on very sparse circulant 0 1 matrices in this paper a particular partition on blocks of generalized circulant 0 1 matrices of composite order with three ones per row is determined using this result it is shown that the bipartite graph associated with a similar matrix in the usual way has genus 1 but a particular case where it holds 0 moreover a lower bound for the permanent of these matrices is achieved

uniform convergence of the free energy of the classical heisenberg model to that of the gaussian model we show that the free energy of the classical heisenberg model converges to the free energy of the gaussian in the low temperature limit the limit is uniform as the field is taken to zero

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

large deviations hypotheses testing and source coding for finite markov chains let x n n geq 1 be a finite markov chain with transition probability matrix of strictly positive entries a large deviation theorem is proved for the empirical transition count matrix and is used to get asymptotically optimal critical regions for testing simple hypotheses about the transition matrix as a corollary the error exponent in the source coding theorem for x n is obtained these results generalize the corresponding results for the independent and identically distributed case

on the number of different permanents of some sparse 0 1 circulant matrices abstract starting from known results about the number of possible values for the permanents of 0 1 circulant matrices with three nonzero entries per row and whose dimension n is prime we prove corresponding results for n power of a prime n product of two distinct primes and n 2 3 h supported by some experimental results we also conjecture that the number of different permanents of n n 0 1 circulant matrices with k nonzero per row is asymptotically equal to n k 2 k o n k 3

on the permanent of certain 0 1 toeplitz matrices abstract we obtain convenient expressions and or efficient algorithms for the permanent of certain very sparse 0 1 toeplitz matrices the classes of matrices considered here include some nontrivial examples of circulants to which none of the previous approaches could be successfully applied

low temperature properties of the hierarchical classical vector model we obtain low temperature properties of the classical vector model in a hierarchical formulation in three or more dimensions we consider the lattice model in a zero or non zero magnetic field where the single site spin variable r v has a density proportional to e lambda phi 2 1 2 for large using renormalization group methods we obtain a convergent expansion for the free energy with zero magnetic field for non zero fields a shift formula is used to obtain the effective action generated by the renormalization group transformation rgt to obtain the pure state zero field free energy and spontaneous magnetization we take the thermodynamic limit together with the zero field limit at a specified rate the spontaneous magnetization m is calculated is non zero and the pure state free energy coincides as expected with the zero field free energy also the sequence of zero field actions does not have a limit but we show that the sequence of actions generated from the original action shifted bym does the limiting action corresponds to a non canonical gaussian fixed point of the rgt

construction of convergent simplicial approximations of quantum fields on riemannian manifolds we construct simplicial approximations of random fields on riemannian manifolds of dimensiond we prove convergence of the fields to the continuum limit for arbitraryd in the gaussian case and ford 2 in the non gaussian case in particular we obtain convergence of the simplicial approximation to the continuum limit for quantum fields on riemannian manifolds with exponential interaction

permanental compounds and permanents of 0 1 circulants abstract it was shown by the author in a recent paper that a recurrence relation for permanents of 0 1 circulants can be generated from the product of the characteristic polynomials of permanental compounds of the companion matrix of a polynomial associated with 0 1 circulants of the given type in the present paper general properties of permanental compounds of companion matrices are studied and in particular of convertible companion matrices i e matrices whose permanental compounds are equal to the determinantal compounds after changing the signs of some of their entries these results are used to obtain formulas for the limit of the n th root of the permanent of the n n 0 1 circulant of a given type as n tends to infinity the root squaring method is then used to evaluate this limit for a wide range of circulant types whose associated polynomials have convertible companion matrices

universal channel coding for general output alphabet we propose two classes of universal codes that are suited to two asymptotic regimes when the output alphabet is possibly continuous the first class has the property that the error probability decays exponentially fast and we identify an explicit lower bound on the error exponent the other class attains the epsilon capacity of the channel and we also identify the second order term in the asymptotic expansion the proposed encoder is essentially based on the packing lemma of the method of types for the decoder we first derive a renyi relative entropy version of clarke and barron s formula the distance between the true distribution and the bayesian mixture which is of independent interest the universal decoder is stated in terms of this formula and quantities used in the information spectrum method the methods contained herein allow us to analyze universal codes for channels with continuous and discrete output alphabets in a unified manner and to analyze their performances in terms of the exponential decay of the error probability and the second order coding rate

information geometry approach to parameter estimation in markov chains we consider the parameter estimation of markov chain when the unknown transition matrix belongs to an exponential family of transition matrices then we show that the sample mean of the generator of the exponential family is an asymptotically efficient estimator further we also define a curved exponential family of transition matrices using a transition matrix version of the pythagorean theorem we give an asymptotically efficient estimator for a curved exponential family primary 62m05

recurrence formulas for permanents of 0 1 circulants abstract a general method is developed for constructing linear recurrence formulas expressing the permanents of 0 1 circulants in terms of permanents of 0 1 circulants of the same type of lower orders the method extends to all 0 1 circulants a result of n metropolis m l stein and p r stein on permanents of 0 1 circulants of a special type

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

on the converse theorem in statistical hypothesis testing for markov chains hypothesis testing for two markov chains is considered under the constraint that the error probability of the first kind is less than or equal to exp rn the error probability of the second kind is minimized the geodesic that connects the two markov chains is defined by analyzing the geodesic the power exponents are calculated and then represented in terms of kullback leibler divergence

on the converse theorem in statistical hypothesis testing simple statistical hypothesis testing is investigated by making use of the divergence geometric method the asymptotic behavior of the minimum value of the error probability of the second kind under the constraint that the error probability of the first kind is bounded above by exp rn is looked for where r is a given positive number if r is greater than the divergence of the two probability measures the so called converse theorem holds it is shown that the condition under which the converse theorem holds can be divided into two separate cases by analyzing the geodesic connecting the two probability measures and as a result an explanation is given for the han kobayashi linear function f sub t x

efficiently computing the permanent and hafnian of some banded toeplitz matrices we present a new efficient method for computing the permanent and hafnian of certain banded toeplitz matrices the method covers non trivial cases for which previous known methods do not apply the main idea is to use the elements of the first row and column which determine the entire toeplitz matrix to construct a digraph in which certain paths correspond to permutations that the permanent and hafnian count since counting paths can be done efficiently the permanent and hafnian for those matrices is easily obtainable

non integrability of measure preserving maps via lie symmetries we consider the problem of characterizing for certain natural number m the local c m non integrability near elliptic fixed points of smooth planar measure preserving maps our criterion relates this non integrability with the existence of some lie symmetries associated to the maps together with the study of the finiteness of its periodic points one of the steps in the proof uses the regularity of the period function on the whole period annulus for non degenerate centers question that we believe that is interesting by itself the obtained criterion can be applied to prove the local non integrability of the cohen map and of several rational maps coming from second order difference equations

nc algorithms for computing the number of perfect matchings in k 3 3 free graph and related problems abstract we show that the problem of computing the number of perfect matchings in k3 3 free graphs is in nc this contrasts with the p completeness of counting the number of perfect matchings in arbitrary graphs as corollaries we obtain nc algorithms for checking if a given k3 3 free graph has a perfect matching and if it has an exact matching our result also opens up the possibility of obtaining an nc algorithm for finding a perfect matching in k3 3 free graphs

lattice systems with a continuous symmetry iii low temperature asymptotic expansion for the plane rotator model we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

computation of sparse circulant permanents via determinants we consider the problem of computing the permanent of circulant matrices we apply some recent results on the computation of the number of perfect matchings of small genus graphs in order to show that the permanent of a circulant matrix with three nonzero entries per row is the linear combination of just four determinants of circulant matrices with the same structure as the original matrix we also show that the same result holds true for a class of circulant matrices with four nonzero entries per row related to the dimer problem with periodic boundary conditions conversely we give hints at the fact that more general circulants do not share similar properties and thus should be dealt with by means of radically different approaches

validity of the spin wave approximation for the free energy of the heisenberg ferromagnet we consider the quantum ferromagnetic heisenberg model in three dimensions for all spins s 1 2 we rigorously prove the validity of the spin wave approximation for the excitation spectrum at the level of the first non trivial contribution to the free energy at low temperatures our proof comes with explicit constructive upper and lower bounds on the error term it uses in an essential way the bosonic formulation of the model in terms of the holstein primakoff representation in this language the model describes interacting bosons with a hard core on site repulsion and a nearest neighbor attraction this attractive interaction makes the lower bound on the free energy particularly tricky the key idea there is to prove a differential inequality for the two particle density which is thereby shown to be smaller than the probability density of a suitably weighted two particle random process on the lattice

error exponents for hypothesis testing of the general source in this correspondence we consider the simple hypothesis testing problems for general sources in the sense of han and verdu recently han established a compact formula for the supremum of achievable exponents for the second kind of error probability under the asymptotic constraint of the form mun e nr n rarr infin on the first kind of error probability mun where r is a given positive number we investigate the same hypothesis testing problems studied by han the aim of the correspondence is to give a new expression for the supremum of achievable error exponents our formula is expressed in terms of the divergences and given in quite different forms from han s expression

two approaches to obtain the strong converse exponent of quantum hypothesis testing for general sequences of quantum states we present two general approaches to obtain the strong converse exponent of simple quantum hypothesis testing for correlated quantum states one approach requires that the states satisfy a certain factorization property typical examples of such states are the temperature states of translation invariant finite range interactions on a spin chain the other approach requires the differentiability of a regularized renyi alpha divergence in the parameter alpha typical examples of such states include temperature states of non interacting fermionic lattice systems and classical irreducible markov chains in all cases we get that the strong converse exponent is equal to the hoeffding anti divergence which in turn is obtained from the regularized renyi divergences of the two states

euclidean invariance in statistical mechanics of classical continuous system it is shown that equilibrium states of classical particles with short range interactions are euclidean invariant whenever their correlation functions have a clustering which is integrable the relation between invariance and clustering is analysed for spatial rotations and internal rotational degrees of freedom the analysis is then extended to the case of long range interactions including the coulomb force and jellium systems

conditional limit theorems under markov conditioning let x 1 x 2 cdots be independent identically distributed random variables taking values in a finite set x and consider the conditional joint distribution of the first m elements of the sample x 1 cdots x n on the condition that x 1 x 1 and the sliding block sample average of a function h cdot cdot defined on x 2 exceeds a threshold alpha eh x 1 x 2 for m fixed and n rightarrow infty this conditional joint distribution is shown to converge m the m step joint distribution of a markov chain started in x 1 which is closest to x l x 2 cdots in kullback leibler information divergence among all markov chains whose two dimensional stationary distribution p cdot cdot satisfies sum p x y h x y geq alpha provided some distribution p on x 2 having equal marginals does satisfy this constraint with strict inequality similar conditional limit theorems are obtained when x 1 x 2 cdots is an arbitrary finite order markov chain and more general conditioning is allowed

lattice systems with a continuous symmetry we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

the strong converse theorem for hypothesis testing the authors present the answer to a so called converse problem by giving the explicit form of the power exponent this strong converse should be regarded as completing a weak converse previously obtained by r e blahut 1974

on the values of permanents of 0 1 circulant matrices with three ones per row abstract starting from previous works concerning permanents of 0 1 circulant matrices with three nonzero entries per row and just using algebraic methods we find some congruences satisfied by the values of such permanents along with simple generalizations to circulants with k ones per row k n k 3 we also give an alternative proof for known relations of fibonacci numbers with some of these values of permanents

on the parity of permanents of circulant matrices abstract starting from previous results concerning determinants and permanents of 0 1 circulant matrices and using theory on finite fields with characteristic 2 we first see denoting by a the n n 0 1 circulant matrix whose 1 s in the first row are exactly in positions i 1 i 2 i k that per a is even iff gcd x i 1 x i 2 x i k 1 x n 1 in z 2 x we then derive some consequences for specific classes of primes n in particular when n is a prime such that the group z n is generated by the residue class 2 we see that for k n and k odd the value per a is always odd similarly for n 7 and n prime with the group of squares in z n generated by the class 2 we see that for k 3 the value per a is always odd contrary to such cases we find that when n is a mersenne prime greater than 3 there are a significant number of circulants a of the above form with k odd and per a even

hypothesis testing with the general source the asymptotically optimal hypothesis testing problem with general sources as the null and alternative hypotheses is studied under exponential type error constraints on the first kind of error probability our fundamental philosophy is to convert all of the hypothesis testing problems to the pertinent computation problems in the large deviation probability theory this methodologically new approach enables us to establish compact general formulas of the optimal exponents of the second kind of error and correct testing probabilities for the general sources including all nonstationary and or nonergodic sources with arbitrary abstract alphabet countable or uncountable these general formulas are presented from the information spectrum point of view

an efficient tree decomposition method for permanents and mixed discriminants abstract we present an efficient algorithm to compute permanents mixed discriminants and hyperdeterminants of structured matrices and multidimensional arrays tensors we describe the sparsity structure of an array in terms of a graph and we assume that its treewidth denoted as is small our algorithm requires o n 2 arithmetic operations to compute permanents and o n 2 n 3 for mixed discriminants and hyperdeterminants we finally show that mixed volume computation continues to be hard under bounded treewidth assumptions

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

decay of correlations and uniqueness of gibbs lattice systems with nonquadratic interaction the aim of this paper is to develop the classical lattice models with unbounded spin to the case of nonquadratic polynomial interaction we demonstrate that the distinct relation between the growths of potentials leads to the uniqueness and the fast decay of correlations for gibbs measure

more inequalities for critical exponents a variety of rigorous inequalities for critical exponents is proved most notable is the low temperature josephson inequalitydv 2 2 others are 1 1 v 1 1 1 d 1 1 for d dv 3 for d 4 and 2m 2m 2 form 2 the hypotheses vary all inequalities are true for the spin 1 2 ising model with nearest neighbor ferromagnetic pair interactions