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

scaling of surface fluctuations and dynamics of surface growth models with power law noise the authors report the results of extensive simulations of three surface growth models in the kpz universality class with a power law noise distribution of the form p eta approximately eta mu 1 in d 2 for all three models the width and height fluctuation distributions obey a scaling relation and exhibit power law tails this has implications for directly observing the presence of power law noise in experiments in addition they present extensive results for both scaling exponents alpha and beta as a function of mu for mu 2 8 although there is some variation in the results for different models for all three models they find anomalous exponents for 2 or mu or 7 while the scaling relation alpha z 2 holds for all mu for a linear model without sideways growth they verify that the presence of power law noise does not lead to anomalous exponents for mu 2

rational extensions of the trigonometric darboux poschl teller potential based on para jacobi polynomials the possibility for the jacobi equation to admit in some cases general solutions that are polynomials has been recently highlighted by calogero and yi who termed them para jacobi polynomials such polynomials are used here to build seed functions of a darboux backlund transformation for the trigonometric darboux poschl teller potential as a result one step regular rational extensions of the latter depending both on an integer index n and on a continuously varying parameter are constructed for each n value the eigenstates of these extended potentials are associated with a novel family of dependent polynomials which are orthogonal on 1 1

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

propagator for finite range potentials the schrodinger equation in integral form is applied to the one dimensional scattering problem in the case of a general finite range nonsingular potential a simple expression for the laplace transform of the transmission propagator is obtained in terms of the associated fredholm determinant by means of matrix methods the particular form of the kernel and the peculiar aspects of the transmission problem play an important role the application to an array of delta potentials is shown

dynamic scaling of the interface in two phase viscous flows in porous media the time evolution and geometry of rough interfaces observed in experiments on immiscible displacement of viscous fluids in porous media are analysed using the concepts of dynamic scaling and self affine fractal geometry the authors find that the development of the interface is governed by dynamic scaling and they have determined the corresponding surface exponents alpha and beta the values of the exponents calculated for the experimental patterns are different from those obtained for a variety of two dimensional models of marginally stable interfaces

exact scaling functions for self avoiding loops and branched polymers it is shown that a recently conjectured form for the critical scaling function for planar self avoiding polygons weighted by their perimeter and area also follows from an exact renormalization group flow into the branched polymer problem combined with the dimensional reduction arguments of parisi and sourlas the result is generalized to higher order multicritical points yielding exact values for all their critical exponents and exact forms for the associated scaling functions

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

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

generalized quantum nonlinear oscillators exact solutions and rational extensions we construct exact solutions and rational extensions to quantum systems of generalized nonlinear oscillators our method is based on a connection between nonlinear oscillator systems and schrodinger models for certain hyperbolic potentials the rationally extended models admit discrete spectrums and corresponding closed form solutions are expressed through jacobi type xm exceptional orthogonal polynomials

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

on crossing event formulas in critical two dimensional percolation several formulas for crossing functions arising in the continuum limit of critical two dimensional percolation models are studied these include watts s formula for the horizontal vertical crossing probability and cardy s new formula for the expected number of crossing clusters it is shown that for lattices where conformal invariance holds they simplify when the spatial domain is taken to be the interior of an equilateral triangle the two crossing functions can be expressed in terms of an equianharmonic elliptic function with a triangular rotational symmetry this suggests that rigorous proofs of watts s formula and cardy s new formula will be easiest to construct if the underlying lattice is triangular the simplification in a triangular domain of schramm s bulk cardy s formula is also studied

on linear equations with general polynomial solutions abstract we provide necessary and sufficient conditions for which an nth order linear differential equation has a general polynomial solution we also give necessary conditions that can directly be ascertained from the coefficient functions of the equation

on the bogomolny schmit conjecture bogomolny and schmit proposed that the critical edge percolation on the square lattice is a good model for the nodal domains of a random plane wave based on this they made a conjecture about the number of nodal domains recent computer experiments showed that the mean number of clusters per vertex and the mean number of nodal domains per unit area are very close but different since the original argument was mostly supported by numerics it was believed that the percolation model is wrong in this paper we give some numerical evidence in favour of the percolation model

toward a conformal field theory for schramm loewner evolutions we discuss the partition function point of view for chordal schramm loewner evolutions and their relationship with correlation functions in conformal field theory both are closely related to crossing probabilities and interfaces in critical models in two dimensional statistical mechanics we gather and supplement previous results with different perspectives point out remaining difficulties and suggest directions for future studies

logarithmic operators and logarithmic conformal field theories logarithmic operators and logarithmic conformal field theories are reviewed prominent examples considered here include c 2 and c 0 logarithmic conformal field theories c 0 logarithmic conformal field theories are especially interesting since they describe some of the critical points of a variety of longstanding problems involving a two dimensional quantum particle moving in a spatially random potential as well as critical two dimensional self avoiding random walks and percolation lack of classification of logarithmic conformal field theories remains a major impediment to progress towards finding complete solutions to these problems

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

logarithmic correlators in nonrelativistic conformal field theory we show how logarithmic terms may arise in the correlators of fields which belong to the representation of the schrodinger virasoro algebra or the affine galilean conformal algebra gca we show that in gca only scaling operator can have a jordan form and rapidity cannot we observe that in both algebras logarithmic dependence appears along the time direction alone

logarithmic operator intervals in the boundary theory of critical percolation we consider the sub sector of the c 0 logarithmic conformal field theory lcft generated by the boundary condition changing bcc operator in two dimensional critical percolation this operator is the zero weight kac operator 1 2 identified with the growing hull of the sle6 process we identify percolation configurations with the significant operators in the theory we consider operators from the first four bcc operator fusions the identity and the stress tensor and its logarithmic partner and its logarithmic partner and the pre logarithmic operator 1 3 we construct several intervals in the percolation model each associated to one of the lcft operators we consider allowing us to calculate crossing probabilities and expectation values of crossing cluster numbers we review the cg which we use as a method of calculating these quantities when the number of bcc operator makes a direct solution to the system of differential equations intractable finally we discuss the case of the six point correlation function which applies to crossing probabilities between the sides of a conformal hexagon specifically we introduce an integral result that allows one to identify the probability that a single percolation cluster touches three alternating sides a hexagon with free boundaries we give results of the numerical integration for the case of a regular hexagon

conformal invariance of voronoi percolation it is proved that in the voronoi model for percolation in dimension 2 and 3 the crossing probabilities are asymptotically invariant under conformal change of metric

families of strictly isospectral potentials for any potential v x which holds n bound states the authors use repeated supersymmetry transformations to construct an n parameter family of strictly isospectral potentials identical eigenenergies reflection and transmission amplitudes for all family members they investigate how the shape and behaviour of various isospectral potentials changes as the n parameters are varied contained in this family are potentials with n widely separated wells each of which holds a single bound state

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

a triangle of interconnected coagulation models a number of new coagulation models depending on a parameter is derived the dependence is considered in two different ways if the parameter takes its maximal value then in the first case we obtain a new discrete kinetic equation we demonstrate that its continuous version is simply the oort hulst s coagulation model in the second case the maximum of the corresponding parameter yields the smoluchowski coagulation equation at the minimal values of both parameters we arrive at another new kinetic equation these three models form a triangle connecting the two known coagulation equations situated in its vertexes smoluchowski and oort hulst equations via an alteration of the parameters also a comparative analysis of these three models is presented as an advantage of the oort hulst approach we compute the coagulation front and establish a connection between the infringement of the mass conservation law and convergence of the coagulation front to infinity

series studies of the potts model i the simple cubic ising model the finite lattice method of series expansion is generalized to the q state potts model on the simple cubic lattice it is found that the computational effort grows exponentially with the square of the number of series terms obtained unlike two dimensional lattices where the computational requirements grow exponentially with the number of terms for the ising q 2 case the authors have extended the low temperature series for the partition functions magnetization and zero field susceptibility to u26 from u20 the high temperature series for the zero field partition function is extended from nu 18 to nu 22 subsequent analysis gives critical exponents in agreement with those from field theory

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

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

the logarithmic triplet theory with boundary the boundary theory for the c 2 triplet model is investigated in detail in particular we show that there are four different boundary conditions that preserve the triplet algebra and check the consistency of the corresponding boundary operators by constructing their ope coefficients explicitly we also compute the correlation functions of two bulk fields in the presence of a boundary and verify that they are consistent with factorization

radford drinfeld and cardy boundary states in the 1 p logarithmic conformal field models we introduce p 1 pseudocharacters in the space of 1 p model vacuum torus amplitudes to complete the distinguished basis in the 2p dimensional fusion algebra to a basis in the whole 3p 1 dimensional space of torus amplitudes and the structure constants in this basis are not necessarily non negative integer numbers we obtain a generalized verlinde formula that gives these structure constants in the context of theories with boundaries we identify the space of vacuum torus amplitudes with the space of ishibashi states then we propose 3p 1 boundary states satisfying the cardy condition

disconjugacy regularity of multi indexed rationally extended potentials and laguerre exceptional polynomials the power of the disconjugacy properties of second order differential equations of schrodinger type to check the regularity of rationally extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re examining the extensions of the isotonic oscillator or radial oscillator potential derived in kth order supersymmetric quantum mechanics or multistep darboux backlund transformation method the function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term whose value is calculated by induction over k the sign of this term being the same as that of the already known highest degree term the potential denominator has the same sign at both extremities of the definition interval a property that is shared by the seed eigenfunction used in the potential construction by virtue of disconjugacy such a property implies the nodeless character of both the eigenfunction and the resulting potential

kazhdan lusztig dual quantum group for logarithimic extensions of virasoro minimal models we derive and study a quantum group gp q that is kazhdan lusztig dual to the w algebra wp q of the logarithmic p q conformal field theory model the algebra wp q is generated by two currents w z and w z of dimension 2p 1 2q 1 and the energy momentum tensor t z the two currents generate a vertex operator ideal r with the property that the quotient wp q r is the vertex operator algebra of the p q virasoro minimal model the number 2pq of irreducible gp q representations is the same as the number of irreducible wp q representations on which r acts nontrivially we find the center of gp q and show that the modular group representation on it is equivalent to the modular group representation on the wp q characters and pseudocharacters the factorization of the gp q ribbon element leads to a factorization of the modular group representation on the center we also find the gp q grothendieck ring which is presumably the logarithmic fusion of the p q model

pair connectedness and shortest path scaling in critical percolation we present high statistics data on the distribution of shortest path lengths between two near by points on the same cluster at the percolation threshold our data are based on a new and very efficient algorithm for d 2 they clearly disprove a recent conjecture by m porto et al 1998 phys rev e 58 r5205 our data also provide upper bounds on the probability that two near by points are on different infinite clusters

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

propagator for narrow potential barriers the schrodinger equation in integral form is applied to the one dimensional scattering problem in the case of a narrow potential barrier since the kernel can be considered in a first approximation separable an explicit expression for the propagator is found by means of the complementary error function the problem of a particle confined in a half space and interacting with a narrow potential barrier is also considered and solved in an approximate way

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

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

propagator for finite range potentials the case of reflection following a previous study on the transmission propagator for a finite range potential the problem of reflection is considered it is found that the laplace transform of the reflection propagator can be expressed in terms of the usual fredholm determinant and of a new similar determinant containing the peculiar characteristics of reflection as an example an array of delta potentials is considered moreover a possible application to the calculation of quantum traversal time is shown

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

a direct electrifying algorithm for backbone identification this paper proposes a new algorithm for identifying backbones in the application of percolation theory this algorithm is based on the current carrying definition of backbone and is carried out on the predetermined spanning cluster it is fairly easy to implement and further parallelize the efficiency is enhanced by the fact that the conductivity of a percolating system can be obtained in the same processing of backbone identification the critical exponents of backbone mass red bonds sites and conductivity obtained by this algorithm are in very good agreement with the existing results

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

on a class of integral equations having application in quantum dynamics a class of fredholm integral equations of the second kind is studied with kernel separable outside the basic interval a b using theorems of matrix algebra the solution for x outside a b is found in terms of the fredholm determinants in a simple and compact form as a particular case the quantum propagator for one dimensional problems is obtained

virasoro representations and fusion for general augmented minimal models in this paper we present explicit results for the fusion of irreducible and higher rank representations in two logarithmically conformal models the augmented c2 3 0 model as well as the augmented yang lee model at c2 5 22 5 we analyse their spectrum of representations which is consistent with the symmetry and associativity of the fusion algebra we also describe the first few higher rank representations in detail in particular we present the first examples of consistent rank 3 indecomposable representations and describe their embedding structure knowing these two generic models we also conjecture the general representation content and fusion rules for general augmented cp q models

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

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

extended krein adler theorem for the translationally shape invariant potentials considering successive extensions of primary translationally shape invariant potentials we enlarge the krein adler theorem to mixed chains of state adding and state deleting darboux backlund transformations it allows us to establish novel bi linear wronskian and determinantal identities for classical orthogonal polynomials

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

a linear second order ode with only polynomial solutions abstract a linear second order ode featuring an arbitrary number of free parameters is identified all solutions of which are polynomials

can the general solution of the second order ode characterizing jacobi polynomials be polynomial the answer to the question posed in the title is affirmative indeed while the second order ode 1 x2 p 2 x p n n 1 p 0 for generic values of the two parameters and and n a non negative integer identifies uniquely up to an overall multiplicative constant characterizing its normalization the jacobi polynomial p n x as its polynomial solution if instead n m and n n with m and n being two non negative integers satisfying the restriction m n n then the general solution of this ode is a polynomial of degree n depending in addition of course to the three parameters n m and n on two arbitrary parameters i e on an additional one besides the overall multiplicative parameter characterizing its normalization

conformal invariance of a model of percolation on random lattices abstract a new model of percolation on random lattices is introduced it allows for simulations on geometries like the sphere for which there do not exist periodic lattices of arbitrary finess horizontal crossing probabilities h are measured on rectangles of various aspect ratios r these measurements agree with cardy s prediction though there are small discrepancies for rectangles with large aspect ratio further crossing probabilities are measured on a cylinder these probe the hypothesis of conformal invariance stated in bull ams 30 1994 1

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 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

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

exceptional orthogonal polynomials exactly solvable potentials and supersymmetry we construct two new exactly solvable potentials giving rise to bound state solutions to the schrodinger equation which can be written in terms of the recently introduced laguerre or jacobi type x1 exceptional orthogonal polynomials these potentials extending either the radial oscillator or the scarf i potential by the addition of some rational terms turn out to be translationally shape invariant as their standard counterparts and isospectral to them

on two dimensional percolation we present new series data for both high and low density bond and site percolation on the square lattice the series have been obtained by the finite lattice method and in all cases extend pre existing series an analysis of these series gives refined estimates of critical points critical exponents and amplitudes for bond and site animals and for the percolation probability and mean size exponents

cluster analysis of the ising model and universal finite size scaling the recent progress in the study of finite size scaling fss properties of the ising model is briefly reviewed we calculate the universal fss functions for the binder parameter g and the magnetization distribution function p m for the ising model on l1 l2 two dimensional lattices with tilted boundary conditions we show that the fss functions are universal for fixed sets of the aspect ratio l1 l2 and the tilt parameter we also study the percolating properties of the ising model giving attention to the effects of the aspect ratio of finite systems we elucidate the origin of the complex structure of p m for the system with large aspect ratio by the multiple percolating cluster argument

universal scaling functions and quantities in percolation models abstract we briefly review recent work on universal finite size scaling functions ufssfs and quantities in percolation models the topics under discussion include a ufssfs for the existence probability also called crossing probability e p the percolation probability p and the probability w n of the appearance of n percolating clusters b universal slope for average number of percolating clusters c ufssfs for a q state bond correlated percolation model corresponding to the q state potts model we also briefly mention some very recent related developments and discuss implications of our results

renormalization group scaling and universality in spanning probability for percolation abstract the spanning probability function for percolation is discussed using ideas from the renormalization group theory we find that apart from a few scale factors the scaling functions are determined by the fixed point and therefore are universal for every system with the same dimensionality spanning rule aspect ratio and boundary conditions being independent of lattice structure and finite interaction length this yields general results concerning the finite size corrections and other corrections to scaling in general dimensions for the special case of the square lattice with free boundaries this theory combined with duality arguments give strong relations among different derivatives of the spanning function with respect to the scaling variables thus yielding several new universal amplitude ratios and allowing systematic study of the corrections to scaling the theoretical predictions are numerically confirmed with excellent accuracy

a modular invariant bulk theory for the c 0 triplet model a proposal for the bulk space of the logarithmic cal w 2 3 triplet model at central charge zero is made the construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory the resulting bulk space is a quotient of the direct sum of projective representations which is isomorphic as a vector space to the direct sum of tensor products of the irreducible representations with their projective covers as a consistency check of our analysis we show that the partition function of the bulk theory is modular invariant and that the boundary state analysis is compatible with the proposed annulus partition functions of this model

logarithmic torus amplitudes abstractfor the example of the logarithmic triplet theory at c 2 the chiral vacuumtorus amplitudes are analysed it is found that the space of these torus amplitudes isspanned by the characters of the irreducible representations as well as a function thatcan be associated to the logarithmic extension of the vacuum representation a fewimplications and generalisations of this result are discussed 1 introduction during the last twenty years much has been understood about the struc ture of rational conformal eld theories rational conformal eld theories are characterisedby the property that they have only nitely many irreducible highest weight representationsof the chiral algebra or vertex operator algebra and that every highest weight represen tation is completely decomposable into irreducible representations the structure of thesetheories is well understood in particular the characters of the irreducible representationstransform into one another under modular transformations 1 see also 2 and the modulars matrix determines the fusion rules via the verlinde formula 3 a general proof for thishas only recently been given in 4 on the other hand it is clear that rational conformal eld theories are rather special andit is therefore important to understand the structure of more general classes of conformal eld theories one such class are the rational logarithmic theories that possess only nitelymany indecomposable representations but for which not all highest weight representationsare completely decomposable the name logarithmic comes from the fact that their chiralcorrelation functions typically have logarithmic branch cuts the rst example of a non rational logarithmic conformal eld theory was found in 5 see also 6 and the rstrational example that shall also concern us in this paper was constructed in 7 for somerecent reviews see 8 9 10 from a physics point of view logarithmic conformal eldtheories appear naturally in various models of statistical physics for example in the theory

can the generalized hypergeometric equation feature several independent polynomial solutions the answer to the question posed in the title is positive it is even possible provided the parameters of the generalized hypergeometric equation are suitably restricted that the general solution of this ordinary differential equation be polynomial

critical percolation the expected number of clusters in a rectangle we show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits in particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant our proof is independent of earlier results and sle techniques and might provide a new approach to establishing conformal invariance of percolation

growth anomaly and its implications abstract the growth mechanism of interfaces in nature may be anomalous in the sense that the interface s roughness is not a universal scaling function this so called growth anomaly can be traced to the non existence of a continuum limit of the growth theory various aspects of the anomaly are discussed and some experimental verifications are suggested

explicit time dependent schrodinger propagators the authors compute explicitly the time dependent schrodinger and heat propagator for the potentials lambda 2m m 1 cosh2 lambda x a delta function potential several cases of periodic delta function potentials and a delta function potential in a square well they also discuss the semiclassical approximation for the delta function potential

a transfer matrix for the backbone exponent of two dimensional percolation rephrasing the backbone of two dimensional percolation as a monochromatic path crossing problem we investigate the latter by a transfer matrix approach conformal invariance links the backbone dimension db to the highest eigenvalue of the transfer matrix t and we obtain the result db 1 6431 0 0006 for a strip of width l t is roughly of size 23l but we manage to reduce it to l we find that the value of db is stable with respect to inclusion of additional blobs tangent to the backbone in a finite number of points

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

w extended kac representations and integrable boundary conditions in the logarithmic minimal models wlm 1 p we construct new yang baxter integrable boundary conditions in the lattice approach to the logarithmic minimal model giving rise to reducible yet indecomposable representations of rank 1 in the continuum scaling limit we interpret these extended kac representations as finitely generated extended feigin fuchs modules over the triplet w algebra the extended fusion rules of these representations are inferred from the recently conjectured virasoro fusion rules of the kac representations in the underlying logarithmic minimal model we also introduce the modules contragredient to the extended kac modules and work out the correspondingly extended fusion algebra our results are in accordance with the kazhdan lusztig dual of tensor products of modules over the restricted quantum universal enveloping algebra at q e i p finally polynomial fusion rings isomorphic with the various fusion algebras are determined and the corresponding grothendieck ring of characters is identified

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

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

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences

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