branched polymers and dimensional reduction we establish an exact relation between self avoiding branched polymers in d 2 continuum dimensions and the hard core continuum gas at negative activity in d dimensions we review conjectures and results on critical exponents for d 2 2 3 4 and show that they are corollaries of our result we explain the connection first proposed by parisi and sourlas between branched polymers in d 2 dimensions and the yang lee edge singularity in d dimensions

conformal invariance and critical dynamics the author discusses the behaviour of dynamic correlation functions under conformal transformations in particular the exact form of the dynamic scaling function at the critical point for an arbitrary isotropic two dimensional model is obtained with either conserved or nonconserved order parameter

renormalized field theory of collapsing directed randomly branched polymers we present a dynamical field theory for directed randomly branched polymers and in particular their collapse transition we develop a phenomenological model in the form of a stochastic response functional that allows us to address several interesting problems such as the scaling behavior of the swollen phase and the collapse transition for the swollen phase we find that by choosing model parameters appropriately our stochastic functional reduces to the one describing the relaxation dynamics near the yang lee singularity edge this corroborates that the scaling behavior of swollen branched polymers is governed by the yang lee universality class as has been known for a long time the main focus of our paper lies on the collapse transition of directed branched polymers we show to arbitrary order in renormalized perturbation theory with ensuremath epsilon expansion that this transition belongs to the same universality class as directed percolation

exact results for 2d directed animals on a strip of finite width proves a conjecture giving the exact number of directed animals of s sites with any root on a strip of finite width of a square lattice the authors also rederive more simply some previous results concerning the connective constant and particular eigenvectors of the transfer matrix

distribution of transverse distances in directed animals we relate x s the average number of sites at a transverse distance x in the directed animals with s sites in d transverse dimensions to the two point correlation function of a lattice gas with nearest neighbour exclusion in d dimensions for large s x s has the scaling form s rds f x rs where rs is the root mean square radius of gyration of animals of s sites we determine the exact scaling function for d 1 to be f r 2 3 erfc r 3 we also show that x 0 s can be determined in terms of the animal number generating function of the directed animals

ϵ expansion for directed animals introduces the problem of directed lattice animals and shows that they are highly anisotropic with two different correlation lengths xi sub and xi perpendicular to parallel and perpendicular to the privileged direction a field theory is introduced and the critical exponents calculated to first order in epsilon dc d where the upper critical dimension dc is 7

yang lee edge singularities from extended activity expansions of the dimer density for bipartite lattices of dimensionality 2 d 7 we have extended in most cases through 24th order the series expansions of the dimer density in powers of the activity in the case of bipartite hyper simple cubic and hyper body centered cubic lattices of dimensionalities 2 d 7 a numerical analysis of these data yields estimates of the exponents characterizing the yang lee edge singularities for lattice ferromagnetic spin models as d varies between the lower and the upper critical dimensionalities our results are consistent with but more extensive and sometimes more accurate than those obtained from the existing dimer series or from the estimates of related exponents for lattice animals branched polymers and fluids we mention also that it is possible to obtain estimates of the dimer constants from our series for the various lattices

dimensional reduction for directed branched polymers dimensional reduction occurs when the critical behaviour of one system can be related to that of another system in a lower dimension we show that this occurs for directed branched polymers dbp by giving an exact relationship between dbp models in d 1 dimensions and repulsive gases at negative activity in d dimensions this implies relations between exponents of the two models d 1 d the exponent describing the singularity of the pressure and d 1 d the correlation length exponent of the repulsive gas it also leads to the relation d 1 1 d where d is the yang lee edge exponent we derive exact expressions for the number of dbp of size n in two dimensions

critical exponents for the percolation problem and the yang lee edge singularity gives details of a calculation of critical exponents for a class of field theory models that have an interaction cubic in the fields the results have already been reported give the exponents to third order in epsilon where epsilon 6 d and d is the dimensionality of space the class of models includes the percolation problem and the yang lee edge singularity so the authors give explicit results for the exponents to order epsilon 3 in these cases by resummation methods based on the symptotic behaviour of the epsilon expansion they obtain numerical estimates for these exponents for a number of interesting values of d

renormalisation group study of fully directed self avoiding walks the author presents a discussion of the critical properties of fully directed self avoiding walks on hypercubic lattices a position space renormalisation group transformation is carried out analytically in two dimensions for any value of the scaling factor b the finite size scaling properties of the transformation are explicitly exhibited including the well known finite size scaling hypothesis v b v approximately 1n b 1 it is shown that the rg transformation in this case is sensitive to v sub thus giving support to existing arguments

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

identity of the universal repulsive core singularity with yang lee edge criticality lattice and continuum fluid models with repulsive core interactions typically display a dominant critical type singularity on the real negative activity axis lai and fisher recently suggested mainly on numerical grounds that this repulsive core singularity is universal and in the same class as the yang lee edge singularities which arise above criticality at complex activities with positive real part a general analytic demonstration of this identification is presented here using a field theory approach with separate representations of the repulsive and attractive parts of the pair interactions

main effects and interactions in mixed and incomplete data frames a mixed data frame mdf is a table collecting categorical numerical and count observations the use of mdf is widespread in statistics and the applications are numerous from abundance data in ec

hard squares with negative activity we show that the hard square lattice gas with activity z 1 has a number of remarkable properties we conjecture that all the eigenvalues of the transfer matrix are roots of unity they fall into groups strings evenly spaced around the unit circle which have interesting number theoretic properties for example the partition function on an m n lattice with periodic boundary condition is identically 1 when m and n are coprime we provide evidence for these conjectures from analytical and numerical arguments

disproving the single level conjecture we consider the size of monotone circuits for quadratic boolean functions that is disjunctions of length 2 monomials our motivation is that a good linear in the number of variables lower bound on the monotone circuit size for a certain type of quadratic function would imply a good even exponential lower bound on the general nonmonotone circuit size to get more insight into the structure of monotone circuits for quadratic functions we consider the so called single level conjecture posed explicitly in the early 1990s the conjecture claims that monotone single level circuits that is circuits which have only one level of and gates for quadratic functions are not much larger than arbitrary monotone circuits in this paper we disprove the conjecture as follows there exist quadratic functions whose monotone circuits have linear size but whose monotone single level circuits require almost quadratic size

constructing o n log n size monotone formulae for the k th threshold function of n boolean variables in this paper we construct formulae for the kth elementary symmetric polynomial of n boolean variables using only conjunction and disjunction which for fixed k are of size o n log n with the construction taking time polynomial in n we also prove theorems involving n log n cdot polynomial in k upper bounds on such formulae our methods involve solving the following combinatorial problem for fixed k and any n construct a collection of r o log n functions f 1 cdots f r from 1 cdots n to 1 cdots k such that any subset of 1 cdots n of order k is mapped 1 1 to 1 cdots k by at least one f i

enumeration of directed site animals on two dimensional lattices studies the problem of directed site animals on the square triangular and hexagonal lattices closed form expressions are proposed for a s the number of animals of size s on the square and triangular lattices these expressions have been checked for s or 33 and s or 10 for the square and triangular lattices respectively by explicit enumeration they imply that a s varies as lambda 2 s 0 for large s where lambda 3 for the square lattice and lambda 4 for the triangular and theta 1 2 for both for the hexagonal lattice a s is found for s or 48 the results are consistent with lambda 2 0252 or 0 0005 and theta 1 2

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

some exact results for moments of 2d directed animals using computer enumerations and the algebraic approximant method of series analysis several new exact results have been found for moments of width perimeter and loops for directed animals on the square and triangular lattices a proof using q series is given for exact solutions for widths on both the square and triangular lattices

series expansions from the corner transfer matrix renormalization group method ii asymmetry and high density hard squares the corner transfer matrix renormalization group method is a method claimed to be subexponential for numerically calculating physical quantities of statistical mechanical models in a previous paper we extended this method to generate series expansions here we show how to extend both the original and series methods to deal with asymmetry in any model we discuss the cases of rotational translational and reflection asymmetry and give some improvements to the method this is demonstrated by an application of the method to generate series for the hard square model in the high density regime producing 51 terms of the partition function and 48 terms of the order parameter these series are analysed producing estimates of the critical point and exponents and showing the likely presence of a confluent singularity with exponent 17 8 in the order parameter

yang lee edge singularity in the one dimensional long range ising model the finite range scaling method has been used to study the yang lee edge singularity problem for the one dimensional ferromagnetic ising model with long range interactions decaying with distance r as r d sigma the phase diagram and the critical exponent nu have been calculated for different values of sigma by using finite ranges l or 9 the critical value sigma c dividing the long range from the short range critical regime has been discussed

flory theory for directed lattice animals and directed percolation the free energies of directed lattice animals in good and theta solvents and the free energy of directed percolation are found by the use of a simple flory type approximation which accounts for the inherent anisotropy of these systems from these free energies the authors obtain the upper critical dimension below which mean field theory breaks down they also calculate closed form dimension dependent expressions for the parallel and perpendicular correlation length exponents which characterise the asymptotic cluster shapes these exponents are in excellent agreement with existing numerical data

robust matrix decomposition with sparse corruptions suppose a given observation matrix can be decomposed as the sum of a low rank matrix and a sparse matrix and the goal is to recover these individual components from the observed sum such additive decompositions have applications in a variety of numerical problems including system identification latent variable graphical modeling and principal components analysis we study conditions under which recovering such a decomposition is possible via a combination of l1 norm and trace norm minimization we are specifically interested in the question of how many sparse corruptions are allowed so that convex programming can still achieve accurate recovery and we obtain stronger recovery guarantees than previous studies moreover we do not assume that the spatial pattern of corruptions is random which stands in contrast to related analyses under such assumptions via matrix completion

rooted spiral trees in dimensions 2 3 and 4 the authors have studied rooted spiral trees in dimensions 2 3 and 4 by exact enumeration and by monte carlo methods numerical estimates of critical exponents thus obtained support the hypothesis that a dimensional reduction by four occurs in this problem a possible mechanism of the origin of this dimensional reduction is proposed

directed lattice animals and the lee yang edge singularity the exponents nu perpendicular to gamma of directed animals in d dimensions are shown to equal those for the lee yang problem in d 1 dimensions for d 2 the author obtains the exact results nu perpendicular to theta 1 2 and for d or 7 the scaling relation theta d 1 nu perpendicular to

restricted strong convexity and weighted matrix completion optimal bounds with noise we consider the matrix completion problem under a form of row column weighted entrywise sampling including the case of uniformentrywise sampling as a special case we analyze the associated random observation operator and prove that with high probability it satisfies a form of restricted strong convexity with respect to weighted frobenius norm using this property we obtain as corollaries a number of error bounds on matrix completion in the weighted frobenius norm under noisy sampling and for both exact and near low rank matrices our results are based on measures of the spikiness and low rankness of matrices that are less restrictive than the incoherence conditions imposed in previous work our technique involves an m estimator that includes controls on both the rank and spikiness of the solution and we establish non asymptotic error bounds in weighted frobenius norm for recovering matrices lying with lq balls of bounded spikiness using information theoretic methods we show that no algorithm can achieve better estimates up to a logarithmic factor over these same sets showing that our conditions on matrices and associated rates are essentially optimal

integrability versus non integrability hard hexagons and hard squares compared in this paper we compare the integrable hard hexagon model with the nonintegrable hard squares model by means of partition function roots and transfer matrix eigenvalues we consider partition functions for toroidal cylindrical and free free boundary conditions up to sizes 40 40 and transfer matrices up to 30 sites for all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis the density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest and equal moduli and exhibits much greater structure than the corresponding density of hard hexagons we also study the special point z 1 of hard squares where all eigenvalues have unit modulus and we give several conjectures for the value at z 1 of the partition functions

robust low rank matrix completion by riemannian optimization low rank matrix completion is the problem where one tries to recover a low rank matrix from noisy observations of a subset of its entries in this paper we propose rmc a new method to deal with the problem of robust low rank matrix completion i e matrix completion where a fraction of the observed entries are corrupted by non gaussian noise typically outliers the method relies on the idea of smoothing the ell 1 norm and using riemannian optimization to deal with the low rank constraint we first state the algorithm as the successive minimization of smooth approximations of the ell 1 norm and we analyze its convergence by showing the strict decrease of the objective function we then perform numerical experiments on synthetic data and demonstrate the effectiveness on the proposed method on the netflix dataset

robust matrix completion this paper considers the problem of estimation of a low rank matrix when most of its entries are not observed and some of the observed entries are corrupted the observations are noisy realizations of a sum of a low rank matrix which we wish to estimate and a second matrix having a complementary sparse structure such as elementwise sparsity or columnwise sparsity we analyze a class of estimators obtained as solutions of a constrained convex optimization problem combining the nuclear norm penalty and a convex relaxation penalty for the sparse constraint our assumptions allow for simultaneous presence of random and deterministic patterns in the sampling scheme we establish rates of convergence for the low rank component from partial and corrupted observations in the presence of noise and we show that these rates are minimax optimal up to logarithmic factors

robust pca via outlier pursuit singular value decomposition svd and principal component analysis pca is one of the most widely used techniques for dimensionality reduction successful and efficiently computable it is nevertheless plagued by a well known well documented sensitivity to outliers recent work has considered the setting where each point has a few arbitrarily corrupted components yet in applications of svd or pca such as robust collaborative filtering or bioinformatics malicious agents defective genes or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted we present an efficient convex optimization based algorithm that we call outlier pursuit which under some mild assumptions on the uncorrupted points satisfied e g by the standard generative assumption in pca problems recovers the exact optimal low dimensional subspace and identifies the corrupted points such identification of corrupted points that do not conform to the low dimensional approximation is of paramount interest in bioinformatics financial applications and beyond our techniques involve matrix decomposition using nuclear norm minimization however our results setup and approach necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition since we develop an approach to recover the correct column space of the uncorrupted matrix rather than the exact matrix itself in any problem where one seeks to recover a structure rather than the exact initial matrices techniques developed thus far relying on certificates of optimality will fail we present an important extension of these methods which allows the treatment of such problems

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

exact results for the generating function of directed column convex animals on the square lattice the mathematical properties of the generating function s x for the number sn of directed column convex lattice animals on a square lattice with a given directed site perimeter n are investigated in particular it is shown that s x can be expressed exactly in terms of algebraic hypergeometric functions a detailed investigation of the asymptotic behaviour of sn as n to infinity is carried out by applying the darboux method to the hypergeometric formula for s x it is also demonstrated that sn satisfies a four term recurrence relation finally it is noted that the techniques used to analyse the lattice animal generating function s x can be applied to any other generating function which satisfies a cubic algebraic equation

the conjunctive complexity of quadratic boolean functions abstract the minimal number of conjuctions in monotone circuits for quadratic boolean functions i e disjunctions of quadratic monomials x i x j is investigated single level circuits which have only one level of conjuctions are compared with arbitrary monotone circuits the computation of the single level complexity is shown to be np complete for almost all quadratic functions almost optimal circuits can be computed in polynomial time the single level conjecture is disproved i e a quadratic function is defined whose single level complexity is larger than its conjuctive complexity

on directed interacting animals and directed percolation we study the phase diagram of fully directed lattice animals with nearest neighbour interactions on the square lattice this model comprises several interesting ensembles directed site and bond trees bond animals strongly embeddable animals as special cases and its collapse transition is equivalent to a directed bond percolation threshold precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n 17 the crossover region in the vicinity of the collapse transition is analysed in detail and the crossover exponent is determined directly from the singular part of the free energy we show using scaling arguments and an exact relation due to dhar that is equal to the fisher exponent governing the size distribution of large directed percolation clusters

low rank matrix recovery from errors and erasures this paper considers the recovery of a low rank matrix from an observed version that simultaneously contains both 1 erasures most entries are not observed and 2 errors values at a constant fraction of unknown locations are arbitrarily corrupted we provide a new unified performance guarantee on when minimizing nuclear norm plus l1 norm succeeds in exact recovery our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns by specializing this one single result in different ways we recover up to poly log factors as corollaries all the existing results in exact matrix completion and exact sparse and low rank matrix decomposition our unified result also provides the first guarantees for 1 recovery when we observe a vanishing fraction of entries of a corrupted matrix and 2 deterministic matrix completion

the growth of ramified clusters by particle evaporation and condensation a reversible cluster growth model in which particles sites at singly bonded tip positions can evaporate with equal probability and condense on another or the same cluster has been investigated using computer simulations both two dimensional square lattice and three dimensional cubic lattice models have been investigated in which particles can follow random walk or ballistic trajectories or can move to any vacant site in the system dw 2 1 or 0 where dw is the dimensionality of the particle trajectory at low densities the clusters are fractal with fractal dimensionalities that are equal or almost equal to those associated with lattice animals irrespective of the value of dw the exponents z and z that describe the growth of the mean cluster size s t and the decay of the number of clusters respectively have values that increases slightly as dw decreases in the high density limit the clusters are compact with rough surfaces in both the 0 and 1 limits the cluster size distribution ns t is broad and can be described in terms of the scaling form ns t s 2f ss t the dependence of the cluster radii of gyration rg on their sizes s can be described by the simple scaling form rg s1 dg pd d d s where d is the dimensionality of the lattice and d is the fractal dimensionality of the clusters

minimax lower bounds for noisy matrix completion under sparse factor models this paper examines fundamental error characteristics for a general class of matrix completion problems where the matrix of interest is a product of two a priori unknown matrices one of which is sparse and the observations are noisy our main contributions come in the form of minimax lower bounds for the expected per element squared error for this problem under several common noise models specifically we analyze scenarios where the corruptions are characterized by additive gaussian noise or additive heavier tailed laplace noise poisson distributed observations and highly quantized e g one bit observations as instances of our general result our results establish that the error bounds derived in soni et al 2016 for complexity regularized maximum likelihood estimators achieve up to multiplicative constants and logarithmic factors the minimax error rates in each of these noise scenarios provided that the nominal number of observations is large enough and the sparse factor has on an average at least one non zero per column

enumeration of directed animals on an infinite family of lattices we prove algebraic equations satisfied by the area generating function for directed animals on an infinite family of regular non planar two dimensional graphs

critical and multicritical semi random 1 d dimensional lattices and hard objects in d dimensions we investigate models of 1 d d lorentzian semi random lattices with one random space like direction and d regular time like ones we prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions this allows for an exact solution of a variety of new models including critical and multicritical generalized 1 1 d lorentzian surfaces with fractal dimensions df k 1 k 1 2 3 as well as a new model of 1 2 d critical tetrahedral complexes with fractal dimension df 12 5 critical exponents and universal scaling functions follow from this solution we finally establish a general connection between 1 d d lorentzian lattices and directed site lattice animals in 1 d dimensions

cellular automata and statistical mechanical models we elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata transient and stationary properties of probabilistic automata are linked to surface and bulk properties respectively of restricted statistical mechanical systems it is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata some exact results are derived including duality transformations exact mappings disorder and linear solutions many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties this approach is considered to be a first step toward the design of fully parallel probabilistic systems whose computational abilities rely on the cooperative behavior of their components

on the complexity of slice functions abstract by a result of berkowitz 1982 the monotone circuit complexity of slice functions cannot be much larger than the circuit combinational complexity of these functions for arbitrary complete bases this result strengthens the importance of the theory of monotone circuits we show in this paper that monotone circuits for slice functions can be understood as special circuits called set circuits here disjunction and conjunction are replaced by set union and set intersection all the main methods known for proving lower bounds on the monotone complexity of boolean functions fail to work in their present form for slice functions furthermore we show that the canonical slice functions of the boolean convolution the nechiporuk boolean sums and the clique function can be computed with a linear number of gates

structural properties of potts model partition functions and chromatic polynomials for lattice strips abstract the q state potts model partition function equivalent to the tutte polynomial for a lattice strip of fixed width l y and arbitrary length l x has the form z g q v j 1 n z g c z g j z g j l x where v is a temperature dependent variable the special case of the zero temperature antiferromagnet v 1 is the chromatic polynomial p g q using coloring and transfer matrix methods we give general formulas for c x g j 1 n x g c x g j for x z p on cyclic and mobius strip graphs of the square and triangular lattice combining these with a general expression for the unique coefficient c z g j of degree d in q c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we determine the number of z g j s with coefficient c d in z g q v for these cyclic strips of width l y to be n z l y d 2d 1 l y d 1 1 2l y l y d for 0 d l y and zero otherwise for both cyclic and mobius strips of these lattices the total number of distinct eigenvalues z g j is calculated to be n z l y 2l y l y results are also presented for the analogous numbers n p l y d and n p l y for p g q we find that n p l y 0 n p l y 1 1 m l y 1 motzkin number n z l y 0 c l y the catalan number and give an exact expression for n p l y our results for n z l y and n p l y apply for both the cyclic and mobius strips of both the square and triangular lattices we also point out the interesting relations n z l y 2 n da tri l y and n p l y 2 n da sq l y where n da n denotes the number of directed lattice animals on the lattice we find the asymptotic growths n z l y l y 1 2 4 l y and n p l y l y 1 2 3 l y as l y some general geometric identities for potts model partition functions are also presented

size and shape of directed lattice animals uses position space renormalisation and exact enumeration methods to study configurational properties of directed lattice animals clusters of directed bonds in which the tail of a new bond must be added to the head of existing bonds exponents are found for the divergence of correlation lengths parallel and perpendicular to the preferred axis in addition an exponent is found that characterises the singularity of the directed animal generating function and critical points are located

the hard hexagon partition function for complex fugacity we study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems we find that the partition function per site computed by baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane we also obtain a new algebraic equation for the low density partition function per site