x where x 0 the asymptotic behaviour of p t0 n has been described in a recent paper of vatutin and wachtel probab theory relat fields 143 177 217 2009 and here we build on that result to give three estimates for p tx n which hold uniformly as n in the regions x o cn x o cn and x cn respectively

large deviations for random walks under subexponentiality the big jump domain for a given one dimensional random walk sn with a subexponential step size distribution we present a unifying theory to study the sequences xn for which as n uniformly for x xn we also investigate the stronger local analogue our theory is self contained and fits well within classical results on domains of partial attraction and local limit theory when specialized to the most important subclasses of subexponential distributions that have been studied in the literature we reproduce known theorems and we supplement them with new results

rigorous generalization of young s law for heterogeneous and rough substrates we consider a sos type model of interfaces on a substrate which is both heterogeneous and rough we first show that for appropriate values of the parameters the differential wall tension that governs wetting on such a substrate satisfies a generalized law which combines both cassie and wenzel laws then in the case of an homogeneous substrate we show that this differential wall tension satisfies either the wenzel s law or the cassie s law according to the values of the parameters

notes on random walks in the cauchy domain of attraction the goal of these notes is to fill some gaps in the literature about random walks in the cauchy domain of attraction which has been in many cases left aside because of its additional technical difficulties we prove here several results in that case a fuk nagaev inequality and a local version of it a large deviation theorem two types of local large deviation theorems we also derive two important applications of these results a sharp estimate of the tail of the first ladder epochs and renewal theorems extending standard renewal theorems to the case of random walks most of our techniques carry through to the case of random walks in the domain of attraction of an alpha stable law with alpha in 0 2 so we also present results in that case since many of them seem to be missing in the literature

random walks with non convolution equivalent increments and their applications abstract this paper mainly presents some global and local asymptotic estimates for the tail probabilities of the supremum and overshoot of a random walk in the intermediate case where the related distributions of the increments of the random walk may not belong to the convolution equivalent distribution class some of the obtained results can include the classical results for this the paper first introduces some new distribution classes using the transform of distributions and investigates their properties and relations with some other existing distribution classes based on the above results some equivalent conditions for the global and local asymptotics of the transform of the distribution of the supremum of the above random walk are given applying these results to risk theory and infinitely divisible laws the paper obtains some asymptotic estimates for the ruin probability and the local ruin probability of the renewal risk model with non convolution equivalent claims and the global and local asymptotics of an infinitely divisible law with a non convolution equivalent levy measure

large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy tailed innovations in this paper we consider the stochastic recurrence equation y t a t y t 1 b t for an i i d sequence of pairs a t b t of nonnegative random variables where we assume that b t is regularly varying with index k 0 and ea k t x c 1 np y 1 x as x holds for some constant c 1 0 for k 1 we also study the large deviation probabilities p s n es n x x x n for some sequence x n whose growth depends on the heaviness of the tail of the distribution of y 1 we show that the relation p s n es n x c 2 np y 1 x holds uniformly for x x n and some constant c 2 0 then we apply the large deviation results to derive bounds for the ruin probability u p sup n 1 s n es n n u for any 0 we show that u c 3 up y 1 u 1 1 1 for some constant c 3 0 in contrast to the case of i i d regularly varying y t s when the above results hold with c 1 c 2 c 3 1 the constants c 1 c 2 and c 3 are different from 1

fluctuations for a φ interface model with repulsion from a wall we consider a interface model on a one dimensional lattice with repulsion from a hard wall we suppose that the repulsion is of the form c 1 where c 0 and denotes the height of the interface from the wall we prove convergence of the equilibrium fluctuations around the hydrodynamic limit to the solution of a spde with singular drift if c 0 the system becomes the funaki olla interface model with reflection at the wall whose equilibrium fluctuations converge to the solution of a spde with reflection we give a new proof of this result using the characterization of such solution as the diffusion generated by an infinite dimensional dirichlet form obtained in a previous paper our method is based on a study of integration by parts formulae w r t the equilibrium measure of the interface model and allows to avoid the proof of the so called boltzmann gibbs principle we also obtain convergence of finite dimensional distributions of non equilibrium fluctuations around the stationary hydrodynamic limit 0

on the spectrum of semi classical witten laplacians and schrodinger operators in large dimension we investigate the low lying spectrum of witten laplacians on forms of arbitrary degree in the semi classical limit and uniformly in the space dimension we show that under suitable assumptions implying that the phase function has a unique local minimum one obtains a number of clusters of discrete eigenvalues at the bottom of the spectrum moreover we are able to count the number of eigenvalues in each cluster we apply our results to certain sequences of schrodinger operators having strictly convex potentials and show that some well known results of semi classical analysis hold also uniformly in the dimension

on the u invariant of function fields of curves over complete discretely valued fields abstract let k be a complete discretely valued field with residue field if char k 0 char 2 and 2 d we prove that there exists an integer n depending on d such that the u invariant of any function field in one variable over k is bounded by n the method of proof is via introducing the notion of uniform boundedness for the p torsion of the brauer group of a field and relating the uniform boundedness of the 2 torsion of the brauer group to the finiteness of the u invariant we prove that the 2 torsion of the brauer group of function fields in one variable over k is uniformly bounded

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

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

cyclic algebras over p adic curves abstract in this paper we study division algebras over the function fields of curves over q p the first and main tool is to view these fields as function fields over nonsingular s which are projective of relative dimension 1 over the p adic ring z p a previous paper showed such division algebras had index bounded by n 2 assuming the exponent was n and n was prime to p in this paper we consider algebras of prime degree and hence exponent q p and show these algebras are cyclic we also find a geometric criterion for a brauer class to have index q

classification theorems for hermitian forms the rost kernel and hasse principle over fields with cd2 k 3 abstract in this paper we prove classification theorems for hermitian forms over some central simple algebras with involution over a field k with cd 2 k 3 we apply these results to show the triviality of the kernel of the rost invariant for the classical algebraic groups associated to such hermitian forms over k we also deduce a hasse principle for algebraic groups defined over function fields of curves over p adic fields thus proving a conjecture due to colliot thelene parimala suresh for a large class of groups

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

noncyclic division algebras over fields of brauer dimension one abstract let k be a complete discretely valued field of rank one with residue field q p it is well known that period equals index in br k we prove that when p 2 there exist noncyclic k division algebras of every 2 power degree divisible by four otherwise every k division algebra is cyclic this gives the first published example of a field whose brauer dimension and cyclic length are not equal

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

shells of twisted flag varieties and the rost invariant we introduce two new general methods to compute the chow motives of twisted flag varieties and settle a 20 year old conjecture of markus rost about the rost invariant for groups of type e 7

equilibrium fluctuations for nabla_ varphi interface model we study the large scale space time fluctuations of an interface which is modeled by a massless scalar field with reversible langevin dynamics for a strictly convex interaction potential we prove that on a large space time scale these fluctuations are governed by an infinite dimensional ornstein uhlenbeck process its effective diffusion type covariance matrix is characterized through a variational formula

metastability in a condensing zero range process in the thermodynamic limit zero range processes with decreasing jump rates are known to exhibit condensation where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value we study such a process on a one dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed super critical particle density we show that the process exhibits metastability with respect to the condensate location i e the suitably accelerated process of the rescaled location converges to a limiting markov process on the unit torus this process has stationary independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice our result extends previous work for fixed lattices and diverging density in beltran and landim probab theory relat fields 152 3 4 781 807 2012 and we follow the martingale approach developed there and in subsequent publications besides additional technical difficulties in estimating error bounds for transition rates the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells corresponding to a typical set of configurations with a particular condensate location the total exit rates from individual wells turn out to diverge in the limit which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells which is of a general nature and can be adapted to other models

dynamical large deviations for the boundary driven weakly asymmetric exclusion process 1 introduction the analysis of the large deviations is asymptotic as the number of degrees of freedom diverges for the stationary nonequilibrium states of interacting particle systems have recently proved to be an important step in the physical description of such states and a rich source of mathematical problems referring to 4 8 for two recent reviews on this topic we briefly outline its basic points we discuss only stochastic lattice gases for which the underlying random fluctuations ensure the necessary ergodicity for a rigorous analysis the stationary nonequilibrium states are characterized by a flow of mass through the system and the corresponding dynamics are not reversible the main difference with respect to reversible equilibrium states follows in equilibrium the invariant measure which determines the thermodynamic properties is given for free by the gibbs distribution specified by the hamiltonian on the contrary in nonequilibrium states the construction of the stationary state requires the solution of a dynamical problem since we are interested only in the macroscopic description only the thermodyamic observables are relevant for lattice gases there is only one of such observable which is the empirical density in equilibrium states the thermodynamic functional like the free energy can then be identified 6 22 24 with the large deviation rate function for the empirical density when the

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

semiclassical limit of liouville field theory liouville field theory lft for short is a two dimensional model of random surfaces which is for instance involved in 2d string theory or in the description of the fluctuations of metrics in 2d liouville quantum gravity this is a probabilistic model that consists in weighting the classical free field action with an interaction term given by the exponential of a gaussian multiplicative chaos the main input of our work is the study of the semiclassical limit of the theory which is a prescribed asymptotic regime of lft of interest in physics literature see cite witten and references therein we derive exact formulas for the laplace transform of the liouville field in the case of flat metric on the unit disk with dirichlet boundary conditions as a consequence we prove that the liouville field concentrates on the solution of the classical liouville equation with explicit negative scalar curvature we also characterize the leading fluctuations which are gaussian and massive and establish a large deviation principle though considered as an ansatz in the whole physics literature it seems that it is the first rigorous probabilistic derivation of the semiclassical limit of lft on the other hand we carry out the same analysis when we further weight the liouville action with heavy matter operators this procedure appears when computing the n points correlation functions of lft

hydrodynamic limit for the ginzburg landau φ interface model with boundary conditions hydrodynamic limit for the ginzburg landau interface model was established in 6 under periodic boundary conditions this paper studies the same problem on a bounded domain imposing dirichlet boundary conditions a nonlinear partial differential equation of second order with boundary conditions is derived as a macroscopic equation the main tools are h 1 method used in 6 and the higher integrability of gradients in 2

tightness of probabilities on c lbrack 0 1 rbrack mathscr y and d lbrack 0 1 rbrack mathscr y let c mathscr i c lbrack 0 1 rbrack mathscr i be the space of all continuous mappings of lbrack 0 1 rbrack to mathscr i where mathscr l is the topological dual of the schwartz space mathscr i of all rapidly decreasing functions let c be the banach space of all continuous functions on lbrack 0 1 rbrack for each varphi in mathscr i pi varphi is defined by pi varphi x in c mathscr i rightarrow x varphi in c given a sequence of probability measures p n on c mathscr i such that for each varphi in mathscr i p n pi 1 varphi is tight in c we prove that p n itself is tight in c mathscr i a similar result is proved for the space of all right continuous mappings of lbrack 0 1 rbrack to mathscr i

large deviations of the front in a one dimensional model of x y 2x we investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction x y 2x on the integer lattice in which y particles do not move while x particles move as independent simple continuous time random walks of total jump rate 2 for a wide class of initial conditions we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval 0 where is the velocity of the front given by the law of large numbers we also give more precise estimates for the rate of decay of the slowdown probabilities our results indicate a gapless property of the generator of the process as seen from the front as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state

on the convergence of entropy for stationary exclusion processes with open boundaries we prove that for a wide class of stochastic lattice gases in contact with reservoirs despite long range correlations the leading order term of the gibbs shannon entropy in the nonequilibrium stationary state is given by the local equilibrium entropy

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

nonlinear elastic free energies and gradient young gibbs measures we investigate in a fairly general setting the limit of large volume equilibrium gibbs measures for elasticity type hamiltonians with clamped boundary conditions the existence of a quasiconvex free energy forming the large deviations rate functional is shown using a new interpolation lemma for partition functions the local behaviour of the gibbs measures can be parametrized by young measures on the space of gradient gibbs measures in view of the unboundedness of the state space the crucial tool here is an exponential tightness estimate that holds for a vast class of potentials and the construction of suitable compact sets of gradient gibbs measures

lattice gases and exactly solvable models we detail the construction of a family of lattice gas automata based on a model of t hooft proceeding by use of symmetry principles to define first the kinematics of the model and then the dynamics a spurious conserved quantity appears we use it to effect a radical transformation of the model into one whose spacetime configurations are equivalent to the two dimensional states of an exactly solvable statistical mechanics model the symmetric eight vertex model with parameters restricted to a disorder variety we comment on the implications of this identification for the original lattice gas

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

period index and u invariant questions for function fields over complete discretely valued fields let k be a complete discretely valued field with residue field and f the function field of a curve over k let p be the characteristic of and l a prime not equal to p if the brauer l dimensions of all finite extensions of are bounded by d and the brauer l dimensions of all extensions of of transcendence degree at most 1 are bounded by d 1 then it is known that the brauer l dimension of f is at most d 2 lieblich in j reine angew math 659 1 41 2011 saltman in j ramanujan math soc 12 25 47 1997 harbater et al in invent math 178 231 263 2009 in this paper we give a bound for the brauer p dimension of f in terms of the p rank of as an application we show that if is a perfect field of characteristic 2 then any quadratic form over f in at least 9 variables is isotropic this leads to the fact that every element in h 3 f mathbb z 2 mathbb z is a symbol if is a finite field of characteristic 2 u f 8 is a result of heath brown leep heath brown in compos math 146 271 287 2010 leep in j reine angew math 2013 to appear

scaling limits of random planar maps with a unique large face we study random bipartite planar maps defined by assigning nonnegative weights to each face of a map we prove that for certain choices of weights a unique large face having degree proportional to the total number of edges in the maps appears when the maps are large it is furthermore shown that as the number of edges n of the planar maps goes to infinity the profile of distances to a marked vertex rescaled by n 1 2 is described by a brownian excursion the planar maps with the graph metric resealed by n 1 2 are then shown to converge in distribution toward aldous brownian tree in the gromov hausdorff topology in the proofs we rely on the bouttier di francesco guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large

the wulff construction and asymptotics of the finite cluster distribution for two dimensional bernoulli percolation we consider two dimensional bernoulli percolation at densityp p c and establish the following results

approximation of the effective conductivity of ergodic media by periodization this paper is concerned with the approximation of the effective conductivity a associated to an elliptic operator xa x x where for x d d 1 a x is a bounded elliptic random symmetric d d matrix and takes value in an ergodic probability space x writing an x the periodization of a x on the torus tdn of dimension d and side n we prove that for almost all we extend this result to non symmetric operators x a e x x corresponding to diffusions in ergodic divergence free flows a is d d elliptic symmetric matrix and e x an ergodic skew symmetric matrix and to discrete operators corresponding to random walks on d with ergodic jump rates