demixing can occur in binary hard sphere mixtures with negative nonadditivity a binary fluid mixture of nonadditive hard spheres characterized by a size ratio gamma sigma 2 sigma 1 infinity a demixing transition with a critical consolute point at a packing fraction scaling as eta approximately d2 d is found even for slightly negative nonadditivity if delta 1 8 ln gamma 2 arguments concerning the stability of the demixing with respect to freezing are provided

a note on almost regular matrices abstract we generalize and prove hofmeister s modification of a conjecture of runge characterizing almost regular graphs and discuss related matters

error scaling laws for linear optimal estimation from relative measurements in this paper we study the problem of estimating vector valued variables from noisy ldquorelativerdquo measurements which arises in sensor network applications the problem can be posed in terms of a graph whose nodes correspond to variables and edges to noisy measurements of the difference between two variables the optimal minimum variance linear unbiased estimate of the node variables with an arbitrary variable as the reference is considered this paper investigates how the variance of the estimation error of a node variable grows with the distance of the node to the reference node a classification of graphs namely dense or sparse in rd 1 les d les 3 is established that determines this growth rate in particular if a graph is dense in 1 d 2 d or 3 d a node variable s estimation error is upper bounded by a linear logarithmic or bounded function of distance from the reference corresponding lower bounds are obtained if the graph is sparse in 1 d 2 d and 3 d these results show that naive measures of graph density such as node degree are inadequate predictors of the estimation error being true for the optimal linear unbiased estimate these scaling laws determine algorithm independent limits on the estimation accuracy achievable in large graphs

hitting and commute times in large random neighborhood graphs in machine learning a popular tool to analyze the structure of graphs is the hitting time and the commute distance resistance distance for two vertices u and v the hitting time huv is the expected time it takes a random walk to travel from u to v the commute distance is its symmetrized version cuv huv hvu in our paper we study the behavior of hitting times and commute distances when the number n of vertices in the graph tends to infinity we focus on random geometric graphs e graphs knn graphs and gaussian similarity graphs but our results also extend to graphs with a given expected degree distribution or erdos renyi graphs with planted partitions we prove that in these graph families the suitably rescaled hitting time huv converges to 1 dv and the rescaled commute time to 1 du 1 dv where du and dv denote the degrees of vertices u and v in these cases hitting and commute times do not provide information about the structure of the graph and their use is discouraged in many machine learning applications

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

the smallest possible interaction radius for synchronization of self propelled particles a central and fundamental issue in the theory of complex systems is to understand how local rules lead to collective behavior of the whole system this paper will investigate a typical collective behavior synchronization of a self propelled particle system modeled by the nearest neighbor rules while connectivity of the dynamic neighbor graphs associated with the underlying systems is crucial for synchronization it is widely known that the verification of such dynamical connectivity is at the core of theoretical analysis ideally conditions used for synchronization should be imposed on the model parameters and the initial states of the particles one crucial model parameter is the interaction radius and we are interested in the following natural and basic question what is the smallest interaction radius for synchronization in this paper we will show that in a certain sense the smallest possible interaction radius approximately equals sqrt log n pi n with n being the population size w

covering problems for brownian motion on spheres bounds are given on the mean time taken by a strong markov process to visit all of a finite collection of subsets of its state space these bounds are specialized to brownian motion on the surface of the unit sphere sigma p in r p this leads to bounds on the mean time taken by this brownian motion to come within a distance varepsilon of every point on the sphere and bounds on the mean time taken to come within varepsilon of every point or its opposite the second case is related to the grand tour a technique of multivariate data analysis that involves a search of low dimensional projections in both cases the bounds are asymptotically tight as varepsilon rightarrow 0 on sigma p for p geq 4

monte carlo simulation of hard hyperspheres in six seven and eight dimensions for low to moderate densities the pair correlation function of hard hyperspheres in six seven and eight dimensions is obtained from monte carlo simulations the value of the pair correlation function at contact is compared with the results from molecular dynamics calculations and a variety of theoretical approaches remarkably good agreement is found with the simple closed form equations of y song e a mason and r m stratt j phys chem 93 6916 6919 1989 the monte carlo results for the equation of state are compared with the theoretical expressions of m baus and j l colot phys rev a 36 3912 1987 m luban and j p j michels phys rev a 41 6796 1990 and high order virial expansions in addition in seven dimensions comparisons are made with the exact py solution provided by m robles m l de haro and a santos j chem phys 120 9113 2004 very good agreement was observed between theory and computer simulation in all dimensions

local density fluctuations hyperuniformity and order metrics questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges the purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d our specific objectives are to study the variance in the number of points contained within a regularly shaped window ensuremath omega of arbitrary size and to further illuminate our understanding of hyperuniform systems i e point patterns that do not possess infinite wavelength fluctuations for large windows hyperuniform systems are characterized by a local variance that grows only as the surface area rather than the volume of the window we derive two formulations for the number variance i an ensemble average formulation which is valid for statistically homogeneous systems and ii a volume average formulation applicable to a single realization of a general point pattern in the large system limit the ensemble average formulation which includes both real space and fourier representations enables us to show that a homogeneous point pattern in a hyperuniform state is at a critical point of a type with appropriate scaling laws and critical exponents but one in which the direct correlation function rather than the pair correlation function is long ranged we also prove that the non negativity of the local number variance does not add a new realizability condition on the pair correlation the volume average formulation is superior for certain computational purposes including optimization studies in which it is desired to find the particular point pattern with an extremal or targeted value of the variance we prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one dimensional hyperuniform patterns we also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one two and three dimensions for spherical windows enabling us to rank order the spatial patterns our results suggest that the local variance may serve as a useful order metric for general point patterns contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension we show that for d 3 the body centered cubic lattice has a smaller variance than the face centered cubic lattice finally for certain hyperuniform disordered point patterns we evaluate the direct correlation function structure factor and associated critical exponents exactly

bounds for the spectral radius of nonnegative matrices abstract we give sharp upper and lower bounds for the spectral radius of a nonnegative matrix with all row sums positive using its average 2 row sums and characterize the equality cases if the matrix is irreducible we compare these bounds with the known bounds using the row sums by examples we also apply these bounds to various matrices associated with a graph including the adjacency matrix the signless laplacian matrix and some distance based matrices some known results are generalized and improved

exact solution of the percus yevick equation for a hard core fluid in odd dimensions it is shown that the percus yevick integral equation for the pair distribution function of a fluid interacting with a hard core potential can be solved not only in one and three dimensions where the solution is well known but more generally in all odd dimensions the nonlinear integral equation is reduced to an algebraic equation of order d 3 for odd dimensions d greater than three as an example the direct correlation function in five dimensions is derived explicitly

on the cover time and mixing time of random geometric graphs the cover time and mixing time of graphs has much relevance to algorithmic applications and has been extensively investigated recently with the advent of ad hoc and sensor networks an interesting class of random graphs namely random geometric graphs has gained new relevance and its properties have been the subject of much study a random geometric graph g n r is obtained by placing n points uniformly at random on the unit square and connecting two points iff their euclidean distance is at most r the phase transition behavior with respect to the radius r of such graphs has been of special interest we show that there exists a critical radius r o p t such that for any r r o p tg n r has optimal cover time of q nlogn with high probability and importantly r o p email protected r c o n where r c o n denotes the critical radius guaranteeing asymptotic connectivity moreover since a disconnected graph has infinite cover time there is a phase transition and the corresponding threshold width is o r c o n on the other hand the radius required for rapid mixing r r a p i email protected r c o n and in particular r r a p i email protected 1 poly logn we are able to draw our results by giving a tight bound on the electrical resistance and conductance of g n r via certain constructed flows

convergence rates for markov chains this is an expository paper that presents various ideas related to nonasymptotic rates of convergence for markov chains such rates are of great importance for stochastic algorithms that are widely used in statistics and in computer science they also have applications to analysis of card shuffling and other areas in this paper we attempt to describe various mathematical techniques that have been used to bound such rates of convergence in particular we describe eigenvalue analysis random walks on groups coupling and minorization conditions connections are made to modern areas of research wherever possible elements of linear algebra probability theory group theory and measure theory are used but efforts are made to keep the presentation elementary and accessible

radial distribution function for sticky hard core fluids following heuristic arguments analytic expressions for the radial distribution functiong r of one and three dimensional sticky hard core fluids i e square well fluids in a scaled limit of infinite depth and vanishing width are proposed the expressions are derived in terms of the simplest pade approximant of a function defined in the laplace space that is consistent with the following physicaly requirements y r e r k b t g r is finite at the contact point and the isothermal compressibility is finite in the case of sticky hard rods the expression obtained is exact while in the case of sticky hard spheres it coincides with the solution of the percus yevick equation

structure of hard hypersphere fluids in odd dimensions the structural properties of single component fluids of hard hyperspheres in odd space dimensionalities d are studied with an analytical approximation method that generalizes the rational function approximation earlier introduced in the study of hard sphere fluids s b yuste and a santos phys rev a 43 5418 1991 the theory makes use of the exact form of the radial distribution function to first order in density and extends it to finite density by assuming a rational form for a function defined in laplace space the coefficients being determined by simple physical requirements fourier transform in terms of reverse bessel polynomials constitute the mathematical framework of this approximation from which an analytical expression for the static structure factor is obtained in its most elementary form the method recovers the solution of the percus yevick closure to the ornstein zernike equation for hyperspheres at odd dimensions the present formalism allows one to go beyond by yielding solutions with thermodynamic consistency between the virial and compressibility routes to any desired equation of state excellent agreement with available computer simulation data at d 5 and d 7 is obtained

packing hyperspheres in high dimensional euclidean spaces we present a study of disordered jammed hard sphere packings in four five and six dimensional euclidean spaces using a collision driven packing generation algorithm we obtain the first estimates for the packing fractions of the maximally random jammed mrj states for space dimensions d 4 5 and 6t o be mrj0 46 0 31 and 0 20 respectively to a good approximation the mrj density obeys the scaling form mrj c1 2 d c2d 2 d where c1 2 72 and c2 2 56 which appears to be consistent with the highdimensional asymptotic limit albeit with different coefficients calculations of the pair correlation function g2r and structure factor sk for these states show that short range ordering appreciably decreases with increasing dimension consistent with a recently proposed decorrelation principle which among other things states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d as in three dimensions where mrj0 64 the packings show no signs of crystallization are isostatic and have a power law divergence in g2r at contact with power law exponent 0 4 across dimensions the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2r has its first minimum additionally we obtain estimates for the freezing and melting packing fractions for the equilibrium hard sphere fluid solid transition f0 32 and m0 39 respectively for d 4 and f0 20 and m0 25 respectively for d 5 although our results indicate the stable phase at high density is a crystalline solid nucleation appears to be strongly suppressed with increasing dimension

low temperature and high temperature approximations for penetrable sphere fluids comparison with monte carlo simulations and integral equation theories the two body interaction in dilute solutions of polymer chains in good solvents can be modeled by means of effective bounded potentials the simplest of which being that of penetrable spheres pss in this paper we construct two simple analytical theories for the structural properties of ps fluids a low temperature lt approximation that can be seen as an extension to pss of the well known solution of the percus yevick py equation for hard spheres and a high temperature ht approximation based on the exact asymptotic behavior in the limit of infinite temperature monte carlo simulations for a wide range of temperatures and densities are performed to assess the validity of both theories it is found that despite their simplicity the ht and lt approximations exhibit a fair agreement with the simulation data within their respective domains of applicability so that they complement each other a comparison with numerical solutions of the py and the hypernetted chain approximations is also carried out the latter showing a very good performance except inside the core at low temperatures

walks and the spectral radius of graphs abstract given a graph g write g for the largest eigenvalue of its adjacency matrix g for its clique number and wk g for the number of its k walks we prove that the inequalities w q r g w q g r g g 1 g w r g hold for all r 0 and odd q 0 we also generalize a number of other bounds on g and characterize semiregular and pseudo regular graphs in spectral terms

toy model for the mean field theory of hard sphere liquids we investigate a toy model of liquid based on simplified hypernetted chain equations in very large spatial dimension d the model does not exhibit a phase transition but several regimes of behavior when d infinity can be observed in different intervals of the density