central limit theorem for a random walk with random obstacles in mathrm r d for some positive constants r and ro with r ro for a probability measure v on the space t we write p ffv d q p we call the process x t p a random walk with random obstacles let t be the set of all q ee tr such that p x l e has an unbounded support and let u be a gibbs state associated with a pair potential d which is translation invariant one of the main results of this paper is the central limit theorem for x t pa where 4w is a conditional probability measure of the gibbs state u given the event t that is 4t a i1 in a previous paper 10 we considered a system of infinitely many hard balls with the same diameter r moving discontinuously in rd we constructed the markov process 4t which describes the system this process has a gibbs state as a stationary measure we showed the ergodicity of the stationary markov

the universal relation between scaling exponents in first passage percolation it has been conjectured in numerous physics papers that in ordinary rst passage percolation on integer lattices the uctuation exponent and the wandering exponent are related through the universal relation 2 1 irrespective of the dimension this is sometimes called the kpz relation between the two exponents this article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense

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

superdiffusivity in first passage percolation in standard first passage percolation on bbb z d with d geq 2 the time minimizing paths from a point to a plane at distance l are expected to have transverse fluctuations of order l xi it has been conjectured that xi d geq 1 2 with the inequality strict superdiffusivity at least for low d and with xi 2 2 3 we prove versions of xi d geq 1 2 for all d and xi 2 geq 3 5

critical behavior for maximal flows on the cubic lattice let f0 and fm be the top and bottom faces of the box 0 k 0 l 0 m in z3 to each edge e in the box we assign an i i d nonnegative random variable t e representing the flow capacity of e denote by k l m the maximal flow from f0 to fm in the box let pc denote the critical value for bond percolation on z3 it is known that k l m is asymptotically proportional to the area of f0 as m k l when the probability that t e 0 exceeds pc but is of lower order if the probability is strictly less than pc here we consider the critical case where the probability that t e 0 is exactly equal to pc and prove that mathop text lim limits k l m to infty frac 1 kl phi k l m 0 text a text s and in l 1 the limiting behavior of related to surfaces on z3 are also considered in this paper

on similarity coefficients for 2 2 tables and correction for chance this paper studies correction for chance in coefficients that are linear functions of the observed proportion of agreement the paper unifies and extends various results on correction for chance in the literature a specific class of coefficients is used to illustrate the results derived in this paper coefficients in this class e g the simple matching coefficient and the dice sorenson coefficient become equivalent after correction for chance irrespective of what expectation is used the coefficients become either cohen s kappa scott s pi mak s rho goodman and kruskal s lambda or hamann s eta depending on what expectation is considered appropriate both a multicategorical generalization and a multivariate generalization are discussed

variational problems with percolation dilute spin systems at zero temperature we study the asymptotic behavior of dilute spin lattice energies by exhibiting a continuous interfacial limit energy computed using the notion of convergence and techniques mixing geometric measure theory and percolation while scaling to zero the lattice spacing the limit is not trivial above a percolation threshold since the lattice energies are not equi coercive a suitable notion of limit magnetization must be defined which can be characterized by two phases separated by an interface the macroscopic surface tension at this interface is characterized through a first passage percolation formula which highlights interesting connections between variational problems and percolation issues a companion result on the asymptotic description on energies defined on paths in a dilute environment is also given

overall properties of a discrete membrane with randomly distributed defects a prototype for variational percolation problems with surface energies is considered the description of the macroscopic properties of a two dimensional discrete membrane with randomly distributed defects in the spirit of the weak membrane model of blake and zisserman quadratic springs that may break at a critical length of the elongation after introducing energies depending on suitable independent identically distributed random variables this is done by exhibiting an equivalent continuum energy computed using convergence geometric measure theory and percolation arguments we show that below a percolation threshold the effect of the defects is negligible and the continuum description is given by the dirichlet integral while above that threshold an additional griffith fracture term appears in the energy which depends only on the defect probability through the chemical distance on the weak cluster of defects

rounding effects of quenched randomness on first order phase transitions frozen in disorder in an otherwise homogeneous system is modeled by interaction terms with random coefficients given by independent random variables with a translation invariant distribution for such systems it is proven that ind 2 dimensions there can be no first order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter discontinuities which would amount to a continuous symmetry breaking in systems which are stochastically invariant under the action of a continuous subgroup ofo n are suppressed by the randomness in dimensionsd 4 specific implications are found in the random field ising model for which we conclude that ind 2 dimensions at all h the gibbs state is unique for almost all field configurations and in the random bond potts model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point the results are explained by the argument of imry and ma 1 the proofs involve the analysis of fluctuations of free energy differences which are shown using martingale techniques to be gaussian on the suitable scale

maximum entropy principles for disordered spins we transform nonstationary independent random fields with exponential radon nikodym factors and study the asymptotics of the transformed processes as applications we deduce conditional limit theorems for such random fields and we study a curie weiss type mean field model of a quenched mixed magnetic crystal this model has quenched site disorder and frustration but non random coupling constants we find a continuous phase transition with critical exponents equal to those of the classical curie weiss theory

fluctuations of extensive functions of quenched random couplings an extensive quantity is a family of functions v of random parameters indexed by the finite regionsv subsets of d over which v are additive up to corrections satisfying the boundary estimate stated below it is shown that unless the randomness is nonessential in the sense that lim v v has a unique value in the absolute i e not just probabilistic sense the variance of such a quantity grows as the volume ofv of particular interest is the free energy of a system with random couplings for such v bounds are derived also for the generating functione et in a separate application variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment

multi particle diffusion limited aggregation we consider a stochastic aggregation model on mathbb z d start with particles distributed according to the product bernoulli measure with parameter mu in addition start with an aggregate at the origin non aggregated particles move as continuous time simple random walks obeying the exclusion rule whereas aggregated particles do not move the aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it this evolution is called multi particle diffusion limited aggregation our main result states that if on d 1 the initial density of particles is large enough then with positive probability the aggregate has linearly growing arms that is there exists a constant c 0 so that at time t the aggregate contains a point of distance at least ct from the origin for all t the key conceptual element of our analysis is the introduction and study of a new growth process consider a first passage percolation process called type 1 starting from the origin whenever type 1 is about to occupy a new vertex with positive probability instead of doing it it gives rise to another first passage percolation process called type 2 which starts to spread from that vertex each vertex gets occupied only by the process that arrives to it first this process may have three phases extinction type 1 gets eventually surrounded by type 2 coexistence infinite clusters of both types emerge and strong survival type 1 produces an infinite cluster which entraps all type 2 clusters understanding the various phases of this process is of mathematical interest on its own right we establish the existence of a strong survival phase and use this to show our main result

multidimensional random subshifts of finite type and their large deviations i introduce random multidimensional subshifts of finite type which generalize models of spin glasses and establish the almost sure large deviations bounds for gibbs measures there the paper is sequel to ekw where the corresponding results were obtained for deterministic multidimensional subshifts of finite type

large deviations for increasing sequences on the plane we prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among poisson points on the plane the rate function for lower tail deviations is derived from a 1977 result of logan and shepp about young diagrams of random permutations for the upper tail we use a coupling with hammersley s particle process and convex analytic techniques along the way we obtain the rate function for the lower tail of a tagged particle in a totally asymmetric hammersley s process

a simplified proof of the relation between scaling exponents in first passage percolation in a recent breakthrough work chatterjee 5 proved a long standing conjecture that relates the transversal exponentand the fluctuation exponentin first passage percolation on z d the purpose of this paper is to introduce a new and intuitive idea that replaces the main argument of 5 and gives an alternative proof of this relation specifically we show that under the assumption that exponents defined in 5 exist one has the relation 2 1 one advantage of our argument is that it does not require the nearly gamma assumption of 5

entropy for translation invariant random cluster measures we study translation invariant random cluster measures with techniques from large deviation theory and convex analysis in particular we prove a large deviation principle with rate function given by a specific entropy and a dobrushinlanfordruelle variational principle that characterizes translation invariant random cluster measures as the solutions of the variational equation for free energy consequences of these theorems include inequalities for edge and cluster densities of translationinvariant random cluster measures 1 introduction the random cluster model was introduced around 1972 by fortuin and kasteleyn 18 as a family of models that contains the bernoulli bond percolation model the ising model and the q state potts models as special cases it has developed into an important and very successful tool for studying the ising and potts models our paper focuses on the properties of the random cluster measures themselves our starting point is grimmett s 23 development of elements of a thermodynamic formalism for this model beginning with a definition of infinite volume random cluster measures in terms of a dlr dobrushinlanfordruelle equation just as for gibbsian systems natural next questions concern the validity of large deviation principles and variational characterizations of infinite volume measures these questions we answer in the present paper and derive some consequences here is a brief overview of the dlr approach of 23 the random cluster measures of interest live on the graph formed by putting edges between nearest neighbor vertices of the d dimensional hypercubic lattice d the edge that connects two vertices x and y that are l 1 distance one apart is

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

inequalities between kappa and kappa like statistics for k k tables the paper presents inequalities between four descriptive statistics that can be expressed in the form p e p 1 e p where p is the observed proportion of agreement of a k k table with identical categories and e p is a function of the marginal probabilities scott s is an upper bound of goodman and kruskal s and a lower bound of both bennett et al s and cohen s we introduce a concept for the marginal probabilities of the k k table called weak marginal symmetry using the rearrangement inequality it is shown that bennett et al s is an upper bound of cohen s if the k k table is weakly marginal symmetric

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

a quantitative central limit theorem for the effective conductance on the discrete torus we study a random conductance problem on a d dimensional discrete torus of size l 0 the conductances are independent identically distributed random variables uniformly bounded from above and below by positive constants the effective conductance al of the network is a random variable depending on l that converges almost surely to the homogenized conductance ahom our main result is a quantitative central limit theorem for this quantity as l in particular we prove there exists some 0 such that dk ld 2 a ahom g l d 2 logd l where dk is the kolmogorov distance and gis a standard normal variable the main achievement of this contribution is the precise asymptotic description of the variance of al 2015 wiley periodicals inc

vertex operators solvable lattice models and metaplectic whittaker functions we show that spherical whittaker functions on an n fold cover of the general linear group arise naturally from the quantum fock space representation of u q widehat mathfrak sl n introduced by kashiwara miwa and stern kms we arrive at this connection by reconsidering the solvable lattice models known as metaplectic ice whose partition functions are metaplectic whittaker functions first we show that a certain hecke action on metaplectic whittaker coinvariants agrees up to twisting with a hecke action of ginzburg reshetikhin and vasserot arising in quantum affine schur weyl duality this allows us to expand the framework of kms by drinfeld twisting to introduce gauss sums into the quantum wedge which are necessary for connections to metaplectic forms our main theorem interprets the row transfer matrices of this ice model as half vertex operators on quantum fock space that intertwine with the action of u q widehat mathfrak sl n in the process we introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to whittaker functions on an n fold metaplectic cover of text gl r these resemble llt polynomials or ribbon symmetric functions introduced by lascoux leclerc and thibon and in fact the metaplectic symmetric functions are up to twisting specializations of supersymmetric llt polynomials defined by lam indeed lam constructed families of symmetric functions from heisenberg algebra actions on the fock space commuting with the u q widehat mathfrak sl n action the heisenberg algebra is independent of drinfeld twisting of the quantum group we explain that half vertex operators agree with lam s construction and this interpretation allows for many new identities for metaplectic symmetric and whittaker functions including cauchy identities while both metaplectic symmetric functions and llt polynomials can be related to vertex operators on the quantum fock space only metaplectic symmetric functions are connected to solvable lattice models

normal approximation for a random elliptic equation we consider solutions of an elliptic partial differential equation in mathbb r d with a stationary random conductivity coefficient that is also periodic with period l boundary conditions on a square domain of width l are arranged so that the solution has a macroscopic unit gradient we then consider the average flux that results from this imposed boundary condition it is known that in the limit l rightarrow infty this quantity converges to a deterministic constant almost surely our main result is that the law of this random variable is very close to that of a normal random variable if the domain size l is large we quantify this approximation by an error estimate in total variation the error estimate relies on a second order poincare inequality developed recently by chatterjee

on direct convergence and periodicity for transition probabilities of markov chains in random environments we study direct convergence of the products p theta 0 cdots p theta n of random stochastic matrices these products can be interpreted as the transition probabilities of nonhomogeneous markov chains selected at random by a stationary environmental sequence theta n in other words a markov chain in a random environment rather than make assumptions analogous to irreducibility and aperiodicity for homogeneous markov chains we introduce equivalence relations that allow convergence results on the equivalence classes the classical decomposition into a cycle of periodic sets is not possible in general so the periodicity in the title is meant only to be suggestive we also examine the frequency of times of positive probability of return to a state or set

harvest rate of reconfigurable pipelines for a reconfigurable architecture the harvest rate is the expected percentage of defect free processors that can be connected into the desired topology the authors give an analytical estimation for the harvest rate of reconfigurable multipipelines based on the following model there are n pipelines each with m stages where each stage of a pipeline is defective with identical independent probability 0 5 and spare wires are provided for reconfiguration by formulating the shifting reconfiguration as weighted chains in a partial ordered set they prove when n spl theta m the harvest rate is between 34 and 72

information theoretic measures for clusterings comparison variants properties normalization and correction for chance information theoretic measures form a fundamental class of measures for comparing clusterings and have recently received increasing interest nevertheless a number of questions concerning their properties and inter relationships remain unresolved in this paper we perform an organized study of information theoretic measures for clustering comparison including several existing popular measures in the literature as well as some newly proposed ones we discuss and prove their important properties such as the metric property and the normalization property we then highlight to the clustering community the importance of correcting information theoretic measures for chance especially when the data size is small compared to the number of clusters present therein of the available information theoretic based measures we advocate the normalized information distance nid as a general measure of choice for it possesses concurrently several important properties such as being both a metric and a normalized measure admitting an exact analytical adjusted for chance form and using the nominal 0 1 range better than other normalized variants

diffusion of directed polymers in a random environment we consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time in spatial dimensionsd 2 we establish that the behavior is diffusive with probability one the diffusion constant is not renormalized by the interaction

large deviations and the maximum entropy principle for marked point random fields we establish large deviation principles for the stationary and the individual empirical fields of poisson and certain interacting random fields of marked point particles in d the underlying topologies are induced by a class of not necessarily bounded local functions and thus finer than the usual weak topologies our methods yield further that the limiting behaviour of conditional poisson distributions as well as certain distributions of gibbsian type is governed by the maximum entropy principle we also discuss various applications and examples

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

directed polymers in a random medium abstract the problem of a polymer in a random medium can be formulated as follows consider a regular lattice with on each bond ij a random energy ij the energy e w of a self avoiding walk w on this lattice is the sum of the energies of all the bonds visited by the walk then the problem is to compute the partition function z w exp e w t several recent results will be presented 1 the mean field version of this problem can be solved exactly and one observes a low temperature phase similar to the spin glass phase 2 for finite dimensional lattices one can derive bounds on the transition temperature in high enough dimension 3 for hierarchical lattices the problem can be reduced to finding stable laws when one combines random variables in a non linear way for example z z 1 z 2 z 3 z 4

entropy limit theorems and variational principles for disordered lattice systems we study infinite volume limits and gibbs states of disordered lattice systems with bounded and continuous potentials our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes we find that many familiar results of invariant potentials such as large deviation theorems variational principles and equivalence of ensembles continue to hold for disordered models with suitably modified statements

limit theorems for maximum flows on a lattice we independently assign a non negative value as a capacity for the quantity of flows per unit time with a distribution f to each edge on the mathbf z d lattice we consider the maximum flows through the edges from a source to a sink in a large cube in this paper we show that the ratio of the maximum flow and the size of the source is asymptotic to a constant this constant is denoted by the flow constant by the max flow and min cut theorem this is equivalent to a statement about the asymptotic behavior of the minimal value assigned to any surface on the large cube we can also show that there exists such a surface that is proportional to the size of the faces of the cube

probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d 1 2 3 via a monte carlo procedure in the disorder in order to probe with high precision the tails of the ground state energy distribution of disordered spin systems k orner katzgraber and hartmann have recently proposed an importance sampling monte carlo markov chain in the disorder in this paper we combine their monte carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution p d e 0 for the directed polymer in a random medium of dimension d 1 2 3 in d 1 we check the validity of the algorithm by a direct comparison with the exact result namely the tracy widom distribution in dimensions d 2 and d 3 we measure the negative tail up to ten standard deviations which correspond to probabilities of order p d e 0 ensuremath sim 10 ensuremath 22 our results are in agreement with zhang s argument stating that the negative tail exponent ensuremath eta d of the asymptotic behavior mathrm ln phantom rule 0 2em 0ex p d e 0 ensuremath sim ensuremath ensuremath mid e 0 ensuremath mid ensuremath eta d as e 0 ensuremath rightarrow ensuremath ensuremath infty is directly related to the fluctuation exponent ensuremath theta d which governs the fluctuations ensuremath delta e 0 l ensuremath sim l ensuremath theta d of the ground state energy e 0 for polymers of length l via the simple formula ensuremath eta d 1 1 ensuremath ensuremath theta d throughout the paper we comment on the similarities and differences with spin glasses

a family of association coefficients for metric scales four types of metric scales are distinguished the absolute scale the ratio scale the difference scale and the interval scale a general coefficient of association for two variables of the same metric scale type is developed some properties of this general coefficient are discussed it is shown that the matrix containing these coefficients between any number of variables is gramian the general coefficient reduces to specific coefficients of association for each of the four metric scales two of these coefficients are well known the product moment correlation and tucker s congruence coefficient applications of the new coefficients are discussed

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

metastates in disordered mean field models ii the superstates we continue to investigate the size dependence of disordered mean field models with finite local spin space in more detail illustrating the concept of superstates as recently proposed by bovier and gayrard we discuss various notions of convergence for the behavior of the paths t tn t 0 1 in the thermodynamic limit n here n is the gibbs measure in the finite volume 1 n and is the disorder variable in particular we prove refined convergence statements in our concrete examples the hopfield model with finitely many patterns having continuous paths and the curie weiss random field ising model having singular paths

large deviations for the chemical distance in supercritical bernoulli percolation the chemical distance d x y is the length of the shortest open path between two points x and y in an infinite bernoulli percolation cluster in this work we study the asymptotic behavior of this random metric and we prove that for an appropriate norm depending on the dimension and the percolation parameter the probability of the event 0 x d 0 x x 1 e 1 e exponentially decreases when x 1 tends to infinity from this bound we also derive a large deviation inequality for the corresponding asymptotic shape result

central limit theorem for first passage percolation time across thin cylinders we prove that first passage percolation times across thin cylinders of the form 0 n hn hn d 1 obey gaussian central limit theorems as long as hn grows slower than n1 d 1 it is an open question as to what is the fastest that hn can grow so that a gaussian clt still holds under the natural but unproven assumption about existence of fluctuation and transversal exponents and strict convexity of the limiting shape in the direction of 1 0 0 we prove that in dimensions 2 and 3 the clt holds all the way up to the height of the unrestricted geodesic we also provide some numerical evidence in support of the conjecture in dimension 2

schur polynomials and the yang baxter equation we describe a parametrized yang baxter equation with nonabelian parameter group that is we show that there is an injective map g mapsto r g from rm gl 2 mathbb c times rm gl 1 mathbb c to end v otimes v where v is a two dimensional vector space such that if g h in g then r 12 g r 13 gh r 23 h r 23 h r 13 gh r 12 g here r i j denotes r applied to the i j components of v otimes v otimes v the image of this map consists of matrices whose nonzero coefficients a 1 a 2 b 1 b 2 c 1 c 2 are the boltzmann weights for the non field free six vertex model constrained to satisfy a 1 a 2 b 1 b 2 c 1 c 2 0 this is the exact center of the disordered regime and is contained within the free fermionic eight vertex models of fan and wu as an application we show that with boundary conditions corresponding to integer partitions the six vertex model is exactly solvable and equal to a schur polynomial s times a deformation of the weyl denominator this generalizes and gives a new proof of results of tokuyama and hamel and king

cohen s linearly weighted kappa is a weighted average of 2 2 kappas an agreement table with n 3 ordered categories can be collapsed into n 1 distinct 2 2 tables by combining adjacent categories vanbelle and albert stat methodol 6 157 163 2009c showed that the components of cohen s weighted kappa with linear weights can be obtained from these n 1 collapsed 2 2 tables in this paper we consider several consequences of this result one is that the weighted kappa with linear weights can be interpreted as a weighted arithmetic mean of the kappas corresponding to the 2 2 tables where the weights are the denominators of the 2 2 kappas in addition it is shown that similar results and interpretations hold for linearly weighted kappas for multiple raters

metastates in disordered mean field models random field and hopfield models we rigorously investigate the size dependence of disordered mean field models with finite local spin space in terms of metastates thereby we provide an illustration of the framework of metastates for systems of randomly competing gibbs measures in particular we consider the thermodynamic limit of the empirical metastate 1 n sum nolimits n 1 n delta mu eta eta where n is the gibbs measure in the finite volume 1 n and the frozen disorder variable is fixed we treat explicitly the hopfield model with finitely many patterns and the curie weiss random field ising model in both examples in the phase transition regime the empirical metastate is dispersed for largen moreover it does not converge for a e but rather in distribution for whose limits we given explicit expressions we also discuss another notion of metastates due to aizenman and wehr

phase transition in the 3d random field ising model we show that the three dimensional ising model coupled to a small random magnetic field is ordered at low temperatures this means that the lower critical dimension dl for the theory isdl 2 settling a long controversy on the subject our proof is based on an exact renormalization group rg analysis of the system this analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of imry and ma the rg acts in the space of ising models and in the space of random field distributions driving the former to zero temperature and the latter to zero variance

optimal flow through the disordered lattice author s aldous d abstract consider routing traffic on the n n torus simultaneously between all source destination pairs to minimize the cost ec e f 2 e where f e is the volume of flow across edge e and the c e form an i i d random environment we prove existence of a rescaled n limit constant for minimum cost by comparison with an appropriate analogous problem about minimum cost flows across a m m subsquare of the lattice institute of mathematical statistics 2007

correction added in proof we extend sanov s theorem on i i d large deviations to independent but not identically distributed random variables and study the generalization of relative entropy that appears as the rate function

large deviations and maximum entropy principle for interacting random fields on mathbb z d we present a new approach to the principle of large deviations for the empirical field of a gibbsian random field on the integer lattice zd this approach has two main features first we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities and thus obtain estimates which are finer and more widely applicable second we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field the limits being those predicted by the maximum entropy principle this result implies a general version of the equivalence of gibbs ensembles stating that every microcanonical limiting state is a grand canonical equilibrium state we also prove a converse to the last statement and discuss some applications 0 introduction as is well known the study of the asymptotic probabilities of large fluctuations of time averages or space averages away from the mean is based on two fundamental principles the principle of large deviations and the maximum entropy principle the former provides the exact rate of exponential decay of the fluctuation probabilities whereas the latter predicts the limiting conditional distribution under the condition that the fluctuations are large it is obvious that these principles are intimately related in this paper we investigate these principles in the case of interacting random fields on the integer lattice 7zd the setup is the following first we let e 4 be any measurable space we shall assume throughout that e a is standard borel but we shall avoid making any explicit topological assumptions on e so we do not assume that e is polish next we let s zd be the d dimensional integer lattice and q f e gas the associated product space in lattice models of statistical mechanics e is called the state space or single spin space and q f the configuration space we let j s f denote the set of all probability

free energy and some sample path properties of a random walk with random potential we study the asymptotic behavior of the free energy for a model defined by sinai of one dimensional random walk with random potential in particular we obtain a central limit theorem and a strong law of large numbers for this free energy we use some results on the free energy to study some sample path properties of this random walk which are related respectively to its recurrence and localization some exponents describing the recurrence and localization are found

microcanonical distributions for lattice gases in this article a large deviation principle cf theorem 1 3 for the empirical distribution functional is applied to prove a rather general version of boltzmann s principle cf theorem 3 5 for models with shift invariant finite range potentials the final section contains an application of these considerations to the two dimensional ising model at sub critical temperature

homogenization of surface and length energies for spin systems abstract we study the homogenization of lattice energies related to ising systems of the form e e u i j c i j e u i u j with u i a spin variable indexed on the portion of a cubic lattice e z d by computing their limit in the framework of surface energies in a bv setting we introduce a notion of homogenizability of the system c i j e that allows to treat periodic almost periodic and random statistically homogeneous models the latter in dimension two when the coefficients are positive ferromagnetic energies in which case the limit energy is finite on bv 1 and takes the form f u u 1 d h d 1 is the normal to u 1 where is characterized by an asymptotic formula in the random case can be expressed in terms of first passage percolation characteristics the result is extended to coefficients with varying sign under the assumption that the areas where the energies are antiferromagnetic are well separated finally we prove a dual result for discrete curves

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

some paradoxical results for the quadratically weighted kappa the quadratically weighted kappa is the most commonly used weighted kappa statistic for summarizing interrater agreement on an ordinal scale the paper presents several properties of the quadratically weighted kappa that are paradoxical for agreement tables with an odd number of categories n it is shown that if one of the raters uses the same base rates for categories 1 and n categories 2 and n 1 and so on then the value of quadratically weighted kappa does not depend on the value of the center cell of the agreement table since the center cell reflects the exact agreement of the two raters on the middle category this result questions the applicability of the quadratically weighted kappa to agreement studies if one wants to report a single index of agreement for an ordinal scale it is recommended that the linearly weighted kappa instead of the quadratically weighted kappa is used

ribbon tableaux hall littlewood functions quantum affine algebras and unipotent varieties we introduce a new family of symmetric functions which are q analogs of products of schur functions defined in terms of ribbon tableaux these functions can be interpreted in terms of the fock space representation fq of uq sln and are related to hall littlewood functions via the geometry of flag varieties we present a series of conjectures and prove them in special cases the essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of fq using ribbon tableaux

vertex operators and hall littlewood symmetric functions abstract we consider vertex operators on space v with a parameter t their components form an associative algebra which is a generalization of the clifford algebra a distinguished orthogonal basis of v is proved to be the hall littlewood symmetric functions we show that kostka foulkes polynomials or certain kazhdan lusztig polynomials for the affine weyl group of type a are matrix coefficients on the space v we also obtain certain generating functions for the product of hall littlewood functions and the kostka foulkes polynomials

a kraemer type rescaling that transforms the odds ratio into the weighted kappa coefficient this paper presents a simple rescaling of the odds ratio that transforms the association measure into the weighted kappa statistic for a 2 2 table

exponent inequalities for the bulk conductivity of a hierarchical model the bulk conductivity p of the bond lattice in d is considered where the bonds have conductivity 1 with probabilityp or e 0 with probability 1 p various representations of the derivatives of p are developed these representations are used to analyze the behavior of p for e 0 near the percolation thresholdp c when the conducting backbone is assumed to have a hierarchical node link blob nlb structure this model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds exact asymptotics of frac d 2 sigma dp 2 for the nlb model are proven under some technical assumptions the proof employs a novel technique whereby frac d 2 sigma dp 2 for the nlb model with e 0 andp nearp c is computed using perturbation theory for p for two and three component resistor lattices aroundp 1 with a sequence of e s converging to 1 as one goes deeper in the hierarchy these asymptotics establish convexity of p for the nlb model nearp c and that its critical exponentt obeys the inequalities 1 t 2 ford 2 3 while 2 t 3 ford 4 the upper boundt 2 ind 3 which is realizable in the nlb class virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model

large deviations for lattice systems i parametrized independent fields we prove large deviation theorems for empirical measures of independent random fields whose distributions depend measurably on an auxiliary parameter this dependence respects the action of the shift group and a large deviation principle holds whenever a certain ergodicity condition is satisfied we also investigate the entropy functions for these processes especially in relation to the usual relative entropy

the phase boundary in dilute and random ising and potts ferromagnets the authors present an approach to dilute ising and potts models based on the fortuin kasteleyn random cluster representation which is simultaneously rigorous intuitive and surprisingly simple their analysis yields with no dimensional restrictions or other caveats the following asymptotic form of the phase boundary for the regular dilute model in which bonds have constant ferromagnetic coupling j with probability p and are vacant with probability 1 p the critical temperature scales as exp j ktc p approximately mod p pc mod implying that the crossover exponent is phi 1 if the constant couplings are replaced by a distribution f j with mass near j 0 quite different crossover behaviour is observed for example if f j approximately jalpha then for p near pc tc p approximately mod p pc mod 1 alpha

2 x 2 kappa coefficients measures of agreement or association summary two general but different contexts in which kappa might be used are defined agreement and association two models one for agreement and one for utility of association are defined yielding different kappa coefficients and different sampling theory asymptotic results are derived for both models small sample evaluations are presented for the model for agreement

large deviation principles for euclidean functionals and other nearly additive processes we prove a large deviation principle for a process indexed by cubes of the multidimensional integer lattice or euclidean space under approximate additivity and regularity hypotheses the rate function is the convex dual of the limiting logarithmic moment generating function in some applications the rate function can be expressed in terms of relative entropy the general result applies to processes in euclidean combinatorial optimization statistical mechanics and computational geometry examples include the length of the minimal tour the traveling salesman problem the length of the minimal matching graph the length of the minimal spanning tree the length of the k nearest neighbors graph and the free energy of a short range spin glass model

the ergodic theory of markov chains in random environments a general formulation of the stochastic model for a markov chain in a random environment is given including an analysis of the dependence relations between the environmental process and the controlled markov chain in particular the problem of feedback assuming stationary environments the ergodic theory of markov processes is applied to give conditions for the existence of finite invariant measure equilibrium distributions and to obtain ergodic theorems which provide results on convergence of products of random stochastic matrices coupling theory is used to obtain results on direct convergence of these products and the structure of the tail field state properties including classification and communication properties are discussed

directed polymers in a random environment some results on fluctuations we consider a polymer model on where to each edgee is associated a random variable v e a polymer configuration is represented by a directed pathr and has a weight exp e r e with 1 t the inverse temperature we extend some rigorous results that have been obtained for the ground state of this model to finite temperatures in particular we obtain some upper and lower bounds on sample to sample free energy fluctuations and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations

divergence of shape fluctuations in two dimensions we consider stochastic growth models such as standard first passage percolation on z d where to leading order there is a linearly growing deterministic shape under natural hypotheses we prove that for d 2 the shape fluctuations grow at least logarithmically in all directions although this bound is far from the expected power law behavior with exponent 1 3 it does prove divergence with additional hypotheses we obtain inequalities involving and the related exponent which is expected to equal 2 3 for d 2 combining these inequalities with previously known results we obtain for standard first passage percolation the bounds 1 8 for d 2 and 3 4 for all d

replica cluster variational method we present a general formalism to make the replica symmetric and replica symmetry breaking ansatz in the context of kikuchi s cluster variational method cvm using replicas and the message passing formulation of cvm we obtain a variational expression of the replicated free energy of a system with quenched disorder both averaged and on a single sample and make the hierarchical ansatz using functionals of functions of fields to represent the messages we obtain a set of integral equations for the message functionals the main difference with the bethe case is that the functionals appear in the equations in implicit form and are not positive definite thus standard iterative population dynamic algorithms cannot be used to determine them in the simplest cases the solution could be obtained iteratively using fourier transforms

quenched large deviations for interacting diffusions in random media the aim of the paper is to establish a large deviation principle ldp for the empirical measure of mean field interacting diffusions in a random environment the point is to derive such a result once the environment has been frozen quenched model the main theorem states that a ldp holds for every sequence of environment satisfying appropriate convergence condition with a rate function that does not depend on the disorder and is different from the rate function in the averaged model similar results concerning the empirical flow and local empirical measures are provided

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media

nondifferentiability of the time constants of first passage percolation we study the paths of minimal cost for first passage percolation in two dimensions and obtain an exponential bound on the tail probability of the ratio of the lengths of the shortest and longest of these this inequality permits us to answer a long standing question of hammersley and welsh on the shift differentiability of the time constant specifically we show that for subcritical bernoulli percolation the time constant is not shift differentiable when p is close to one half

bulk transport properties and exponent inequalities for random resistor and flow networks the properties of random resistor and flow networks are studied as a function of the density p of bonds which permit transport it is shown that percolation is sufficient for bulk transport in the sense that the conductivity and flow capacity are bounded away from zero wheneverp exceeds an appropriately defined percolation threshold relations between the transport coefficients and quantities in ordinary percolation are also derived assuming critical scaling these relations imply upper and lower bounds on the conductivity and flow exponents in terms of percolation exponents the conductivity exponent upper bound so derived saturates in mean field theory

existence of hydrodynamics for the totally asymmetric simple k exclusion process in a totally asymmetric simple k exclusion process particles take nearest neighbor steps to the right on the lattice z under the constraint that each site contain at most k particles we prove that such processes satisfy hydrodynamic limits under euler scaling and that the limit of the empirical particle profile is the entropy solution of a scalar conservation law with a concave flux function our technique requires no knowledge of the invariant measures of the process which is essential because the equilibria of asymmetric k exclusion are unknown but we cannot calculate the flux function precisely the proof proceeds via a coupling with a growth model on the two dimensional lattice in addition to the basic k exclusion with constant exponential jump rates we treat the site disordered case where each site has its own jump rate randomly chosen but frozen for all time the hydrodynamic limit under site disorder is new even for the simple exclusion process the case k 1 our proof makes no use of the markov property so at the end of the paper we indicate how to treat the case with arbitrary waiting times

boson fermion correspondence for hall littlewood polynomials the boson fermion correspondence is given for the case of the hall littlewood polynomials which is a t deformation of the correspondence realizing the schur polynomials

refined cauchy littlewood identities and six vertex model partition functions iii deformed bosons abstract we study hall littlewood polynomials using an integrable lattice model of t deformed bosons working with row to row transfer matrices we review the construction of hall littlewood polynomials of the a n root system within the framework of this model introducing appropriate double row transfer matrices we extend this formalism to hall littlewood polynomials based on the b c n root system and obtain a new combinatorial formula for them we then apply our methods to prove a series of refined cauchy and littlewood identities involving hall littlewood polynomials the last two of these identities are new and relate infinite sums over hyperoctahedrally symmetric hall littlewood polynomials with partition functions of the six vertex model on finite domains

assessing a fuzzy extension of rand index and related measures this empirical study extends the results of hullermeier rifqi henzgen and senge 2012 it examines the ability of a generalization of the rand index and four related measures of similarity to recover the cluster structure of the data in the framework of fuzzy c means clustering the index range is also used as a criterion statistic a monte carlo simulation is conducted for both the null case and where the data have a well defined cluster structure the fuzzy extension of the related measures is not so effective for imbalanced data on the contrary whether the index is dice fowlkes and mallows hurbert and arabie or jaccard it provides reliable results for noise data or for data containing fairly balanced clusters the criticisms of the rand index in the context of crisp clustering can also be extended to its fuzzy version

statistics and meta statistics in the concept of complexity abstract a crucial distinction between different classes of complexity measures is found to consist in their statistical structure while first order statistics always provides one or another kind of a measure of randomness second order meta statistics provides more subtle measures of complexity an example of a second order measure is constructed and applied to specific cases some formal and conceptual issues concerning the general relevance of meta statistical concepts of complexity are pointed out

large deviation estimates for a conditional probability distribution applications to random interaction gibbs measures let x i y i d be independent identically distributed random variables with arbitrary distribution we show that for almost every y i i the conditional law of the empirical field given y i i satisfies to large deviation inequalities this applies to the study of gibbs measures with random interaction in the case of some mean field models as well as of short range summable interaction we show that the pressure is nonrandom and is given by a variational formula these random gibbs measures have the same large deviation rate which does not depend on the particular realization of the interaction their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula

concentration inequalities for functions of gibbs fields with application to diffraction and random gibbs measures we derive useful general concentration inequalities for functions of gibbs fields in the uniqueness regime we also consider expectations of random gibbs measures that depend on an additional disorder field and prove concentration w r t the disorder field both fields are assumed to be in the uniqueness regime allowing in particular for non independent disorder fields the modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the dobrushin contraction coefficient and are explicitly computable on the basis of these inequalities we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in euclidean space restricted to a finite volume here we also allow for thermal dislocations of the scatterers around their equilibrium positions extending recent results for independent scatterers we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean the bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set

the affineq schur algebra abstract we introduce an analogue of the q schur algebra associated to coxeter systems of type a n 1 we give two constructions of this algebra the first construction realizes the algebra as a certain endomorphism algebra arising from an affine hecke algebra of type a r 1 where n r this generalizes the original q schur algebra as defined by dipper and james and the new algebra contains the ordinary q schur algebra and the affine hecke algebra as subalgebras using this approach we can prove a double centralizer property the second construction realizes the affine q schur algebra as the faithful quotient of the action of a quantum group on the tensor power of a certain module analogous to the construction of the ordinary q schur algebra as a quotient of u g l n