percolation processes in monomer polyatomic mixtures in this paper the percolation of mixtures of monomers and polyatomic species kk mers on a square lattice is studied by means of a finite size scaling analysis the critical exponents and the scaling collapsing of the fraction of percolating lattices are found a phase diagram separating a percolating from a non percolating region is determined the main features of the phase diagram are discussed in order to predict its evolution for larger kk mer sizes

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

on the universality of crossing probabilities in two dimensional percolation six percolation models in two dimensions are studied percolation by sites and by bonds on square hexagonal and triangular lattices rectangles of widtha and heightb are superimposed on the lattices and four functions representing the probabilities of certain crossings from one interval to another on the sides are measured numerically as functions of the ratioa b in the limits set by the sample size and by the conventions and on the range of the ratioa b measured the four functions coincide for the six models we conclude that the values of the four functions can be used as coordinates of the renormalization group fixed point

re examination of 3d percolation threshold estimates a traditional monte carlo simulation of a 1000 1000 1000 simple cubic lattice gave a bond percolation threshold near 0 2494 and a site percolation threshold near 0 3116 with probable error bars near 10 4

percolation of interacting classical dimers on the square lattice abstract we study the percolation properties of the interacting classical dimer model on the square lattice by means of monte carlo simulations and finite size scaling analysis we define ising clusters based on the dimer configuration the percolation point of the clusters coincides with the critical point of the kosterlitz thouless transition of the dimer model which is t c 0 654 2 furthermore we find that the largest cluster at the kosterlitz thouless point is a fractal with fractal dimension d c 1 874 2 which coincides with the critical exponent describing the critical behavior of the dimer dimer correlation function which is theoretically predicted to be 15 8

definition of percolation thresholds on self affine surfaces we study the percolation transition on a two dimensional substrate with long range self affine correlations we find that the position of the percolation threshold on a correlated lattice is no longer unique and depends on the spanning rule employed numerical results are provided for spanning across the lattice in specified horizontal or vertical either or both directions

finite size scaling in strips antiperiodic boundary conditions the exact values of the universal amplitude a relating the correlation length xi approximately l a to the strip width l at the critical point of ising and 3 state potts models are obtained for the case of antiperiodic or twisted boundary conditions predictions are also made for the finite size scaling behaviour of the interfacial tension

stochastic renormalization group in percolation i fluctuations and crossover abstract a generalization of the renormalization group which describes order parameter fluctuations in finite systems is developed in the specific context of percolation this stochastic renormalization group srg expresses statistical self similarity through a non stationary branching process the srg provides a theoretical basis for analytical or numerical approximations both at and away from criticality whenever the correlation length is much larger than the lattice spacing regardless of the system size for example the srg predicts order parameter distributions and finite size scaling functions for the complete crossover between phases for percolation the simplest srg describes structural quantities conditional on spanning such as the total cluster mass or the minimum chemical distance between two boundaries in these cases the central limit theorem for independent random variables holds at the stable off critical fixed points while a fractal central limit theorem describing long range correlations holds at the unstable critical fixed point this first part of a series of articles explains these basic concepts and a general theory of crossover subsequent parts will focus on limit theorems and comparisons of small cell srg approximations with simulation results

crossing probabilities in one two or three directions for percolation on a cubic lattice the probabilities of crossing in exactly one two or three directions have been investigated using monte carlo simulations of percolation on cubic lattices with free boundary conditions the crossing probability in exactly one given direction shows the least influence of finite size effects and is best suited to obtain the percolation threshold pc finite size effects are most important for simultaneous crossing in three directions crossing probabilities at pc for large lattices are the same for site and bond percolation the crossing probability in a given direction regardless of crossing in other directions r1 is 0 28 exact results for small lattices agree with simulation results

convergence of threshold estimates for two dimensional percolation using a recently introduced algorithm for simulating percolation in microcanonical fixed occupancy samples we study the convergence with increasing system size of a number of estimates for the percolation threshold for an open system with a square boundary specifically for site percolation on a square lattice we show that the convergence of the average probability estimate is described by a nontrivial correction to scaling exponent as predicted previously and measure the value of this exponent to be 0 90 ifmmode pm else textpm fi 0 02 for the median and cell to cell estimates of the percolation threshold we verify that convergence does not depend on this exponent having instead a slightly faster convergence with a trivial analytic leading exponent

adaptive subspace detectors we use the theory of generalized likelihood ratio tests glrts to adapt the matched subspace detectors msds of scharf 1991 and of scharf and frielander 1994 to unknown noise covariance matrices in so doing we produce adaptive msds that may be applied to signal detection for radar sonar and data communication we call the resulting detectors adaptive subspace detectors asds these include kelly s 1987 glrt and the adaptive cosine estimator ace of kaurt and scharh see ibid vol 47 p 2538 41 1999 and of scharf and mcwhorter see proc 30th asilomar conf signals syst comput pacific grove ca 1996 for scenarios in which the scaling of the test data may deviate from that of the training data we then present a unified analysis of the statistical behavior of the entire class of asds obtaining statistically identical decompositions in which each asd is simply decomposed into the nonadaptive matched filter the nonadaptive cosine or t statistic and three other statistically independent random variables that account for the performance degrading effects of limited training data

percolation on two and three dimensional lattices in this work we apply a highly efficient monte carlo algorithm recently proposed by newman and ziff to treat percolation problems the site and bond percolations are studied on a number of lattices in two and three dimensions quite good results for the wrapping probabilities correlation length critical exponent and critical concentration are obtained for the square simple cubic hexagonal close packed and hexagonal lattices by using relatively small systems we also confirm the universal aspect of the wrapping probabilities regarding site and bond dilution

fast monte carlo algorithm for site or bond percolation we describe in detail an efficient algorithm for studying site or bond percolation on any lattice the algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system we demonstrate our algorithm by using it to investigate a number of issues in percolation theory including the position of the percolation transition for site percolation on the square lattice the stretched exponential behavior of spanning probabilities away from the critical point and the size of the giant component for site percolation on random graphs

image denoising via sparse and redundant representations over learned dictionaries we address the image denoising problem where zero mean white and homogeneous gaussian additive noise is to be removed from a given image the approach taken is based on sparse and redundant representations over trained dictionaries using the k svd algorithm we obtain a dictionary that describes the image content effectively two training options are considered using the corrupted image itself or training on a corpus of high quality image database since the k svd is limited in handling small image patches we extend its deployment to arbitrary image sizes by defining a global image prior that forces sparsity over patches in every location in the image we show how such bayesian treatment leads to a simple and effective denoising algorithm this leads to a state of the art denoising performance equivalent and sometimes surpassing recently published leading alternative denoising methods

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

universality at the three dimensional percolation threshold the fraction of samples spanning a lattice at its percolation threshold is found via simulations of bond and site bond percolation to have a universal value of about 0 42 in three dimensions the ratio of shift of the finite to infinite size threshold to width of the threshold distribution is also compatible with universality lattices with up to 10013 sites and or very good statistics were needed to obtain clear results in these monte carlo tests preliminary studies with smaller lattices were inconclusive

universality of the crossing probability for the potts model for q 1 2 3 4 the universality of the crossing probability pi hs of a system to percolate only in the horizontal direction was investigated numerically by a cluster monte carlo algorithm for the q state potts model for q 2 3 4 and for percolation q 1 we check the percolation through fortuin kasteleyn clusters near the critical point on the square lattice by using representation of the potts model as the correlated site bond percolation model it was shown that probability of a system to percolate only in the horizontal direction pi hs has the universal form pi hs a q q z for q 1 2 3 4 as a function of the scaling variable z b q l 1 nu q p p c q l zeta q here p 1 exp beta is the probability of a bond to be closed a q is the nonuniversal crossing amplitude b q is the nonuniversal metric factor nu q is the correlation length index and zeta q is the additional scaling index the universal function q x approximately equal exp z the nonuniversal scaling factors were found numerically

continuum percolation thresholds in two dimensions a wide variety of methods have been used to compute percolation thresholds in lattice percolation the most powerful of these methods consists of microcanonical simulations using the union find algorithm to efficiently determine the connected clusters and in two dimensions using exact values from conformal field theory for the probability at the phase transition that various kinds of wrapping clusters exist on the torus we apply this approach to percolation in continuum models finding overlaps between objects with real valued positions and orientations in particular we find precise values of the percolation transition for disks squares rotated squares and rotated sticks in two dimensions and confirm that these transitions behave as conformal field theory predicts the running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality

largest cluster in subcritical percolation the statistical behavior of the size or mass of the largest cluster in subcritical percolation on a finite lattice of size n is investigated below the upper critical dimension presumably d c 6 it is argued that as stackrel ensuremath rightarrow n ensuremath infty the cumulative distribution function converges to the fisher tippett or gumbel distribution e ensuremath e ensuremath z in a certain weak sense when suitably normalized the mean grows as s ensuremath xi mathrm log n where s ensuremath xi p is a crossover size the standard deviation is bounded near s ensuremath xi ensuremath pi sqrt 6 with persistent fluctuations due to discreteness these predictions are verified by monte carlo simulations on d 2 square lattices of up to 30 million sites which also reveal finite size scaling the results are explained in terms of a flow in the space of probability distributions as stackrel ensuremath rightarrow n ensuremath infty the subcritical segment of the physical manifold 0lpl p c approaches a line of limit cycles where the flow is approximately described by a renormalization group from the classical theory of extreme order statistics

test of universal finite size scaling in two dimensional site percolation traditional two scale factor universality concerning the number of clusters within a correlation volume near the percolation threshold is reconfirmed by a comparison of site percolation on square and triangular lattices we also test universality at the threshold e g of the ratio where is the correlation length and l the lattice size at the threshold the result is sensitive to the system s shape and aspect ratio boundary conditions the algorithm the detailed units measuring and l and possibly other factors as yet unexplored

precise percolation thresholds of two dimensional random systems comprising overlapping ellipses abstract this work explores the percolation thresholds of continuum systems consisting of randomly oriented overlapping ellipses high precision percolation thresholds for various homogeneous ellipse systems with different aspect ratios are obtained from extensive monte carlo simulations based on the incorporation of vieillard baron s contact function of two identical ellipses with our efficient algorithm for continuum percolation in addition we generalize vieillard baron s contact function from identical ellipses to unequal ellipses and extend the monte carlo algorithm to heterogeneous ellipse systems where the ellipses have different dimensions and or aspect ratios based on the concept of modified excluded area a general law is verified for precise prediction of percolation threshold for many heterogeneous ellipse systems in particular the study of heterogeneous ellipse systems gains insight into the apparent percolation threshold symmetry observed earlier in systems comprising unequal circles consiglio et al 2004

shape dependent universality in percolation abstract the shape dependent universality of the excess percolation cluster number and cross configuration probability on a torus is examined besides the aspect ratio of the torus the universality class depends upon the twist in the periodic boundary conditions which may be introduced when triangular lattices are used in simulations

bond and site percolation in three dimensions we simulate the bond and site percolation models on a simple cubic lattice with linear sizes up to l 512 and estimate the percolation thresholds to be p c mathrm bond 0 248 phantom rule 0 16em 0ex 811 phantom rule 0 16em 0ex 82 10 and p c mathrm site 0 311 phantom rule 0 16em 0ex 607 phantom rule 0 16em 0ex 7 2 by performing extensive simulations at these estimated critical points we then estimate the critical exponents 1 ensuremath nu 1 1410 15 ensuremath beta ensuremath nu 0 477 phantom rule 0 16em 0ex 05 15 the leading correction exponent y i ensuremath 1 2 2 and the shortest path exponent d mathrm min 1 3756 3 various universal amplitudes are also obtained including wrapping probabilities ratios associated with the cluster size distribution and the excess cluster number we observe that the leading finite size corrections in certain wrapping probabilities are governed by an exponent ensuremath approx ensuremath 2 rather than y i ensuremath approx ensuremath 1 2

precise determination of the bond percolation thresholds and finite size scaling corrections for the sc fcc and bcc lattices extensive monte carlo simulations were performed to study bond percolation on the simple cubic s c face centered cubic f c c and body centered cubic b c c lattices using an epidemic kind of approach these simulations provide very precise values of the critical thresholds for each of the lattices pc s c 0 2488126 0 0000005 pc f c c 0 1201635 0 0000010 and pc b c c 0 1802875 0 0000010 for p close to pc the results follow the expected finite size and scaling behavior with values for the fisher exponent 2 189 0 002 the finite size correction exponent 0 64 0 02 and the scaling function exponent 0 445 0 01 confirmed to be universal

percolation threshold parameters of fluids extensive monte carlo simulations on three qualitatively different model supercritical fluids square well fluid lennard jonesium and primitive water have been performed to examine percolation threshold parameters for continuum correlated models and their relation to general results valid for random lattice models random site percolation simple cubic lattice has therefore been considered as well two different bond criteria the configurational and self bound ones defining a cluster have been used in addition to the percolation threshold occupation probability p c and the percolation threshold fluid density ensuremath rho c the correlation length exponent ensuremath nu and the wrapping probability at the percolation threshold r w c have also been evaluated it is found that parameters ensuremath nu and r w c exhibit not only strong temperature dependence but also unlike the case of lattice systems dependence on the nature of the system considered and the employed definition of the cluster

numerical results for crossing spanning and wrapping in two dimensional percolation using a recently developed method to simulate percolation on large clusters of distributed machines 1 we have numerically calculated crossing spanning and wrapping probabilities in two dimensional site and bond percolation with exceptional accuracy our results are fully consistent with predictions from conformal field theory we present many new results that await theoretical explanation particularly for wrapping clusters on a cylinder we therefore provide possibly the most up to date reference for theoreticians working on crossing spanning and wrapping probabilities in two dimensional percolation

finite size effect in seven dimensional site percolation on a hypercubic lattice with l7 sites and l up to 21 the site percolation threshold for the spanning of a cluster from top to bottom is found to approach its asymptotic value with a 1 l law in contrast to the ising curie temperature in five dimensions

lattice shapes and scaling functions for bond random percolation on honeycomb lattices we use a histogram monte carlo simulation method to calculate the scaling functions of the existence probability ep and the percolation probability p of the bond random percolation model on honeycomb lattices with aspect ratios of 0 5 and 2 0 we find that such different aspect ratios give quite different scaling functions near the critical region however they give a consistent critical point critical exponents and the thermodynamic order parameter from renormalization group calculations some interesting theoretical problems related to this work are discussed

dimension reduction for hypothesis testing in worst case scenarios this paper considers a one among many detection problem where one has to discriminate between pure noise and one among l alternatives that are known up to an amplitude factor two issues linked to high dimensionality arise in this framework first the computational complexity associated to the generalized likelihood ratio glr with the constraint of sparsity one inflates linearly with l which can be an obstacle when multiple data sets have to be tested second standard procedures based on dictionary learning aimed at reducing the dimensionality may suffer from severe power losses for some alternatives thus suggesting a worst case scenario strategy in the case where the learned dictionary has k 1 column we show that the exact solution of the resulting detection problem which can be formulated as a minimax problem can be obtained by quadratic programming because it allows a better sampling of the diversity of the alternatives the case k 1 is expected to improve the detection performances over the case k 1 the worst case analysis of this case which is more involved leads us to propose two minimax learning algorithms numerical results show that these algorithms indeed allow to increase performances over the k 1 case and are in fact comparable to the glr using the full set of alternatives while being computationally simpler

critical probability and scaling functions of bond percolation on two dimensional random lattices we locate the critical probability of bond percolation on two dimensional random lattices as because of the symmetry with respect to permutation of the two axes for random lattices we expect that for an aspect ratio of unity and sufficiently large lattices the probability of horizontal spanning equals the probability of vertical spanning this is confirmed by our monte carlo simulations we show that the ideas of universal scaling functions and non universal metric factors can be extended to random lattices by studying the existence probability and the percolation probability p on finite square planar triangular and random lattices with periodic boundary conditions using a histogram monte carlo method our results also indicate that the metric factors may be the same between random lattices and planar triangular lattices provided that the aspect ratios are 1 and

universality of 3d percolation exponents and first order corrections to scaling for conductivity exponents abstract by using a fast nested dissection algorithm we compare the results of finite size scaling at p c and of p scaling l const on large cubic random resistor networks up to 500 500 500 the p scaling for conductivity of both site and bond networks leads to an exponent t 2 00 1 the finite size scaling leads to the ratio of this conductivity exponent to the coherence length exponent t 2 283 3 combining these results we estimate 0 876 6 in excellent agreement with a value proposed by ballesteros et al the first order correctional exponent is found to be 1 0 2

detection tests using sparse models with application to hyperspectral data the problem of finding efficient methods for the detection of unknown sparse signals buried in noise is addressed we present two detection tests adapted to sparse signals based on the maximum a posteriori map estimate of the sparse vector of parameters the first is the posterior density ratio test which computes the ratio of the a posteriori distribution under each hypothesis of the data model the second is a likelihood ratio test in which the map replaces the maximum likelihood ml estimate the behaviors and the relative differences between these tests are investigated through a detailed study of their structural characteristics the proposed approaches are compared to the generalized likelihood ratio test glr showing successful results in the case of a simple model first and then for a model in which sparsity is promoted through the use of a highly redundant dictionary

boundary conditions and scaling functions of percolation models we use a histogram monte carlo simulation method to calculate the scaling functions of the existence probability ep and the percolation probability p of the site percolation model on square lattices with free and periodic boundary conditions we find that different boundary conditions give quite different scaling functions near the critical region however they give the consistent critical point critical exponents and the thermodynamic order parameter from renormalization group calculations similar results are found for other percolation models the implications of our calculated results for some theoretical problems of current interest are discussed

universality of critical existence probability for percolation on three dimensional lattices using a histogram monte carlo simulation method we calculate the existence probability for bond percolation on simple cubic sc and body centred cubic bcc lattices and site percolation on sc lattices with free boundary conditions the spanning rule considered by reynolds stanley and klein is used to define percolating clusters we find that for such systems has very good finite size scaling behaviour and the value of at the critical point is universal and is about

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

some composition determinants abstract we compute several parametric determinants in which rows and columns are indexed by compositions where the entries are either products of binomial coefficients or products of powers these results generalize previous determinant evaluations due to the first and fourth author siam j matrix anal appl 23 2001 459 471 and a polynomial generalization of the power compositions determinant linear multilinear algebra in press and they prove two conjectures of the second author advanced determinant calculus a complement preliminary version

crossing on hyperbolic lattices we divide the circular boundary of a hyperbolic lattice into four equal intervals and study the probability of a percolation crossing between an opposite pair as a function of the bond occupation probability p we consider the 7 3 heptagonal enhanced or extended binary tree ebt the ebt dual and the 5 5 pentagonal lattices we find that the crossing probability increases gradually from 0 to 1 as p increases from the lower p l to the upper p u critical values we find bounds and estimates for the values of p l and p u for these lattices and identify the self duality point p corresponding to where the crossing probability equals 1 2 comparison is made with recent numerical and theoretical results

detection of an anomalous cluster in a network we consider the problem of detecting whether or not in a given sensor network there is a cluster of sensors which exhibit an unusual behavior formally suppose we are given a set of nodes and attach a random variable to each node we observe a realization of this process and want to decide between the following two hypotheses under the null the variables are i i d standard normal under the alternative there is a cluster of variables that are i i d normal with positive mean and unit variance while the rest are i i d standard normal we also address surveillance settings where each sensor in the network collects information over time the resulting model is similar now with a time series attached to each node we again observe the process over time and want to decide between the null where all the variables are i i d standard normal and the alternative where there is an emerging cluster of i i d normal variables with positive mean and unit variance the growth models used to represent the emerging cluster are quite general and in particular include cellular automata used in modelling epidemics in both settings we consider classes of clusters that are quite general for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic by far the most popular method in practice achieves that same rate within a logarithmic factor our results are not limited to the normal location model but generalize to any one parameter exponential family when the anoma lous clusters are large enough

winding clusters in percolation on the torus and the mobius strip using a simulation technique introduced recently we study winding clusters in percolation on the torus and the mobius strip for different aspect ratios the asynchronous parallelization of the simulation makes very large system and sample sizes possible our high accuracy results are fully consistent with predictions from conformal field theory the numerical results for the mobius strip and the number distribution of winding clusters on the torus await theoretical explanation to our knowledge this study is the first of its kind

universality of the excess number of clusters and the crossing probability function in three dimensional percolation extensive monte carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three dimensional percolation on the simple cubic s c face centred cubic f c c and body centred cubic b c c lattices systems l l l0 with l0 l were studied for both bond s c f c c b c c and site f c c percolation the excess number of clusters q b per unit length was confirmed to be a universal quantity with a value q b 0 412 likewise the critical crossing probability in the l0 direction with periodic boundary conditions in thell plane was found to follow a universal exponential decay as a function of r d l 0 l for large r simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f c c and b c c lattices yielding pc f c c d 0 199 2365 0 000 0010 pc b c c d 0 245 9615 0 000 0010 we also report the value pc s c d 0 311 6080 0 000 0004 for site percolation

matched subspace detectors we formulate a general class of problems for detecting subspace signals in subspace interference and broadband noise we derive the generalized likelihood ratio glr for each problem in the class we then establish the invariances for the glr and argue that these are the natural invariances for the problem in each case the glr is a maximal invariant statistic and the distribution of the maximal invariant statistic is monotone this means that the glr test glrt is the uniformly most powerful invariant detector we illustrate the utility of this finding by solving a number of problems for detecting subspace signals in subspace interference and broadband noise in each case we give the distribution for the detector and compute performance curves

scaling and universality in the spanning probability for percolation we discuss the spanning percolation probability function for three different spanning rules in general dimensions with both free and periodic boundary conditions our discussion relies on the renormalization group theory which indicates that apart from a few scale factors the scaling functions for the spanning probability are determined by the fixed point and therefore are universal for every system with the same dimensionality spanning rule aspect ratio and boundary conditions for square and rectangular systems we combine this theory with simple relations between the spanning rules and with duality arguments and find strong relations among different derivatives of the spanning function with respect to the scaling variables thus yielding several universal amplitude ratios and allowing a systematic study of the corrections to scaling both singular and analytic in the system size the theoretical predictions are numerically confirmed with excellent accuracy textcopyright 1996 the american physical society

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

finite size scaling for percolation on elongated lattices in two and three dimensions we derive scaling laws for the percolation properties of an elongated lattice i e those with dimensions of l d ensuremath 1 ifmmode times else texttimes fi mathrm nl in d dimensions where n denotes the aspect ratio of the lattice based on statistical arguments it is shown that in the direction of the extension the percolation threshold scales approximately as mathrm ln n 1 a in both two and three dimensions extensive monte carlo simulations of the site percolation model confirm this scaling behavior it is further shown that the density of the incipient infinite cluster at the percolation threshold scales differently in two and three dimensions

universal scaling functions for site and bond percolations on planar lattices abstract universality and scaling are two important concepts in the theory of critical phenomena it is generally believed that site and bond percolations on lattices of the same dimensions have the same set of critical exponents but they have different scaling functions in this paper we briefly review our recent monte carlo results about universal scaling functions for site and bond percolation on planar lattices we find that by choosing an aspect ratio for each lattice and a very small number of non universal metric factors all scaled data of the existence probability e p and the percolation probability p for site and bond percolations on square plane triangular and honeycomb lattices may fall on the same universal scaling functions we also find that free and periodic boundary conditions share the same non universal metric factors when the aspect ratio of each lattice is reduced by the same factor the non universal metric factors remain the same the implications of such results are discussed

crossing probabilities in critical 2 d percolation and modular forms abstract crossing probabilities for critical 2 d percolation on large but finite lattices have been derived via boundary conformal field theory these predictions agree very well with numerical results however their derivation is heuristic and there is evidence of additional symmetries in the problem this contribution gives a preliminary examination some unusual modular behavior of these quantities in particular the derivatives of the horizontal and horizontal vertical crossing probabilities transform as a vector modular form one component of which is an ordinary modular form and the other the product of a modular form with the integral of a modular form we include consideration of the interplay between conformal and modular invariance that arises

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

generation of percolation cluster perimeters by a random walk a type of self avoiding random walk whish generates the perimeter of two dimensional lattice percolation clusters is given the algorithm has been simulated on a computer yielding the mean perimeter length as a function of occupation probability

surface and edge exponents for the spreading of 3d percolation monte carlo results are presented for the spreading of 3d percolation where the seed consists of an infinite straight line this line can either be in the bulk of the material into which spreading is possible on a planar surface of the material or on a rectangular edge in the last two cases spreading occurs only into angular regions with 180 degrees resp 90 degrees while the mean distance grows at p pc in all three cases with the same exponent the average number of growth sites grows resp vanishes in all three cases with different exponents in particular this implies that at p pc the sites on the surface of a big cube belonging to the maximal cluster within that cube have fractal dimension 1

distributions and power of optimal signal detection statistics in finite case for detecting weak and sparse signals by a set of n input p values the higher criticism hc type statistics the berk jones b j type statistics and the phi divergence statistics have the equivalent asymptotic optimality as n goes to infinity however they can have significantly different performance in practical data analysis where n is always finite and even very small to address this problem in a broader context this paper introduces a general family of goodness of fit statistics called the ggof which unifies a broad signal detection statistics including these optimal ones efficient and accurate analytical calculations for the distributions of the ggof statistics are provided under arbitrary i i d continuous models of the null and the alternative hypotheses based on that a systematic power study reveals that in finite case the number of signals is often more relevant than the signal proportion the hc and the reverse hc have advantages for relatively sparser and denser signals respectively while the b j is more robust a general framework is given to apply the ggof into data analysis based on the generalized linear models an application to the snp set based genome wide association study gwas for crohn s disease shows that these optimal statistics have a good potential for detecting novel disease genes with weak snp effects the calculations have been implemented into an r package settest and published on the cran

percolation of the site random cluster model by monte carlo method we propose a site random cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation to simulate the model on a square lattice we combine the color assignation and the swendsen wang methods to design a highly efficient cluster algorithm with a small critical slowing down phenomenon to verify whether or not it is consistent with the bond random cluster model we measure several quantities such as the wrapping probability re the percolating cluster density p and the magnetic susceptibility per site p as well as two exponents such as the thermal exponent yt and the fractal dimension yh of the percolating cluster we find that for different exponents of cluster weight q 1 5 2 2 5 3 3 5 and 4 the numerical estimation of the exponents yt and yh are consistent with the theoretical values the universalities of the site random cluster model and the bond random cluster model are completely identical for larger values of q we find obvious signatures of the first order percolation transition by the histograms and the hysteresis loops of percolating cluster density and the energy per site our results are helpful for the understanding of the percolation of traditional statistical models

effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory we investigate the effects of anisotropy on the finite size scaling of connectivity and conductivity of continuum percolation in three dimensions we consider a system of size x ifmmode times else texttimes fi y ifmmode times else texttimes fi z in which cubic bodies of size a ifmmode times else texttimes fi b ifmmode times else texttimes fi c are placed randomly we define two aspect ratios to request anisotropy then we expect that the displacement of average connected fraction p averaged over the realizations about the isotropic universal curves will be a function of the two aspect ratios this is accounted by considering an apparent percolation threshold in each direction which leads to 50 of realizations connecting in that direction we find the aspect ratios dependency of the apparent threshold and investigate the finite size scaling transformations for the mean connected fraction and its associated fluctuations moreover we apply a single phase pressure solver to determine the conductivity of various realizations of the system finally we apply the same idea to account for the effect of anisotropy on the conductivity scaling

percolation thresholds and universal formulas a calculation of percolation thresholds of 11 two dimensional and 18 three dimensional lattices is presented among the three dimensional ones are a random lattice and its dual plus a number of aniso tropic regular lattices the results are used to test universal formulas that relate the percolation thresholds of lattices to their dimension and coordination number the evidence suggests that dimension and coordination number are not sufficient to predict percolation thresholds

high temperature series expansions for the q state potts model on a hypercubic lattice and critical properties of percolation we present results for the high temperature series expansions of the susceptibility and free energy of the q state potts model on a d dimensional hypercubic lattice zd for arbitrary values of q the series are up to order 20 for dimension d 1 limit of these series we estimate the percolation threshold pc and critical exponent gamma for bond percolation in different dimensions we also extend the 1 d expansion of the critical coupling for arbitrary values of q up to order d 9

advanced determinant calculus a complement abstract this is a complement to my previous article advanced determinant calculus c krattenthaler advanced determinant calculus seminaire lotharingien combin 42 1999 the andrews festschrift article b42q 67 pp in the present article i share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory g almkvist c krattenthaler j petersson some new formulas for experiment math 12 2003 441 456 moreover i add a list of determinant evaluations which i consider as interesting which have been found since the appearance of the previous article or which i failed to mention there including several conjectures and open problems

similarity of percolation thresholds on the hcp and fcc lattices extensive monte carlo simulations were performed in order to determine the precise values of the critical thresholds for site phcpc s 0 199 255 5 0 000 001 0 and bond phcpc b 0 120 164 0 0 000 001 0 percolation on the hcp lattice to compare with previous precise measurements on the fcc lattice also exact enumeration of the hcp and fcc lattices was performed and yielded generating functions and series for the zeroth first and second moments of both lattices when these series and the values of pc are compared to those for the fcc lattice it is apparent that the site percolation thresholds are different however the bond percolation thresholds are equal within error bars and the series only differ slightly in the higher order terms suggesting the actual values are very close to each other if not identical

minireview new results for old percolation abstract percolation theory deals with the formation of clusters in disordered media at the percolation threshold for the first time an infinite cluster appears however research of the last years showed that at this phase transition some old ideas were wrong there can also be two or three clusters spanning from top to bottom even in large two dimensional lattices and the probability of spanning is not given by simple real space renormalization ideas

on position space renormalization group approach to percolation in a position space renormalization group psrg approach to percolation one calculates the probabilityr p b that a finite lattice of linear sizeb percolates wherep is the occupation probability of a site or bond a sequence of percolation thresholdsp c b is then estimated fromr p c b p c b and extrapolated to the limitb to obtainp c p c recently it was shown that for a certain spanning rule and boundary condition r p c r c is universal and sincep c is not universal the validity of psrg approaches was questioned we suggest that the equationr p c b where isany number in 0 1 provides a sequence ofp c b s thatalways converges top c asb thus there is anenvelope from any point inside of which one can converge top c however the convergence is optimal if r c by calculating the fractal dimension of the sample spanning cluster atp c we show that the same is true aboutany critical exponent of percolation that is calculated by a psrg method thus psrg methods are still a useful tool for investigating percolation properties of disordered systems

excess number of percolation clusters on the surface of a sphere abstract monte carlo simulations were performed in order to determine the excess number of clusters b and the average density of clusters nc for the two dimensional swiss cheese continuum percolation model on a planar l l system and on the surface of a sphere the excess number of clusters for the l l system was confirmed to be a universal quantity with a value b 0 8841 as previously predicted and verified only for lattice percolation the excess number of clusters on the surface of a sphere was found to have the value b 1 215 1 for discs with the same coverage as the flat critical system finally the average critical density of clusters was calculated for continuum systems as nc 0 04075 5

simple cubic random site percolation thresholds for neighborhoods containing fourth nearest neighbors in this paper random site percolation thresholds for a simple cubic sc lattice with site neighborhoods containing next next next nearest neighbors 4nn are evaluated with monte carlo simulations a recently proposed algorithm with low sampling for percolation thresholds estimation bastas et al arxiv 1411 5834 is implemented for the studies of the top bottom wrapping probability the obtained percolation thresholds are p c 4nn 0 31160 12 p c 4nn nn 0 15040 12 p c 4nn 2nn 0 15950 12 p c 4nn 3nn 0 20490 12 p c 4nn 2nn nn 0 11440 12 p c 4nn 3nn nn 0 11920 12 p c 4nn 3nn 2nn 0 11330 12 and p c 4nn 3nn 2nn nn 0 10000 12 where 3nn 2nn and nn stand for next next nearest neighbors next nearest neighbors and nearest neighbors respectively as an sc lattice with 4nn neighbors may be mapped onto two independent interpenetrated sc lattices but with a lattice constant that is twice as large the percolation threshold p c 4nn is exactly equal to p c nn the simplified method of bastas et al allows for uncertainty of the percolation threshold value p c to be reached similar to that obtained with the classical method but ten times faster

potts and percolation models on bowtie lattices we give the exact critical frontier of the potts model on bowtie lattices for the case of q 1 the critical frontier yields the thresholds of bond percolation on these lattices which are exactly consistent with the results given by ziff et al j phys a 39 15083 2006 for the q 2 potts model on a bowtie a lattice the critical point is in agreement with that of the ising model on this lattice which has been exactly solved furthermore we do extensive monte carlo simulations of the potts model on a bowtie a lattice with noninteger q our numerical results which are accurate up to seven significant digits are consistent with the theoretical predictions we also simulate the site percolation on a bowtie a lattice and the threshold is s c 0 5479148 7 in the simulations of bond percolation and site percolation we find that the shape dependent properties of the percolation model on a bowtie a lattice are somewhat different from those of an isotropic lattice which may be caused by the anisotropy of the lattice

on cardy s formula for the critical crossing probability in 2d percolation cardy s formula for the probability r of crossing a rectangular critical percolation system in the vertical direction with free boundaries on the two sides is written explicitly in terms of the aspect ratio r heightlwidth the first three terms are given by n 24 33i r 4 2 e r 3 e 7n 13 e 3zr 3 i which with n 1 nv i r m sufficient to determine x to high accuracy for all r

the number of incipient spanning clusters in two dimensional percolation using methods of conformal field theory we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k i n the limit when k is large the study of the structure of large clusters at the percolation threshold continues to pose interesting problems whose solution sheds light on the nature of the critical state in general recently some attention has been paid to incipient spanning clusters iscs these are clusters which connect two disjoint segments of the boundary of a macroscopically large region langlands et al 1 conjectured that the probability that at least one such cluster exists that is that the segments are connected is invariant under conformal transformations this statement was placed in the context of conformal field theory in 2 where an explicit formula was given for this crossing probability for a review of the status of conformal invariance in critical percolation see 3 more recently aizenman 4 has considered among other things the probability that there exist n distinct iscs connecting the two segmentsy in the case of a rectangular region 0 kl 0 l he has proved that the probability p n k l that the strip is traversed in the direction of length kl b y nindependent clusters satisfies the bounds a e n 2 k

are the tails of percolation thresholds gaussians the probability distribution of percolation thresholds in finite lattices was first believed to follow a normal gaussian behaviour with increasing computer power and more efficient simulational techniques this belief turned to a stretched exponential behaviour instead here based on a further improvement of the monte carlo data we show evidence on square lattices that this question is numerically not yet answered at all

a crossing probability for critical percolation in two dimensions langlands et al considered two crossing probabilities and in their extensive numerical investigations of critical percolation in two dimensions cardy was able to find the exact form of by treating it as a correlation function of boundary operators in the limit of the q state potts model we extend his results to find an analogous formula for which compares very well with the numerical results