integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity this article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices under polynomial growth assumptions we prove that the energy functionals f varepsilon stored in the deformation of an varepsilon scaling of a stochastic lattice converge to a continuous energy functional when varepsilon goes to zero in particular the limiting energy functional is of integral type and deterministic if the lattice is ergodic we also generalize to systems and nonlinear settings well known results on stochastic homogenization of discrete elliptic equations as an application of the main result we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics we finally address some mechanical properties of the limiting models such as frame invariance isotropy and natural states

first passage percolation network flows and electrical resistances we show that the first passage times of first passage percolation on 2 are such that p 0n n decay geometrically as n where may represent any of the four usual first passage time processes the former estimate requires no moment condition on the time coordinates but there exists a geometrically decaying estimate for the latter quantity if and only if the time coordinate distribution has finite moment generating function near the origin here is the time constant and 0 we study the line to line first passage times and describe applications to the maximum network flow through a randomly capacitated subsection of 2 and to the asymptotic behaviour of the electrical resistance of a subsection of 2 when the edges of the subsection are wires in an electrical network with random resistances in the latter case we show for example that if each edge resistance equals 1 or ohms with probabilities p and 1 p respectively then the effective resistance rn across opposite faces of an n by n box satisfies the following a if p 1 2 then p rn 1 as n b if p 1 2 then there exists p such that p p 1 underset raise0 3em hbox smash scriptscriptstyle leqslant mathop lim inf r n limits n to infty underset raise0 3em hbox smash scriptscriptstyle leqslant mathop lim sup r n limits n to infty underset raise0 3em hbox smash scriptscriptstyle leqslant v p 1

on the exponential decay of the n point correlation functions and the analyticity of the pressure the goal of this paper is to provide estimates leading to a direct proof of the exponential decay of the n point correlation functions for certain unbounded models of kac type the methods are based on estimating higher order derivatives of the solution of the witten laplacian equation on 1 forms associated with the hamiltonian of the system we also provide a formula for the taylor coefficients of the pressure that is suitable for a direct proof of the analyticity

delocalization of two dimensional random surfaces with hard core constraints we study the fluctuations of random surfaces on a two dimensional discrete torus the random surfaces we consider are defined via a nearest neighbor pair potential which we require to be twice continuously differentiable on a possibly infinite interval and infinity outside of this interval no convexity assumption is made and we include the case of the so called hammock potential when the random surface is uniformly chosen from the set of all surfaces satisfying a lipschitz constraint our main result is that these surfaces delocalize having fluctuations whose variance is at least of order log n where n is the side length of the torus we also show that the expected maximum of such surfaces is of order at least log n the main tool in our analysis is an adaptation to the lattice setting of an algorithm of richthammer who developed a variant of a mermin wagner type argument applicable to hard core constraints we rely also on the reflection positivity of the random surface model the result answers a question mentioned by brascamp et al on the hammock potential and a question of velenik

coherence resonance and stochastic resonance in directionally coupled rings abstract in coupled systems symmetry plays an important role for the collective dynamics we investigate the dynamical response to noise with and without weak periodic modulation for two classes of ring systems each ring system consists of unidirectionally coupled bistable elements but in one class the number of elements is even while in the other class the number is odd consequently the rings without forcing show at a certain coupling strength either ordering similar to anti ferromagnetic chains or auto oscillations analysing the bifurcations and fixed points of the two ring classes enables us to explain the dynamical response measured to noise and weak modulation moreover by analysing a simplified model we demonstrate that the response is universal for systems having a directional component in their stochastic dynamics in phase space around the origin

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

correlation at low temperature ii asymptotics the present paper is a continuation of ref 4 where the truncated two point correlation function for a class of lattice spin systems was proved to have exponential decay at low temperature under a weak coupling assumption in this paper we compute the asymptotics of the correlation function as the temperature goes to zero this paper thus extends ref 3 in two directions the hamiltonian function is allowed to have several local minima other than a unique global minimum and we do not require translation invariance of the hamiltonian function we are in particular able to handle spin systems on a general lattice

asymptotic properties of multistate random walks ii applications to inhomogeneous periodic and random lattices the previously developed formalism for the calculation of asymptotic properties of multistate random walks is used to study random walks on several inhomogeneous periodic lattices where the periodically repeated unit cell contains a number of inequivalent sites as well as on lattices with a random distribution of inequivalent sites we concentrate on the question whether the random walk properties depend on the spatial arrangement of the sites in the unit cell or only on the number density of the different types of sites specifically we consider lattices with periodic and random arrangements of columns and lattices with periodic and random arrangements of anisotropic scatterers

random walks in asymmetric random environments we consider random walks on zd with transition ratesp x y given by a random matrix ifp is a small random perturbation of the simple random walk we show that the walk remains diffusive for almost all environmentsp ifd 2 the result also holds for a continuous time markov process with a random drift the corresponding path space measures converge weakly in the scaling limit to the wiener process for almost everyp

pde with random coefficients and euclidean field theory in this paper a new proof of an identity of giacomin olla and spohn is given the identity relates the 2 point correlation function of a euclidean field theory to the expectation of the green s function for a pde with random coefficients the euclidean field theory is assumed to have convex potential an inequality of brascamp and lieb therefore implies gaussian bounds on the fourier transform of the 2 point correlation function by an application of results from random pde the previously mentioned identity implies pointwise gaussian bounds on the 2 point correlation function

matrix semigroups with constant spectral radius abstract multiplicative matrix semigroups with constant spectral radius c s r are studied and applied to several problems of algebra combinatorics functional equations and dynamical systems we show that all such semigroups are characterized by means of irreducible ones each irreducible c s r semigroup defines walks on euclidean sphere all its nonsingular elements are similar in the same basis to orthogonal we classify all nonnegative c s r semigroups and arbitrary low dimensional semigroups for higher dimensions we describe five classes and leave an open problem on completeness of that list the problem of algorithmic recognition of c s r property is proved to be polynomially solvable for irreducible semigroups and undecidable for reducible ones

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

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

the localization properties of a random steady flow on a lattice we consider a random stationary vector field on a multidimensional lattice and investigate flow connected subsets of the lattice invariant under the action of the associated flow the subsets of primary interest are cycles and vortices each of which is the set of orbits terminating in the same cycle we prove that with probability 1 each vortex only involves a finite number of sites of the lattice under the assumption of independence of the vector field in different sites we find that with probability 1 the vortices exhaust all possible maximal flowconnected invariant subsets of the lattice if and only if the probability of existence of a cycle is positive thus if cycles exist a particle under the action of the flow only moves within a bounded region i e it is completely localized

a lower bound on the variance of conductance in random resistor networks we study the conductance of random resistor networks ind 2 dimensions it is shown under some technical assumptions that if a network exhibits a nonzero conductivity in the infinite volume limit then the variance of a finite volume conductance grows at least like the volume

submean variance bound for effective resistance of random electric networks we study a model of random electric networks with bernoulli resistances in the case of the lattice mathbb z 2 we show that the point to point effective resistance between 0 and a vertex v has a variance of order at most rm log v frac 2 3 whereas its expected value is of order log v when v goes to infinity when d 2 expectation and variance are of the same order similar results are obtained in the context of p resistance the proofs rely on a modified poincare inequality due to falik and samorodnitsky 7

an invariance principle for isotropic diffusions in random environment we investigate in this work the asymptotic behavior of isotropic diffusions in random environment that are small perturbations of brownian motion when the space dimension is three or more we prove an invariance principle as well as transience our methods also apply to questions of homogenization in random media

a gel fand type spectral radius formula and stability of linear constrained switching systems abstract using ergodic theory in this paper we present a gel fand type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup s restricted to a subset that need not carry the algebraic structure of s this generalizes the berger wang formula using it as a tool we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one sided markov shift associated to s

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

random walks in random environments random walks in random environments and their diffusion analogues have been a source of surprising phenomena and challenging problems especially in the non reversible situation since they began to be studied in the 1970s we review the model available results and techniques and point out several gaps in the understanding of these processes

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

kantorovich distance in the martingale clt and quantitative homogenization of parabolic equations with random coefficients the article begins with a quantitative version of the martingale central limit theorem in terms of the kantorovich distance this result is then used in the study of the homogenization of discrete parabolic equations with random i i d coefficients for smooth initial condition the rescaled solution of such an equation once averaged over the randomness is shown to converge polynomially fast to the solution of the homogenized equation with an explicit exponent depending only on the dimension polynomial rate of homogenization for the averaged heat kernel with an explicit exponent is then derived similar results for elliptic equations are also presented

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

rigorous renormalisation group and disordered systems abstract we consider random walks in a non symmetric random environment we report a recent result based on a renormalisation group approach showing that for d 62 2 these walks are diffusive when the disorder is weak

high dimensional lipschitz functions are typically flat a homomorphism height function on the d dimensional torus mathbb z n d is a function on the vertices of the torus taking integer values and constrained to have adjacent vertices take adjacent integer values a lipschitz height function is defined similarly but may also take equal values on adjacent vertices for each of these models we consider the uniform distribution over all such functions with predetermined values at some fixed vertices boundary conditions our main result is that in high dimensions and with zero boundary values the random function obtained is typically very flat having bounded variance at any fixed vertex and taking at most c log n 1 d values with high probability this result matches up to constants a lower bound of benjamini yadin and yehudayoff our results extend to any dimension d ge2 if one replaces the torus mathbb z n d by an enhanced version of it the torus mathbb z n d times mathbb z 2 d 0 for some fixed d 0 consequently we establish one side of a conjectured roughening transition in two dimensions the full transition is established for a class of tori with nonequal side lengths including for example the n times lfloor frac 1 10 log n rfloor torus in another case of interest we find that when the dimension d is taken to infinity while n remains fixed the random function takes at most r values with high probability where r 5 for the homomorphism model and r 4 for the lipschitz model suitable generalizations are obtained when n grows with d our results have consequences also for the related model of uniform 3 coloring and establish that for certain boundary conditions a uniformly sampled proper 3 coloring of mathbb z n d will be nearly constant on either the even or odd sublattice our proofs are based on the construction of a combinatorial transformation suitable to the homomorphism model and on a careful analysis of the properties of a class of cutsets which we term odd cutsets for the lipschitz model our results rely also on a bijection of yadin this work generalizes results of galvin and kahn refutes a conjecture of benjamini yadin and yehudayoff and answers a question of benjamini haggstrom and mossel

the structure of fluctuations in stochastic homogenization four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications the field and the flux of the solution operator applied to a general deterministic right hand side and the field and the flux of the corrector homogenization is the study of the large scale properties of these objects in case of random coefficients these quantities fluctuate and their fluctuations are a priori unrelated depending on the law of the coefficient field and in particular on the decay of its correlations on large scales these fluctuations may display different scalings and different limiting laws if any in this contribution we identify another crucial intrinsic quantity motivated by h convergence which we refer to as the homogenization commutator and is related to variational quantities first considered by armstrong and smart in the simplified setting of the random conductance model we show what we believe to be a general principle namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability which is expressed in form of a suitable two scale expansion and reveals the pathwise structure of fluctuations in stochastic homogenization in addition we show that the rescaled homogenization commutator converges in law to a gaussian white noise and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method this collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates extensions to the non symmetric continuum setting are also discussed the details of which are postponed to forthcoming works

grad o perturbations of massless gaussian fields we investigate weak perturbations of the continuum massless gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity

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

witten laplacian method for the decay of correlations the aim of this paper is to apply direct methods to the study of integrals that appear naturally in statistical mechanics and euclidean field theory we provide weighted estimates leading to the exponential decay of the two point correlation functions for certain classical convex unbounded models the methods involve the study of the solutions of the witten laplacian equations associated with the hamiltonian of the system

corrigendum addendum to sets of matrices all infinite products of which converge abstract this corrigendum addendum supplies corrected statements and proofs of some results in our paper appearing in linear algebra appl 161 1992 227 263 these results concern special kinds of bounded semigroups of matrices it also reports on progress on the topics of this paper made in the last eight years

on birth and death processes in symmetric random environment we prove a limit theorem for a process in a random one dimensional medium which has been considered before as a model for hopping conduction in a disordered medium to the edge between the two integersj and j 1 a rate j 0 is attached these j j integral are taken as independent identically distributed random variables and represent the medium for given values j x t is a markov chain in continuous time which jumps fromj to j 1 and from j 1 toj at the same rate j we show that in many cases there exists normalizing constants y t which tend to oo witht such that the distribution of x t t or more generally of the whole processx st t s 0 converges to a limit as t the limit process is continuous and self similar

scaling limit for a class of gradient fields with nonconvex potentials we consider gradient fields x x d whose law takes the gibbs boltzmann form z 1 exp x y v v x where the sum runs over nearest neighbors we assume that the potential v admits the representation v log d exp 1 2 2 where is a positive measure with compact support in 0 hence the potential v is symmetric but nonconvex in general while for strictly convex v s the translation invariant ergodic gradient gibbs measures are completely characterized by their tilt a nonconvex potential as above may lead to several ergodic gradient gibbs measures with zero tilt still every ergodic zero tilt gradient gibbs measure for the potential v above scales to a gaussian free field

a characterisation of the gaussian free field we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain markov property is a multiple of the gaussian free field this result holds subject only to a fourth moment assumption

a strong central limit theorem for a class of random surfaces this paper is concerned with d 2 dimensional lattice field models with action v nabla phi cdot where v mathbf r d rightarrow mathbf r is a uniformly convex function the fluctuations of the variable phi 0 phi x are studied for large x via the generating function given by g x mu ln langle e mu phi 0 phi x rangle a in two dimensions g x mu partial 2g x mu partial mu 2 is proportional to ln vert x vert the main result of this paper is a bound on g x mu partial 3 g x mu partial mu 3 which is uniform in vert x vert for a class of convex v the proof uses integration by parts following helffer sjostrand and witten and relies on estimates of singular integral operators on weighted hilbert spaces

finite range decomposition for gaussian measures with improved regularity abstract we consider a family of gradient gaussian vector fields on the torus z l n z d adams koteck muller and independently bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators i e the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter l k we improve this result and show that the decay behaviour of the kernels in fourier space can be controlled then we show the regularity of the integration map that convolves functionals with the partial measures of the finite range decomposition in particular the new finite range decomposition avoids the loss of regularity which arises in the renormalisation group approach to anisotropic problems in statistical mechanics

potential energy landscape and finite state models of array enhanced stochastic resonance noise and coupling can optimize the response of arrays of nonlinear elements to periodic signals we analyze such array enhanced stochastic resonance aesr using finite state transition rate models we simply derive the transition rate matrices from the underlying potential energy function of the corresponding langevin problem our implementation exploits floquet theory and provides useful theoretical and numerical tools our framework both facilitates analysis and elucidates the mechanism of aesr in particular we show how sublinear coupling diminishes aesr but superlinear coupling enhances it

the diffusion limit for reversible jump processes on z d with ergodic random bond conductivities we consider a reversible jump process on d whose jump rates themselves are random we show mean square convergence of this process under diffusion scaling to a limiting brownian motion with a certain diffusion matrix characterizing effective conductivity

the generalized spectral radius and extremal norms the generalized spectral radius also known under the name of joint spectral radius or after taking logarithms maximal lyapunov exponent of a discrete inclusion is examined we present a new proof for a result of barabanov which states that for irreducible sets of matrices an extremal norm always exists this approach lends itself easily to the analysis of further properties of the generalized spectral radius we prove that the generalized spectral radius is locally lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius sufficient conditions for the existence of extremal norms are obtained

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

the additive structure of elliptic homogenization one of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions since the latter are nonlocal functions of the former in this paper we address this problem in a new way in the context of linear elliptic equations in divergence form by showing that certain quantities associated to the energy density of solutions are essentially additive as a result we are able to prove quantitative estimates on the weak convergence of the gradients fluxes and energy densities of the first order correctors under blow down which are optimal in both scaling and stochastic integrability the proof of the additivity is a bootstrap argument completing the program initiated in armstrong et al commun math phys 347 2 315 361 2016 using the regularity theory recently developed for stochastic homogenization we reduce the error in additivity as we pass to larger and larger length scales in the second part of the paper we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables in particular we prove that the first order correctors converge in the large scale limit to a variant of the gaussian free field

quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics we study quantitatively the effective large scale behavior of discrete elliptic equations on the lattice mathbb z d with random coefficients the theory of stochastic homogenization relates the random stationary and ergodic field of coefficients with a deterministic matrix of effective coefficients this is done via the corrector problem which can be viewed as a highly degenerate elliptic equation on the infinite dimensional space of admissible coefficient fields in this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a spectral gap estimate w r t a glauber dynamics on coefficient fields as it is the case for independent and identically distributed coefficients as a main result we prove an optimal decay in time of the semigroup associated with the corrector problem i e of the generator of the process called random environment as seen from the particle as a corollary we recover existence of stationary correctors in dimensions d 2 and prove new optimal estimates for regularized versions of the corrector in dimensions d ge 2 we also give a self contained proof of a new estimate on the gradient of the parabolic variable coefficient green s function which is a crucial analytic ingredient in our approach as an application of these results we prove the first and optimal estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients

a central limit theorem for the effective conductance linear boundary data and small ellipticity contrasts given a resistor network on mathbb z d with nearest neighbor conductances the effective conductance in a finite set with a given boundary condition is the minimum of the dirichlet energy over functions with the prescribed boundary values for shift ergodic conductances linear dirichlet boundary conditions and square boxes the effective conductance scaled by the volume of the box converges to a deterministic limit as the box size tends to infinity here we prove that for i i d conductances with a small ellipticity contrast also a non degenerate central limit theorem holds the proof is based on the corrector method and the martingale central limit theorem a key integrability condition is furnished by the meyers estimate more general domains boundary conditions and ellipticity contrasts will be addressed in a subsequent paper

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

irreducible semigroups with multiplicative spectral radius abstract irreducible multiplicative semigroups of finite dimensional linear operators with some additional properties are studied each of the properties submultiplicativity of the spectral radius its multiplicativity or is constancy as well as a certain property called the rota condition implies that every member of such a semigroup is except for a scalar coefficient similar to the direct sum of an isometry and a nilpotent operator

pattern densities in non frozen planar dimer models in this paper we introduce a family of observables for the dimer model on a bi periodic bipartite planar graph called pattern density fields we study the scaling limit of these objects for non frozen gibbs measures of the dimer model and prove that they converge to a linear combination of a derivative of the gaussian massless free field and an independent white noise

symmetric markov chains in ℤ d how fast can they move consider a reversible markov chain x n which takes values in a subset of d if the steps of the chain are uniformly bounded and the invariant measure satisfies a mild regularity condition varopoulos carne and kesten have obtained estimates on p x n x text 0 lambda n 1 2 which exhibit a gaussian tail in but blow up as n following kesten s approach we derive bounds which are uniform in n in some special cases our main result however is an example which shows that in general the estimates of varopoulos carne and kesten are essentially best possible

annealed estimates on the green function we consider a random uniformly elliptic coefficient field a x on the d dimensional integer lattice mathbb z d we are interested in the spatial decay of the quenched elliptic green function g a x y next to stationarity we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic sobolev inequality holds for the ensemble langle cdot rangle we prove that all stochastic moments of the first and second mixed derivatives of the green function that is langle nabla x g x y p rangle and langle nabla x nabla y g x y p rangle have the same decay rates in x y gg 1 as for the constant coefficient green function respectively this result relies on and substantially extends the one by delmotte and deuschel probab theory relat fields 133 358 390 2005 which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of g that is langle nabla x g x y 2 rangle and langle nabla x nabla y g x y rangle as an application we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast

continuum modeling of flexible structures with application to vibration control in this paper we consider the construction of hybrid models for flexible structures including both the natural spatially distributed dynamics of low mass density structures and the lumped models associated with rigid body dynamics for the purposes of active control of structural vibrations we argue that by using transform methods the standard approach to transfer function or mechanical impedance modeling can be extended to provide a complete formulation of a hybrid model with localized control we proceed to give a systematic procedure for obtaining the required transfer functions and green s functions for the hybrid model dynamics consisting of one dimensional elastic elements such as beams rods and cables together with rigid body lumped models we focus on a well known family of models for elastic elements together with various reductions and indicate the construction of well posed state space models for these systems using the resulting state space we next demonstrate the construction of a complete frequency domain characterization of the elemental system response using transfer functions and green s functions hybrid constructions are discussed for various interconnections of distributed and lumped elements and a simple illustrative example is given we highlight the extent to which the required computations can be performed automatically on a digital computer using a computer algebra system

the generalized spectral radius is strictly increasing using a result linking convexity and irreducibility of matrix sets it is shown that the generalized spectral radius of a compact set of matrices is a strictly increasing function of the set in a very natural sense as an application some consequences of this property in the area of time varying stability radii are discussed in particular using the implicit function theorem sufficient conditions for lipschitz continuity are derived an example is presented of a linearly increasing family of matrix polytopes for which the proximal subgradient of the generalized spectral radius at a certain polytope contains 0 so that the implicit function theorem is not applicable in all cases

correlation structure of the corrector in stochastic homogenization recently the quantification of errors in the stochastic homogenization of divergence form operators has witnessed important progress our aim now is to go beyond error bounds and give precise descriptions of the effect of the randomness in the large scale limit this paper is a first step in this direction our main result is to identify the correlation structure of the corrector in dimension 33 and higher this correlation structure is similar to but different from that of a gaussian free field

noise enhanced fisher information in parallel arrays of sensors with saturation this paper investigates stochastic resonance in parallel arrays of uncoupled saturating devices the fisher information is used to demonstrate the possibility of noise improved parameter estimation for arbitrary parametric signals especially it is shown that improvement by noise always occurs in these arrays for any configuration of the input signal even in optimal configuration the results contribute to establish stochastic resonance in parallel uncoupled arrays as a general mechanism of enhancement by noise which can occur in wide classes of nonlinearities and for various information processing tasks it can supplement other mechanisms of stochastic resonance that take place in isolated nonlinearities but generally in restricted configurations of the input signal

quenched invariance principles for random walks and elliptic diffusions in random media with boundary via a dirichlet form extension theorem and making full use of two sided heat kernel estimates we establish quenched invariance principles for random walks in random environments with a boundary in particular we prove that the random walk on a supercritical percolation cluster or amongst random conductances bounded uniformly from below in a half space quarter space etc converges when rescaled diffusively to a reflecting brownian motion which has been one of the important open problems in this area we establish a similar result for the random conductance model in a box which allows us to improve existing asymptotic estimates for the relevant mixing time furthermore in the uniformly elliptic case we present quenched invariance principles for domains with more general boundaries

high order correctors and two scale expansions in stochastic homogenization in this paper we study high order correctors in stochastic homogenization we consider elliptic equations in divergence form on mathbb z d with the random coefficients constructed from i i d random variables we prove moment bounds on the high order correctors and their gradients under dimensional constraints it implies the existence of stationary correctors and stationary gradients in high dimensions as an application we prove a two scale expansion of the solutions to the random pde which identifies the first and higher order random fluctuations in a strong sense

on matrix semigroups bounded above and below abstract an irreducible norm closed semigroup of complex matrices is simultaneously similar to a semigroup of partial isometries if and only if a the norms of all nonzero members of it are uniformly bounded above and below and b its idempotents commute this is a generalization of the well known result on bounded groups

on the averaged green s function of an elliptic equation with random coefficients we consider a divergence form elliptic difference operator on the lattice mathbb z d with a coefficient matrix that is an i i d perturbation of the identity matrix recently bourgain introduced novel techniques from harmonic analysis to prove the convergence of the feshbach schur perturbation series related to the averaged green s function of this model our main contribution is a refinement of bourgain s approach which improves the key decay rate from 2d epsilon to 3d epsilon the optimal decay rate is conjectured to be 3d as an application we derive estimates on higher derivatives of the averaged green s function which go beyond the second derivatives considered by delmotte deuschel and related works

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

spectral measure and approximation of homogenized coefficients this article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations the originality of this work is the use of a well known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients in particular we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error and conversely combined with results by otto and the first author in low dimension and results by the second author in high dimension this allows us to prove that for any dimension d 2 there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem

towards a nonperturbative renormalization group analysis we prove that a certain class of convex gradient models in high dimensional spaces without the presence of a small parameter renormalizes to a free field as a consequence we establish a certain asymptotic formula for the partition function in some ways this is a realization of gawedzki and kupiainen s idea to use correlation inequalities to augment the rigorous renormalization group methods we use the more particular suggestion of spencer to use certain inequalities of brascamp and lieb and also the formulation of the correlation functions in terms of the solutions to some partial differential equations given by helffer and sjostrand