interval criteria for oscillation of nonlinear second order dynamic equations on time scales interval oscillation criteria are established for a second order nonlinear dynamic equation on time scales by utilizing a generalized riccati technique and an integral averaging technique the theory can be applied to second order dynamic equations regardless of the choice of delta or nabla derivatives

graphical representations for ising systems in external fields a graphical representation based on duplication is developed that is suitable for the study of ising systems in external fields two independent replicas of the ising system in the same field are treated as a single four state ashkin teller model bonds in the graphical representation connect the ashkin teller spins for ferromagnetic systems it is proved that ordering is characterized by percolation in this representation the representation leads immediately to cluster algorithms some applications along these lines are discussed

the high temperature case for the random k sat problem we give a completely rigorous proof that the replica symmetric solution holds at high enough temperature for the random k sat problem the most notable feature of this problem is that the order parameter of the system is a function and not a number

the three dimensional dilute ising magnet the dilute ising model with a p 0 8 fraction of the sites occupied by spins is simulated on l3 systems for l up to 300 using a single cluster algorithm of wolff in the range of reduced temperature 0 002 t tc tc 0 03 the susceptibility appears to fit a simple power with an effective exponent eff 1 36

quenched annealed and functional large deviations for one dimensional random walk in random environment suppose that the integers are assigned random variables i taking values in the unit interval which serve as an environment this environment defines a random walk x n called a rwre which when at i moves one step to the right with probability i and one step to the left with probability 1 i when the i sequence is i i d greven and den hollander 1994 proved a large deviation principle for x n n conditional upon the environment with deterministic rate function we consider in this paper large deviations both conditioned on the environment quenched and averaged on the environment annealed for the rwre in the ergodic environment case the annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy we also give a detailed qualitative description of the resulting rate functions our techniques differ from those of greven and den hollander and allow us to present also a trajectorial quenched large deviation principle

superparamagnetic segmentation by excitable neural systems magnetic modeling for clustering or segmentation purposes can either associate the image data to external quenched fields or to the interactions among a set of auxiliary variables the latter gives rise to superparamagnetic segmentation and is usually done with potts systems we have used the superparamagnetic clustering technique to segment images with the aid of different associated systems results using potts model are comparable to those obtained using excitable fitzhugh nagumo and morris lecar model neurons interactions between the associated system components are a function of the difference of luminosity on a gray scale of neighbor pixels and the difference of membrane potential

absence of self averaging of the order parameter in the sherrington kirkpatrick model we prove that if n is the sherrington kirkpatrick sk hamiltonian and the quantity bar q n n 1 sum left langle s l right rangle h 2 converges in the variance to a nonrandom limit asn then the mean free energy of the model converges to the expression obtained by sk since this expression is known not to be correct in the low temperature region our result implies the non self averaging of the order parameter of the sk model this fact is an important ingredient of the parisi theory which is widely believed to be exact we also prove that the variance of the free energy of the sk model converges to zero asn i e the free energy has the self averaging property

bulk and surface critical behavior of the three dimensional ising model and conformal invariance using a continuous cluster monte carlo algorithm we investigate the critical three dimensional ising model in its anisotropic limit from the ratio of the magnetic correlations in the strong and the weak coupling directions we determine the length ratio relating the isotropic ising model and the anisotropic limit on this basis we simulate the critical ising model on a spherocylinder s2 x r1 i e a curved geometry obtained from a conformal mapping of the infinite space r3 from correlation lengths along the spherocylinder combined with the prediction of conformal invariance we estimate the magnetic and thermal scaling dimensions as x h 0 5182 6 and x t 1 419 7 respectively the behavior of the binder cumulant is also determined in the limit of an infinitely long spherocylinder next free boundary conditions are imposed on the equators of the spherocylinder and thus the geometry s1 x s x r1 is obtained the surface magnetic scaling dimension is estimated as x s h 1 263 5 the consistency of the aforementioned estimations and existing results confirms that the three dimensional ising model is conformally invariant further the precision of these results reveals that as in two dimensions conformal mappings provide a powerful tool to investigate critical phenomena with the continuous cluster algorithm we also perform simulations of systems inside a conventional solid cylinder the surface magnetic correlation length differs within the estimated error margin by a factor pi 2 from that along a half spherocylinder s1 x s x r1 with the same radius

on the partition function of a directed polymer in a gaussian random environment the purpose of this work is the study of the partition function of a dimensional lattice directed polymer in a gaussian random environment being the inverse of temperature in the low dimensional cases we prove that for all the renormalized partition function converges to 0 and the correlation of two independent configurations does not converge to 0 in the high dimensional case a lower tail of has been obtained for small furthermore we express some thermodynamic quantities in terms of the path measure alone

second order oscillation of forced functional differential equations with oscillatory potentials new oscillation criteria are established for second order differential equations containing both delay and advanced arguments of the form k t x t p t x t t a 1x t t q t x s t b 1x s t e t t 0 where a 1 and b 1 k p q e t s are continuous real valued functions k t 0 is nondecreasing t and o are nondecreasing t t t s t t and lim t t t the potentials p q and e are allowed to change sign and the information on the whole half line is not required as opposed to the usual case in most articles among others as an application of the results we are able to deduce that every solution of x t m 1sintx t p12 m 2costx t p6 cos2t m 1 m 2 0 is oscillatory provided that either m 1 orm 2 is sufficiently large

dual monte carlo and cluster algorithms we discuss the development of cluster algorithms from the viewpoint of probability theory and not from the usual viewpoint of a particular model from the viewpoint of probability theory we detail the nature of a cluster algorithm make explicit the assumptions embodied in all clusters of which we are aware and define the construction of free cluster algorithms we also illustrate these procedures by rederiving the swendsen wang algorithm presenting the details of the loop algorithm for a worldline simulation of a quantum ital s 1 2 model and proposing a free cluster version of the swendsen wang replica method for the random ising model how the principle of maximum entropy might be used to aid the construction of cluster algorithms is also discussed

variance reduced estimator of the connected two point function in the presence of a broken z 2 symmetry the exchange or geometric cluster algorithm allows us to define a variance reduced estimator of the connected two point function in the presence of a broken z 2 symmetry we present numerical tests for the improved blume capel model on the simple cubic lattice we perform simulations for the critical isotherm the low temperature phase at vanishing external field and for comparison also the high temperature phase for the connected two point function a substantial reduction of the variance can be obtained allowing us to compute the correlation length with high precision based on these results estimates for various universal amplitude ratios that characterize the universality class of the three dimensional ising model are computed

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

monte carlo study of the random field ising model using a cluster flipping monte carlo algorithm combined with a generalization of the histogram reweighting scheme of ferrenberg and swendsen we have studied the equilibrium properties of the thermal random field ising model on a cubic lattice in three dimensions we have equilibrated systems of l x l x l spins with values of l up to 32 and for these systems the cluster flipping method appears to a large extent to overcome the slow equilibration seen in single spin flip methods from the results of our simulations we have extracted values for the critical exponents and the critical temperature and randomness of the model by finite size scaling for the exponents we find v 1 02 0 06 b 0 06 0 07 y 1 9 0 2 and mean y 2 9 0 2

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

data clustering using a model granular magnet we present a new approach to clustering based on the physical properties of an inhomogeneous ferromagnet no assumption is made regarding the underlying distribution of the data we assign a potts spin to each data point and introduce an interaction between neighboring points whose strength is a decreasing function of the distance between the neighbors this magnetic system exhibits three phases at very low temperatures it is completely ordered all spins are aligned at very high temperatures the system does not exhibit any ordering and in an intermediate regime clusters of relatively strongly coupled spins become ordered whereas different clusters remain uncorrelated this intermediate phase is identified by a jump in the order parameters the spin spin correlation function is used to partition the spins and the corresponding data points into clusters we demonstrate on three synthetic and three real data sets how the method works detailed comparison to the performance of other techniques clearly indicates the relative success of our method

critical slowing down in local dynamics simulations the author reports some universal and non universal aspects of the critical dynamics of the three dimensional ising model obtained in extensive monte carlo finite size simulations the author shows that the time dependence of the magnetization of finite lattices is composed of two kinds of fluctuations at the critical point i phase fluctuations from one metastable minimum of the free energy to the other dominating the long time behaviour of the magnetization ii critical fluctuations inside each minimum decaying on a comparatively short timescale both kinds of fluctuations show up the same critical exponent z 2 10 or 0 02 which is also in excellent agreement with the exponent z of energy fluctuations

the kac version of the sherrington kirkpatrick model at high temperatures we study the kac version of the sherrington kirkpatrick sk model of a spin glass i e a spin glass with long but finite range interaction on mathbb z d and gaussian mean zero couplings we prove that for all 1 the free energy of this model converges to that of the sk model as the range of the interaction tends to infinity moreover we prove that for all temperatures for which the infinite volume gibbs state is unique the free energy scaled by the square root of the volume converges to a gaussian with variance c where 1 is the range of the interaction moreover at least for almost all values of this variance tends to zero as goes to zero the value in the sk model we interpret our finding as a weak indication that at least at high temperatures the sk model can be seen as a reasonable asymptotic model for lattice spin glasses

system size dependence of the autocorrelation time for the swendsen wang ising model abstract we present monte carlo simulation results of the autocorrelation time for the swendsen wang method for the simulation of the ising model we have calculated the exponential and the integrated autocorrelation time at the critical point t c of the two dimensional ising model our results indicate that both autocorrelation times depend logarithmically on the linear system size l instead of a power law the simulations were carried out on the parallel computer of the condensed matter theory group at the university of mainz

monte carlo study of the ising model phase transition in terms of the percolation transition of physical clusters finite squarel l ising lattices with ferromagnetic nearest neighbor interaction are simulated using the swendsen wang cluster algorithm both thermal properties internal energyu specific heatc magnetization m susceptibility and percolation cluster properties relating to the physical clusters namely the fortuin kasteleyn clusters percolation probability p percolation susceptibility p cluster size distributionnl are evaluated paying particular attention to finite size effects it is shown that thermal properties can be expressed entirely in terms of cluster properties p being identical to m in the thermodynamic limit while finite size corrections differ in contrast p differs from even in the thermodynamic limit since a fluctuation in the size of the percolating net contributes to but not to p neartc the cluster size distribution has the scaling properties as hypothesized by earlier phenomenological theories we also present a generalization of the swendsen wang algorithm allowing one to cross over continuously to the glauber dynamics

some comparison and oscillation results for first order differential equations and inequalities with a deviating argument abstract in this paper we are dealing with first order differential equations and inequalities with a deviating argument and we give some new necessary sufficient and necessary and sufficient conditions concerning a the comparison of oscillatory and asymptotic behavior of solutions of 1 first order vector differential equations and corresponding to them operator and scalar differential equations with a deviating argument and 2 first order differential equations and corresponding to them inequalities with a deviating argument and b the existence of a certain type of nonoscillatory solutions of differential equations and inequalities with a deviating argument what is important in our opinion is the fact that some of the obtained results concern not only the cases of constant sign coefficients but also the cases where the coefficients of equations and inequalities under consideration are of non constant sign

cluster monte carlo algorithms for random ising models abstract variations of the swendsen wang and wolff cluster flip algorithms are proposed to perform monte carlo simulations of ising models with random bonds and random fields including systems with frustration the new methods are tested by applying them to small lattices and comparing the results with exact data

the sherrington kirkpatrick model of spin glasses and stochastic calculus the high temperature case we study the fluctuations of free energy energy and entropy in the high temperature regime for the sherrington kirkpatrick model of spin glasses we introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter thus formulating the thermodynamic formalism in terms of random processes the well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments a generalization of a result proved by aizenman lebowitz and ruelle 1

critical dynamics of the swendsen wang algorithm in the three dimensional ising model abstract the correlation times of the three dimensional ising model at the critical point for the swendsen wang dynamics are obtained up to a system size 893 strong corrections to scaling are present in the size dependences of the correlation times of the energy and the size of the largest cluster making the estimation of the true asymptotic dynamic critical exponent difficult when the largest cluster is not flipped the dynamic critical exponent zm defined by the magnetization correlations of the largest cluster is estimated to be 0 339 0 004

oscillation and nonoscillation of advanced differential equations with variable coefficients the oscillation and nonoscillation of the advanced differential equations x t p t x t 0 t t0 and x t i 1npi t x t i 0 t t0 are investigated where p t pi t c t0 0 and i are positive constants at first a sharp sufficient condition for the oscillation of eq is obtained then the result is generalized to eq these results improve the corresponding conclusions derived by ladas and stavroulakis j differential equations 44 1982 134 152 next two examples are given to illustrate the advantages of our results finally the sufficient conditions for these two equations to be nonoscillatory are also obtained

spin glasses and stein s method we introduce some applications of stein s method in the high temperature analysis of spin glasses stein s method allows the direct analysis of the gibbs measure without having to eate a cavity another advantage is that it gives limit theorems with total variation error bounds although the bounds can be suboptimal a surprising byproduct of our analysis is a relatively transparent explanation of the thouless anderson palmer system of equations along the way we develop stein s method for mixtures of two gaussian densities

a multiresolution stochastic level set method for mumford shah image segmentation the mumford shah model is one of the most successful image segmentation models however existing algorithms for the model are often very sensitive to the choice of the initial guess to make use of the model effectively it is essential to develop an algorithm which can compute a global or near global optimal solution efficiently while gradient descent based methods are well known to find a local minimum only even many stochastic methods do not provide a practical solution to this problem either in this paper we consider the computation of a global minimum of the multiphase piecewise constant mumford shah model we propose a hybrid approach which combines gradient based and stochastic optimization methods to resolve the problem of sensitivity to the initial guess at the heart of our algorithm is a well designed basin hopping scheme which uses global updates to escape from local traps in a way that is much more effective than standard stochastic methods in our experiments a very high quality solution is obtained within a few stochastic hops whereas the solutions obtained with simulated annealing are incomparable even after thousands of steps we also propose a multiresolution approach to reduce the computational cost and enhance the search for a global minimum furthermore we derived a simple but useful theoretical result relating solutions at different spatial resolutions

damage spreading in the kawasaki ising model abstract we observe no phase transition in the damage spreading probability at the curie point of the 2d ising model with nearest neighbor spin exchange dynamics furthermore we report values of the growth exponent z close to unity implying speed of light transmission of information and values of the exponents describing the scaling behavior of the damage mass with time and distance close to two finally we discuss the spreading of damage in a one dimensional system

self averaging in finite random copolymers we investigate how martingale techniques can be used to derive information on the extent of self averaging of the free energy for some lattice models of finite random copolymers

simulations of lattice animals and trees the scaling behaviour of randomly branched polymers in a good solvent is studied in two to nine dimensions using as microscopic models lattice animals and lattice trees on simple hypercubic lattices as a stochastic sampling method we use a biased sequential sampling algorithm with re sampling similar to the pruned enriched rosenbluth method perm used extensively for linear polymers essentially we start simulating percolation clusters either site or bond re weigh them according to the animal tree ensemble and prune or branch the further growth according to a heuristic fitness function in contrast to previous applications of perm this fitness function is not the weight with which the actual configuration would contribute to the partition sum but is closely related to it we obtain high statistics of animals with up to several thousand sites in all dimension 2 d 9 in addition to the partition sum number of different animals we estimate gyration radii and numbers of perimeter sites in all dimensions we verify the parisi sourlas prediction and we verify all exactly known critical exponents in dimensions 2 3 4 and 8 in addition we present the hitherto most precise estimates for growth constants in d 3 for clusters with one site attached to an attractive surface we verify for d 3 the superuniversality of the cross over exponent at the adsorption transition predicted by janssen and lyssy but not for d 2 there we find 0 480 4 instead of the conjectured 1 2 finally we discuss the collapse of animals and trees arguing that our present version of the algorithm is also efficient for some of the models studied in this context but showing that it is not very efficient for the classical model for collapsing animals

rigorous lower bound on the dynamic critical exponents of the swendsen wang algorithm we prove the rigorous lower bound z sw v for the dynamic critical exponent of the swendsen wang algorithm for two dimensional q state potts models with q 2 3 4 this implies z sw 0 2 5 1 we present numerical data indicating that z sw 0 55 0 03 0 89 0 05 for q 3 4 95 confidence limits statistical errors only the discrepancy for q 4 appears to be caused by multiplicative logarithmic corrections

geometrical properties of swendsen wang clusters abstract several geometrical properties of swendsen wang clusters are studied numerically for the 2d ising and 3 state potts models at the critical temperature the fractal dimensions of the total mass backbone elastic backbone shortest path and red bonds are determined the results agree with some theoretical predictions

graphical representations and cluster algorithms for critical points with fields a two replica graphical representation and associated cluster algorithm is described that is applicable to ferromagnetic ising systems with arbitrary fields critical points are associated with the percolation threshold of the graphical representation results from numerical simulations of the ising model in a staggered field are presented the dynamic exponent for the algorithm is measured to be less than 0 5

conformal invariance and simulations in curved geometries abstract conformal mappings serve as useful tools for the determination of universal properties of critical models typical applications are subject to a major restriction namely that the pertinent conformal mapping should lead to a geometry that can be investigated by means of numerical methods such as monte carlo simulations since conformal mappings of 3 d systems usually lead to curved geometries which are difficult to investigate numerically most applications have thus far been restricted to 2 d systems we present a solution of this problem for discrete spin models by taking the anisotropic or hamiltonian limit which renders one of the lattice directions continuous such that the dimensionality in effect reduces by one applications to the 3 d ising and percolation models confirm the predictions obtained from the assumption of conformal invariance and lead to accurate numerical results for the scaling dimensions

fluctuations of the free energy in the rem and the p spin sk models we consider the random fluctuations of the free energy in the p spin version of the sherrington kirkpatrick sk model in the high temperature regime using the martingale approach of comets and neveu as used in the standard sk model combined with truncation techniques inspired by a recent paper by talagrand on the p spin version we prove that the random corrections to the free energy are on a scale n p 2 2 only and after proper rescaling converge to a standard gaussian random variable this is shown to hold for all values of the inverse temperature smaller than a critical p we also show that p 2 in 2 as p additionally we study the formal p limit of these models the random energy model here we compute the precise limit theorem for the properly rescaled partition function at all temperatures for 2 in 2 fluctuations are found at an exponentially small scale with two distinct limit laws above and below a second critical value in 2 2 for up to that value the rescaled fluctuations are gaussian while below that there are non gaussian fluctuations driven by the poisson process of the extreme values of the random energies for larger than the critical 2 in 2 the fluctuations of the logarithm of the partition function are on a scale of 1 and are expressed in terms of the poisson process of extremes at the critical temperature the partition function divided by its expectation converges to 1 2

cluster algorithm for hard spheres and related systems in this paper we present a cluster algorithm for the simulation of hard spheres and related systems in this algorithm a copy of the configuration is rotated with respect to a randomly chosen pivot point the two systems are then superposed and clusters of overlapping spheres in the joint system are isolated each of these clusters can be flipped independently a process which generates non local moves in the original configuration a generalization of this algorithm which works perfectly well at small density can be made to work successfully at densities around the solid liquid transition point in the two dimensional hard sphere system

replica symmetry breaking and exponential inequalities for the sherrington kirkpatrick model we provide an extremely accurate picture of the sherrington kirkpatrick model in three cases for high temperature for large external field and for any temperature greater than or equal to 1 and sufficiently small external field we describe the system at the level of the central limit theorem or as physicists would say at the level of fluctuations around the mean field we also obtain much more detailed information in the form of exponential inequalities that express a uniform control over higher order moments we give a complete rigorous proof that at the generic point of the predicted low temperature region there is replica symmetry breaking in the sense that the system is unstable with respect to an infinitesimal coupling between two replicas

oscillations caused by several retarded and advanced arguments in this paper we study the oscillatory behavior of equations of the forms y t i 1npiy t i 0 and y t i 1npiy t i 0 where piand i i 1 2 n are positive constants we prove that each one of the following conditions 1 pi i1 e for some i i 1 2 n 2 i 1npi 1 e where min 1 2 n 3 i 1npi 1 n i 1n i 1 e or 4 1 n i 1n pi i 1 2 21 e implies that every solution of and oscillates a generalization in the case where the coefficientspi i 1 2 n are positive and continuous functions is also presented

cluster monte carlo algorithms abstract the swendsen wang and wolff monte carlo algorithms are described in some detail using the potts model as an example various generalizations are then reviewed and some applications are discussed two complete fortran programs for the algorithms are provided

on the distribution of overlaps in the sherrington kirkpatrick spin glass model this paper describes some of the analytic tools developed recently by ghirlanda and guerra in the investigation of the distribution of overlaps in the sherrington kirkpatrick spin glass model and of parisi s ultrametricity in particular we introduce to this task a simplified but also generalized model on which the gaussian analysis is made easier moments of the hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps under the essential tool of self averaging we describe with full rigour various overlap identities and replica independence that actually hold in a rather large generality the results are presented in a language accessible to probabilists and analysts

intersecting random half spaces toward the gardner derrida formula gardner and derrida have introduced a natural version of the problem of the capacity of the binary perceptron with temperature and they proposed based on physical methods remarkable formulas for this model we give a complete rigorous proof that these formulas are correct at sufficiently high temperature for a much larger class of models

quenched large deviation principle for the overlap of a p spins system in this note we prove a quenched large deviation result for the overlap of a p spins interaction system at high temperature the rate function of the large deviation principle is proved to be deterministic and some of its basic properties are studied our result is based on a pure state result for a multidimensional p spins system combined with a careful application of the gartner ellis theorem

non algebraic domain growth in random magnets a cell dynamical approach the authors develop a novel numerical approach based on a computationally efficient cell dynamical system cds model for studying the kinetics of ordering in systems described by a non conserved order parameter with quenched disorder evolving from unstable initial states they use this model to study the kinetics of domain growth in a coarse grained version of the random exchange ising model their numerical data strongly indicate quantitative agreement with the theoretically predicted asymptotic growth law over a limited range of disorder amplitudes they also compare their observations with those in laboratory experiments and make important predictions regarding dynamical scaling in these systems

numerical studies of critical percolation in three dimensions presents results of high statistics simulations of the spreading of 3d percolation close to the critical point the main results are i a more precise estimate of the spreading resp chemical distance dimension ii a novel heuristic scaling theory for percolation near surfaces and iii more precise values for critical surface and edge exponents in addition the author verifies the very precise results of ziff and stell 1988 for the percolation thresholds and studies in detail corrections to scaling the latter is made possible by simulating bond and site percolation in a number of different geometries in addition in most cases one could measure more than one observable during each simulation which provided quite severe internal consistency constraints in some cases the author found quite important deviations from scaling which are not easily described by a single power or logarithmic correction term

boundary effects in the three dimensional ising model abstract simulations of the standard three dimensional ising model with a variety of boundary conditions for medium and large systems are compared with field theoretical predictions in particular setting boundary spins equal to zero is not the best approximation for the dirichtet boundary condition of zero magnetization at the boundary the simulations for the specific heat agree semiquantitatively with the field theory for periodic and dirichtet boundary conditions

ising model simulation in directed lattices and networks abstract on directed lattices with half as many neighbours as in the usual undirected lattices the ising model does not seem to show a spontaneous magnetisation at least for lower dimensions instead the decay time for flipping of the magnetisation follows an arrhenius law on the square and simple cubic lattice on directed barabasi albert networks with two and seven neighbours selected by each added site metropolis and glauber algorithms give similar results while for wolff cluster flipping the magnetisation decays exponentially with time

static and dynamic critical behavior of thin magnetic ising films this work presents a study of the effect of film thickness on the static and dynamic critical behavior of thin magnetic ising films monte carlo simulations using the wolff algorithm were performed to determine the static and dynamic critical exponents of the films a dimensionality crossover from 2d to 3d due to the finiteness of the films in the static and dynamic critical behavior was observed as the film thickness increases in addition a slight increase in the effective dimension deff and a considerable increase in the critical temperature tc were found small values for the dynamic critical exponents were obtained indicating that the wolff algorithm is a very efficient method for these magnetic systems

hierarchical monte carlo simulation of the ising model abstract we pursue schmidt s recent idea to generate ising spin configurations via a hierarchical monte carlo procedure the method is modified and improved and applied to the two dimensional ising model at criticality the system size dependence is investigated for a 32 32 lattice the autocorrelation functions of the energy and the squared magnetization decay faster than with the standard method by a factor in the range of 3 to 10 on the basis of a number of plausible hypotheses we construct a theoretical framework within which the hierarchical method can be understood

free energy of the spherical mean field model we compute at any temperature the free energy of the multi p spin spherical model when only terms for p even are considered

single cluster monte carlo dynamics for the ising model we present an extensive study of a new monte carlo acceleration algorithm introduced by wolff for the ising model it differs from the swendsen wang algorithm by growing and flipping single clusters at a random seed in general it is more efficient than swendsen wang dynamics ford 2 giving zero critical slowing down in the upper critical dimension monte carlo simulations give dynamical critical exponentszw 0 33 0 05 and 0 44 0 10 ind 2 and 3 respectively and numbers consistent withzw 0 ind 4 and mean field theory we present scaling arguments which indicate that the wolff mechanism for decorrelation differs substantially from swendsen wang despite the apparent similarities of the two methods

on nonoscillation of mixed advanced delay differential equations with positive and negative coefficients for a mixed advanced delay differential equation with variable delays and coefficients email protected t a t x g t b t x h t 0 t t 0 where a t 0 b t 0 g t t h t t explicit nonoscillation conditions are obtained

large deviations and mean field theory for asymmetric random recurrent neural networks in this article we study the asymptotic dynamics of a noisy discrete time neural network with random asymmetric couplings and thresholds more precisely we focus our interest on the limit behaviour of the network when its size grows to infinity with bounded time in the case of gaussian connection weights we use the same techniques as ben arous and guionnet see 3 to prove that the image law of the distribution of the neurons activation states by the empirical measure satisfies a temperature free large deviation principle moreover we prove that if the connection weights satisfy a general condition of domination by gaussian tails then the distribution of the activation potential of each neuron converges weakly towards an explicit gaussian law the characteristics of which are contained in the mean field equations stated by cessac doyon quoy samuelides see 4 6 furthermore under this hypothesis we obtain a law of large numbers and a propagation of chaos result finally we show that many classical distributions on the couplings fulfill our general condition thus this paper provides rigorous mean field results for a large class of neural networks which is currently investigated in neural network literature

rigorous results for mean field models for spin glasses in this paper i explain why i believe that it is important to prove rigorous results about mean field models for spin glasses and why i think it is difficult i also describe at a high level what mathematicians have been able to prove

rigorous results for the hopfield model with many patterns we perform a thorough investigation of the main aspects of the hopfield model with many patterns advances are made toward the validity of the replica symmetric solution strong evidence of the validity of this solution is given over the entire domain where this validity is conjectured complete proof is given in a subregion that contains strictly the ergodic region

thermal operators in ising percolation we discuss a new cluster representation for the internal energy and the specific heat of the d dimensional ising model obtained by studying the percolation mapping of an ising model with an arbitrary set of antiferromagnetic links such a representation relates the thermal operators to the topological properties of the fortuin kasteleyn fk clusters of ising percolation and is a powerful tool to get new exact relations on the topological structure of fk clusters of the ising model defined on an arbitrary graph

the sherrington kirkpatrick model a challenge for mathematicians the sherrington kirkpatrick sk model for spin glasses is deceptively simple to state yet its rigorous study represents a considerable challenge we report here some modest progresses obtained through elementary methods even in the supposedly simple high temperature region a number of basic questions remain unsolved

replica exchange algorithm and results for the three dimensional random field ising model the random field ising model with gaussian disorder is studied using a different monte carlo algorithm the algorithm combines the advantages of the replica exchange method and the two replica cluster method and is much more efficient than the metropolis algorithm for some disorder realizations three dimensional systems of size 24 3 are studied each realization of disorder is simulated at a value of temperature and uniform field that is adjusted to the phase transition region for that disorder realization energy and magnetization distributions show large variations from one realization of disorder to another for some realizations of disorder there are three well separated peaks in the magnetization distribution and two well separated peaks in the energy distribution suggesting a first order transition

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

rigorous low temperature results for the mean field p spins interaction model we prove that just below the critical temperature the mean field p spins interaction model for p suitably large spontaneously decomposes into different states the asymptotic overlaps between any two different states are zero under a mild unproven hypothesis on the weight distribution of these states we prove that they are pure states this situation is called in physics one level of symmetry breaking

optimization of monte carlo simulations abstract monte carlo simulations of phase transitions and critical phenomena are hampered by long relaxation times and large fluctuations the computational effort for the collection of data points in a simulation often varies from data point to data point by orders of magnitude the optimal investment of computing time at different data points is a priori not clear we propose a method to calculate distributions of statistical weight and computing time which minimize the error of the parameters to be determined the method is discussed for the example of a finite size simulation in statistical physics we calculate explicitly the optimal distributions for algebraic dependencies it is shown that the optimal distributions depend on d z x where d is the dimension of the system z is the dynamical critical exponent and x is the scaling exponent of the relative variance of the data depending on one may save 60 80 of the total computing time compared to distributions with equal relative weight

critical dynamics of cluster algorithms in the dilute ising model autocorrelation times for thermodynamic quantities att c are calculated from monte carlo simulations of the site diluted simple cubic ising model using the swendsen wang and wolff cluster algorithms our results show that for these algorithms the autocorrelation timesdecrease when reducing the concentration of magnetic sites from 100 down to 40 this is of crucial importance when estimating static properties of the model since the variances of these estimators increase with autocorrelation time the dynamical critical exponents are calculated for both algorithms observing pronounced finite size effects in the energy autocorrelation data for the algorithm of wolff we conclude that when applied to the dilute ising model cluster algorithms become even more effective than local algorithms for whichincreasing autocorrelation times are expected

oscillations of delay differential equations with variable coefficients consider the delay differential equation t p t x t 0 where p t c t0 r and is a positive constant we show that every solution of this equation oscillates if tt p t dt 1 e for sufficiently large t and t0 p t exp tt p s ds 1 e 1 dt

high temperature regime for a multidimensional sherrington kirkpatrick model of spin glass comets and neveu have initiated in 5 a method to prove convergence of the partition function of disordered systems to a log normal random variable in the high temperature regime by means of stochastic calculus we generalize their approach to a multidimensional sherrington kirkpatrick model with an application to the heisenberg model of uniform spins on a sphere of d see 9 the main tool that we use is a truncation of the partition function outside a small neighbourhood of the typical energy path

cluster monte carlo simulation of the transverse ising model we formulate a cluster monte carlo method for the anisotropic limit of ising models on d 1 dimensional lattices which in effect are equivalent with d dimensional quantum transverse ising models using this technique we investigate the transverse ising models on the square triangular kagome honeycomb and simple cubic lattices the monte carlo data are analyzed by finite size scaling in each case we find as expected that the critical behavior fits well in the d 1 dimensional ising universality class for the transverse ising model on the square lattice we determine the binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or classical direction and that in the other two directions matching this universal function with the case of the isotropic ising model yields the length ratio relating the isotropic ising model with the anisotropic limit the efficiency of the present algorithm is reflected by the precision of its results which improves significantly on earlier analyses

high dimensional lattice gases we investigate the critical behaviour of hard core lattice gases in four five and six dimensions by means of monte carlo simulations in order to suppress critical slowing down we use a geometrical cluster monte carlo algorithm in particular nearest neighbour exclusion lattice gases on simple hypercubic lattices are investigated these models undergo ising like ordering transitions where the majority of the lattice gas particles settle on one of two sublattices a finite size scaling analysis of the simulation data confirms that these lattice gases display classical critical behaviour the results agree with the renormalization predictions at and above the upper critical dimensionality in particular the predicted value of the binder cumulant is confirmed

the simple cubic lattice gas with nearest neighbour exclusion ising universality abstract the lattice gas with nearest neighbour exclusion on the simple cubic lattice is studied by means of statistically accurate monte carlo simulations with an efficient cluster algorithm our results for critical exponents are y h 2 47 1 and y t 1 60 2 these results agree well with the three dimensional ising universality class we explain the discrepancy with an earlier study the critical activity is z c 1 0559 1 the size distribution of the clusters indicates that the percolation threshold of the cluster formation process of the monte carlo algorithm coincides with the critical point

interval oscillation of second order functional differential equations with oscillatory potentials abstract interval oscillation criteria are established for second order differential equations of the form e k t x t p t x g t q t x g t 1 x g t e t t 0 where 1 k p q e and g are continuous real valued functions k t 0 is nondecreasing g t is nondecreasing lim t g t we consider g t t and g t t separately illustrative examples are included

on the stochastic calculus method for spins systems in this note we show how to generalize the stochastic calculus method introduced by comets and neveu comm math phys 166 1995 549 564 for two models of spin glasses namely the sk model with external field and the perceptron model this method allows to derive quite easily some fluctuation results for the free energy in those two cases

finding long cycles in graphs we analyze the problem of discovering long cycles inside a graph we propose and test two algorithms for this task the first one is based on recent advances in statistical mechanics and relies on a message passing procedure the second follows a more standard monte carlo markov chain strategy special attention is devoted to hamiltonian cycles of nonregular random graphs of minimal connectivity equal to 3

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

non equilibrium critical relaxation of the wolff dynamics of the ising model abstract the non equilibrium critical relaxation of the single cluster dynamics is studied for the ferromagnetic ising model on the square and cubic lattices power law behavior is observed in the energy relaxation process starting from an all up spin configuration from this power decay the dynamical exponent z is estimated to be 0 50 0 05 and 1 3 0 1 for the square and cubic lattices respectively where the time is defined by the number of cluster flip trials if the time scale is normalized by cluster mass they are 0 25 0 05 and 0 3 0 1 respectively

critical behaviour of non equilibrium q state systems we present two classes of non equilibrium models with critical behaviour each model is characterized by an integer q 1 and is defined on configurations of q valued spins on regular lattices the definitions of the models are very similar to the updating rules in wolff s algorithm for the potts model but both classes break detailed balance except for q 2 and q infinity in the first case both models reduce to the ising model while one of them reduces to percolation more precisely to the general epidemic process for q infinity locations of the critical point and critical exponents are estimated in two dimensions

annealing markov chain monte carlo with applications to ancestral inference abstract markov chain monte carlo mcmc the metropolis hastings algorithm has been used for many statistical problems including bayesian inference likelihood inference and tests of significance though the method generally works well doubts about convergence often remain here we propose mcmc methods distantly related to simulated annealing our samplers mix rapidly enough to be usable for problems in which other methods would require eons of computing time they simulate realizations from a sequence of distributions allowing the distribution being simulated to vary randomly over time if the sequence of distributions is well chosen then the sampler will mix well and produce accurate answers for all the distributions even when there is only one distribution of interest these annealing like samplers may be the only known way to get a rapidly mixing sampler these methods are essential for attacking very hard problems which arise in areas such as statistical genetics we illustrate the methods wi

a new approach to critical exponents in phase transitions of spin systems abstract we investigate a method to obtain critical exponents using standard numerical simulation techniques for spin systems we show that by analyzing the inverse of the logarithmic derivative of the susceptibility with respect to temperature the accuracy in determining can be increased significantly the method is tested on the two and three dimensional ising models but this alternative approach is thought to be most useful for spin systems on non periodic structures like quasi crystals and fractal lattices where the finite size scaling method is difficult to apply

fast simulation of the ising model using cellular automata a totalistic ising cellular automaton ica is proposed which can reproduce the thermodynamics of the ising model the automaton rules are preset in the usual way but some random mixing of these existing rules is subsequently permitted the model employs a vectorized multispin coding algorithm to reach spin updating speeds comparable with the best metropolis monte carlo performances and is characterized by a nonlinear dynamical critical exponent whose value seems less than two