the projected area of the three dimensional self avoiding polygons in a plane calculates the exponent theta s governing the growth of the projected area of the three dimensional self avoiding polygons in a plane the series result suggests that theta s 2vsap where vsap is the correlation length exponent of the three dimensional self avoiding polygons

self avoiding polygons on the square l and manhattan lattices transfer matrix techniques are used to extend the self avoiding polygon generating function on the square lattice to terms in x46 corresponding to 46 step polygons these techniques are then extended to apply to directed square lattices such as the l and manhattan lattice and the self avoiding polygon generating function to x48 is found for these lattices series analysis confirms that the specific heat exponent alpha 1 2 for the self avoiding walk problem and gives the following estimates for the connective constants mu sq 2 638155 or 0 000025 mu l 1 5657 or 0 0019 and mu man 1 7328 or 0 0005 some evidence for a correction to scaling exponent delta approximately 0 84 is found from square lattice series

confidence intervals for a binomial proportion and asymptotic expansions p z 2n 1 2 p 1 p 1 2 is currently in near universal use we first show that the coverage properties of the wald interval are persistently poor and defy virtually all conventional wisdom we then proceed to a theoretical comparison of the standard interval and four additional alternative intervals by asymptotic expansions of their coverage probabilities and expected lengths the four additional interval methods we study in detail are the score test interval wilson the likelihood ratio test interval a jeffreys prior bayesian interval and an interval suggested by agresti and coull the asymptotic expansions for coverage show that the first three of these alternative methods have coverages that fluctuate about the nominal value while the agresti coull interval has a somewhat larger and more nearly conservative coverage function for the five interval methods we also investigate asymptotically their average coverage relative to distributions for p supported within 0 1 in terms of expected length asymptotic expansions show that the agresti coull interval is always the longest of these the remaining three are rather comparable and are shorter than the wald interval except for p near 0 or 1 these analytical calculations support and complement the findings and the recommendations in brown cai and dasgupta statist sci 2001 16 101 133

inclusion exclusion relations on the triangular lattice the combinatorial factors required for performing finite lattice series expansions on the triangular lattice are discussed series expansions can be obtained from a linear combination of free energies for finite convex hexagonal regions an algorithm for calculating the combinatorial weights is given a number of results are presented showing that many cancellations occur and that not all convex hexagons of less than a given size are required for a given degree of computational complexity as given by the size of the largest transfer matrix the expansion using hexagons of less than a certain perimeter and exploiting the full triangular symmetry gives 50 more series terms than the earlier approach which adds second neighbour bonds to the square lattice and only exploits square lattice symmetries

counting elements and geodesics in thompson s group f we present two quite different algorithms to compute the number of elements in the sphere of radius n of thompson s group f with standard generating set the first of these requires exponential time and polynomial space but additionally computes the number of geodesics and is generalisable to many other groups the second algorithm requires polynomial time and space and allows us to compute the size of the spheres of radius n with n 1500 using the resulting series data we find that the growth rate of the group is bounded above by 2 62167 this is very close to guba s lower bound of 3 5 2 indeed numerical analysis of the series data strongly suggests that the growth rate of the group is exactly 3 5 2

monte carlo test of a hyperscaling relation for the two dimensional self avoiding walk the authors simulated self avoiding walks on the square lattice with fixed endpoints by means of a dynamic monte carlo algorithm from these data they obtain an evaluation of the effective coordination number mu and the critical exponents alpha and v they can therefore test the hyperscaling relation 2 alpha dv with a careful estimate of systematic and statistical errors

the number of convex polygons on the square and honeycomb lattices a subset of the set of self avoiding polygons sap embeddable on the square lattice which display the property of convexity is defined an algorithm for their enumeration is developed and from the available series coefficients the exact generating function is found the singularity structure appears similar to that of the unsolved sap problem but with different critical exponents and critical points the enumeration of convex polygons allows the extension of the existing series for the square lattice sap by one term for the honeycomb lattice similar results have been obtained despite a less natural definition of convexity

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

polygons in restricted geometries subjected to infinite forces we consider self avoiding polygons in a restricted geometry namely an infinite l m tube in these polygons are subjected to a force f parallel to the infinite axis of the tube when the force stretches the polygons while when the force is compressive we obtain and prove the asymptotic form of the free energy in both limits we conjecture that the asymptote is the same as the limiting free energy of hamiltonian polygons polygons which visit every vertex in a box we investigate such polygons and in particular use a transfer matrix methodology to establish that the conjecture is true for some small tube sizes

bounds on self avoiding walks on directed square lattices the manhattan lattice is the covering lattice of the l lattice and the author has studied self avoiding walks on this pair of lattices upper and lower bounds on the connective constant mu have been obtained as have series analysis estimates

collapse of two dimensional linear polymers we study the collapse transition of a two dimensional very long polymer the model we consider is a lattice model where the chain is represented by a self avoiding walk with nearest neighbor attraction by using the transfer matrix technique we calculate exactly the thermal and geometrical properties of the polymer on strips of finite width we then use finite size scaling to determine the values of the tricritical point exponentsv1 0 55 0 01vu 1 15 0 15 1 v1 1 80 0 05 we compare these results to the other values already proposed in the literature

universality of surface exponents of self avoiding walks on a manhattan lattice the authors use phenomenological renormalization group techniques to study the bulk and surface properties of self avoiding walks on a manhattan lattice they find that as is the case for the bulk exponents the underlying bond directionality does not change the universality class of the surface exponents relative to undirected lattices they also obtain an estimate for the position of the binding transition

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

self avoiding rings on the triangular lattice the finite lattice method of series expansion is used to extend the enumeration of self avoiding polygons on the triangular lattice through to rings of 35 steps the authors also give the enumeration of these triangular polygons grouped by perimeter and area of up to 21 unit triangles for perimeters up to 24 steps

universal amplitude in the sizes of rings in two dimensions for self avoiding rings of n steps in two dimensions the limiting value as n to infinity of the combination npn r2 nxcn where pn is the number of distinct rings r2 n is their mean square radius of gyration and xc is the critical fugacity is equal to a calculable lattice dependent number times a universal amplitude this latter quantity is calculated exactly using methods of conformal invariance the value is in good agreement with the results of enumeration studies

monte carlo series analysis of irreversible self avoiding walks ii the growing self avoiding walk for pt i see ibid vol 18 p 1515 1985 the authors study the scaling behaviour of the recently introduced growing saw they apply a ratio type analysis of monte carlo generated enumeration data to extract the critical indices for the two dimensional walk they find the usual saw values for these indices for the two dimensional walk the true asymptotic behaviour can only be studied for extremely long chains however on the basis of this data they can exclude the possibility of an upper critical dimension dc 3

phenomenological renormalisation of the self avoiding walk in two dimensions phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self avoiding walk problem on a square lattice a transfer matrix technique is developed for the polymer problem the results indicate that flory s value v 0 75 is true in two dimensions to extremely high accuracy

size and area of square lattice polygons we use the finite lattice method to calculate the radius of gyration the first and second area weighted moments of self avoiding polygons on the square lattice the series have been calculated for polygons up to perimeter 82 analysis of the series yields high accuracy estimates confirming theoretical predictions for the value of the size exponent and certain universal amplitude combinations furthermore a detailed analysis of the asymptotic form of the series coefficients provide the firmest evidence to date for the existence of a correction to scaling exponent 3 2

convergence and extrapolation in finite size scaling renormalization abstract finite size renormalization technique is studied by examining a test model which is exactly solvable and the problem of lattice animals and self avoiding walks bulk estimates of the leading irrelevant rg eigenvalue are found and compared with independent estimates from finite size data we introduce an extrapolation technique which is applicable to non monotonic sequences of exponent approximants

kinetic growth walks and trails on oriented square lattices hull percolation and percolation hulls the authors have studied kinetic growth walks on the manhattan lattice and kinetic growth trails on the l lattice they find that the directional properties of these lattices prevent the kinetic growth walks from being trapped except at the origin as a consequence kgws are found to be in the skw universality class for these lattices extensive numerical calculations support this fact moreover it has been shown that the recently introduced hull percolation is equivalent to the kinetic growth walks on the manhattan lattice and the bond percolation hull on the square lattice at the percolation threshold is equivalent to the kinetic growth trails on the l lattice their numerical estimates agree nicely with the original estimates and thus support the equivalence

algebraic techniques for enumerating self avoiding walks on the square lattice the authors describe a new algebraic technique for enumerating self avoiding walks on the rectangular lattice the computational complexity of enumerating walks of n steps is of order 3n 4 times a polynomial in n and so the approach is greatly superior to direct counting techniques they have enumerated walks of up to 39 steps as a consequence they are able to accurately estimate the critical point critical exponent and critical amplitude

tests of hyperuniversality for self avoiding walks a universal combination of amplitudes for the end to end distance of n step self avoiding walks and for the number of n step self avoiding polygons is estimated numerically for various lattices in d 2 universality of this amplitude combination is confirmed to good accuracy while in d 3 the data are consistent with universality but the error limits are rather large for finite width square lattice strips the authors calculate universal critical point correlation amplitudes by fixed fugacity monte carlo simulations in addition these amplitudes are calculated analytically either exactly or approximately by cardy s conformal mapping approach numerical values agree well with the theoretical predictions

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

the size and number of rings on the square lattice the authors have calculated the number of self avoiding polygons on the square lattice to 56 steps and the caliper size to 54 steps analysis of the generating function permits estimates of the connective constant mu 2 638 1585 or 10 6 and the critical exponents alpha 0 500 06 or 0 000 06 and v 0 753 or 0 007 the singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent delta 1 5 as predicted by nienhuis 1982 1984 the confluent part however maps into the additive analytic term due to the value of the exponent alpha