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

fractals in surface growth with power law noise the authors present a microscopic description of interface growth with power law noise distribution in the form p eta eta 1 approximately 1 mu which exhibits non universal roughening for the mu d 1 case in d 1 dimensions the existence of a fractal pattern in the bulk of the aggregate is explained leading trivially to the proof of the identity alpha z 2 for the roughening and the dynamical scaling exponents alpha and z respectively investigations on the distribution of step sizes of the discretized interface and the saturated growth speed further support the arguments

the loewner equation maps and shapes an approach called schramm loewner evolution sle provides a new method for dealing with a wide variety of scale invariant problems in two dimensions this approach is based upon an older method called loewner evolution le which connects analytic and geometrical constructions in the complex plane in this paper the bases of le and sle are described and some simple applications are discussed in relatively non technical form a bibliography of the subject is presented

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

classification of doubly periodic minimal surfaces of genus zero we prove that if the quotient surface of a properly embedded doubly periodic minimal surface in 3 has genus zero then the surface is one of the classical scherk examples

non gaussian limiting behavior of the percolation threshold in a large system we study short range percolation models in a finite box we define the percolation threshold as a random variable obtained from a stochastic procedure used in actual numerical calculations and study the asymptotic behavior of these random variables as the size of the box goes to infinity we formulate very general conditions under which in two dimensions rescaled threshold variables cannot converge to a gaussian and determine the asymptotic behavior of their second moments in terms of a widely used definition of correlation length we also prove that in all dimensions the finite volume percolation thresholds converge in probability to the percolation threshold of the infinite system the convergence result is obtained by estimating the rate of decay of the limiting distribution function s tail in terms of the correlation length exponent the proofs use exponential estimates of crossing probabilities substantial parts of the proofs apply in all dimensions

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

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools

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

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

random positive operator valued measures we introduce several notions of random positive operator valued measures povms and we prove that some of them are equivalent we then study statistical properties of the effect operators for the canonical examples starting from the limiting eigenvalue distribution we derive the large system limit for several quantities of interest in quantum information theory such as the sharpness the noise content and the probability range finally we study different compatibility criteria and we compare them for generic povms

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

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

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

extrinsic killing spinors under intrinsic and extrinsic curvature assumptions on a riemannian spin manifold and its boundary we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic killing spinors of non negative killing constant as a corollary we prove that a complete ricci flat spin manifold with mean convex boundary isometric to a round sphere is necessarily a flat disc

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

complete diagrammatics of the single ring theorem using diagrammatic techniques we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices the formalism allows us to map two distinct areas of free random variables hermitian positive definite operators and non normal r diagonal operators we also rederive the haagerup larsen theorem and show how its recent extension to the eigenvector correlation function appears naturally within this approach

correlation length and its critical exponent for percolation processes some critical exponent inequalities are given involving the correlation length of site percolation processes on d in particular it is shown thatv 2 d which implies that the critical exponentv cannot take its mean field value for the three dimensional percolation processes

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

critical percolation on the torus we compute the various crossing probabilities defined by r langlands p pouliot and y saint aubin for the critical percolation on the torus

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

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

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

resolvents of r diagonal operators we consider the resolvent a 1 of any diagonal operator a in a ii 1 factor our main theorem theorem 1 1 gives a universal asymptotic formula for the norm of such a resolvent en route to its proof we calculate the transform of the operator c 2 where c is voiculescu s circular operator and we give an asymptotic formula for the negative moments of a 2 for any diagonal a we use a mixture of complex analytic and combinatorial techniques each giving finer information where the other can give only coarse detail in particular we introduce partition structure diagrams in section 4 a new combinatorial structure arising in free probability

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

immersed hypersurfaces with constant weingarten curvature in a recent paper korevaar 1 5 used the alexandrov reflection principle to show that closed embedded hypersurfaces in n 1 lhn 1 or the upper hemisphere of s 1 are umbilic spheres provided a certain function f of the principal curvatures t t 22 is constant he only had to assume that f is positive on the positive cone 212i 0 vi and that f is elliptic on the component f of 21 f 0 which contains here f is said to be elliptic if 0f d21 0 for all i 1 inn this generalization of earlier sphere theorems see 9 and 10 for references cannot be extended to hypersurface immersions in view of recent counterexamples 1 11 however assuming additional curvature conditions walter derived in 10 global results for hypersurface immersions in a space n l c of constant curvature c which have a constant higher mean curvature function h here h is the r th symmetric function of the principal curvatures it was shown that such hypersurfaces are of constant mean curvature h1 provided they have nonnegative sectional curvature and non negative principal curvatures as a consequence they have to be isoparametric with at most two distinct principal curvatures and can therefore be completely classified here we show that it is not necessary to assume all principal curvatures to be non negative moreover we extend waiter s result to general symmetric functions f f 2 let m be a smooth connected and compact manifold without boundary then we have the following result

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

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

almost all quantum channels are equidistant in this work we analyze properties of generic quantum channels in the case of large system size we use random matrix theory and free probability to show that the distance between two independent random channels converges to a constant value as the dimension of the system grows larger as a measure of the distance we use the diamond norm in the case of a flat hilbert schmidt distribution on quantum channels we obtain that the distance converges to 12 2 giving also an estimate for the maximum success probability for distinguishing the channels we also consider the problem of distinguishing two random unitary rotations in this work we analyze properties of generic quantum channels in the case of large system size we use random matrix theory and free probability to show that the distance between two independent random channels converges to a constant value as the dimension of the system grows larger as a measure of the distance we use the diamond norm in the case of a flat hilbert schmidt distribution on quantum channels we obtain that the distance converges to 12 2 giving also an estimate for the maximum success probability for distinguishing the channels we also consider the problem of distinguishing two random unitary rotations

r transforms of free joint distributions and non crossing partitions abstract we introduce an extension of the notion of r transform defined by d voiculescu to free joint distributions i e normalized linear functionals on algebras of non commutative polynomials in several indeterminates we point out that the r transform has good behavior with respect to the operations of free product and free convolution of joint distributions we prove that the explicit computation of the inverse r transform of a joint distribution is done via a formula of summation over the lattice of non crossing partitions which shows that the r transform is the operator theoretic counterpart of the free cumulants considered in the combinatorial approach of r speicher moreover in connection to the equivariance of the multidimensional r transform under rotations we show that a natural free analogue of a classical result about rotations of independent random variables is holding

the distribution of percolating concentrations in finite systems abstract simulation of finite cubic lattices suggests that even in the limit of large systems the distribution of percolation thresholds deviates from a gaussian thus confirming ziff s two dimensional result

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

on the asymptotic distribution of the commutator and anticommutator of random matrices we study in the limit of infinite matrix order the normalized eigenvalue counting measures of the commutator and anticommutator of two hermitian or real symmetric matrices rotated independently one respect to another by the random unitary or orthogonal haar distributed matrix we establish the convergence with probability 1 to a limiting nonrandom measure we obtain and analyze the functional equations for the stieltjes transforms of the limiting measures

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

the o n model on the annulus we use coulomb gas methods to derive an explicit form for the scaling limit of the partition function of the critical o n model on an annulus with free boundary conditions as a function of its modulus this correctly takes into account the magnetic charge asymmetry and the decoupling of the null states it agrees with an earlier conjecture based on bethe ansatz and quantum group symmetry and with all known results for special values of n it gives new formulae for percolation the probability that a cluster connects the two opposite boundaries and for self avoiding loops the partition function for a single loop wrapping non trivially around the annulus the limit n 0 also gives explicit examples of partition functions in logarithmic conformal field theory

strong haagerup inequalities for free r diagonal elements abstract in this paper we generalize haagerup s inequality u haagerup an example of a nonnuclear c algebra which has the metric approximation property invent math 50 1978 1979 279 293 on convolution norm in the free group to a very general context of r diagonal elements in a tracial von neumann algebra moreover we show that in this holomorphic setting the inequality is greatly improved from its original form we give combinatorial proofs of two important special cases of our main result and then generalize these techniques en route we prove a number of moment and cumulant estimates for r diagonal elements that are of independent interest finally we use our strong haagerup inequality to prove a strong ultracontractivity theorem generalizing and improving the one in p biane free hypercontractivity comm math phys 184 1997 457 474

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

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

crossing formulae for critical percolation in an annulus an exact formula is given for the probability that there exists a spanning cluster between opposite boundaries of an annulus in the scaling limit of critical percolation the entire distribution function for the number of distinct spanning clusters is also derived these results are found using coulomb gas methods their forms are compared with the expectations of conformal field theory

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

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

spectrum of the product of independent random gaussian matrices we show that the eigenvalue density of a product x x1x2 xm of m independent nxn gaussian random matrices in the limit n infinity is rotationally symmetric in the complex plane and is given by a simple expression rho z z 1 mpisigma 2 m z 2 2 m for z sigma the parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the gaussian fluctuations this form of the eigenvalue density is highly universal it is identical for products of gaussian hermitian non hermitian and real or complex random matrices it does not change even if the matrices in the product are taken from different gaussian ensembles we present a self contained derivation of this result using a planar diagrammatic technique additionally we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill pastur lindeberg s condition we provide a numerical evidence supporting this conjecture

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

r diagonal dilation semigroups this paper addresses extensions of the complex ornstein uhlenbeck semigroup to operator algebras in free probability theory if a 1 a k are free fancyscript r diagonal operators in a ii1 factor then d t a i 1 cdots a i n e nt a i 1 cdots a i n defines a dilation semigroup on the non self adjoint operator algebra generated by a 1 a k we show that d t extends in two different ways to a semigroup of completely positive maps on the von neumann algebra generated by a 1 a k moreover we show that d t satisfies an optimal ultracontractive property d t colon l 2 to l infty sim t 1 for small t 0

the probability distribution of the percolation threshold in a large system we show that the distribution of the percolation threshold in a large finite system does not converge to a gaussian when the size of the system goes to infinity provided that the two widely accepted definitions of correlation length are equivalent the shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length the result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v all results are rigorously proven in the 2d case generalizations for three dimensions are also discussed

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

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

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

sample variance in free probability abstract let x 1 x 2 x n denote i i d centered standard normal random variables then the law of the sample variance q n i 1 n x i x 2 is the 2 distribution with n 1 degrees of freedom it is an open problem in classical probability to characterize all distributions with this property and in particular whether it characterizes the normal law in this paper we present a solution of the free analogue of this question and show that the only distributions whose free sample variance is distributed according to a free 2 distribution are the semicircle law and more generally so called odd laws by which we mean laws with vanishing higher order even cumulants in the way of proof we derive an explicit formula for the free cumulants of q n which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of r cyclicity

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

hypersurfaces with constant scalar curvature in space forms let rn l c be an n 1 dimensional riemannian manifold with constant sectional curvature c we also call it space form when c 1 r l c s l i e n 1 dimensional unit sphere space when c o rn t c e n l i e n 1 dimensional euclidean space let m be an n dimensional compact hypersurface in rn l c and el e a local orthonormal frame field on m col con its dual coframe field then the second fundamental form of m is

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

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