Critical Exp onents Hyp erscaling and Universal Amplitude Ratios for Two and ThreeDimensional SelfAvoiding Walks Bin Li Debt and Equity Markets Group Merril l Lynch World Financial Center New York NY USA Internet BLIMLCOM Neal Madras Department of Mathematics and Statistics York University Keele Street North York Ontario MJ P CANADA Internet MADRASNEXUSYORKUCA Alan D Sokal Department of Physics New York University hep-lat/9409003 6 Sep 1994 Washington Place New York NY USA Internet SOKALNYUEDU September KEY WORDS Selfavoiding walk p olymer critical exp onent hyperscaling universal amplitude ratio second virial co ecient interpenetration ratio renor malization group twoparameter theory Monte Carlo pivot algorithm KarpLuby algorithm Formerly with the Department of Physics New York University Abstract We make a highprecision Monte Carlo study of two and threedimensional selfavoiding walks SAWs of length up to steps using the pivot algo rithm and the KarpLuby algorithm We study the critical exp onents and as well as several universal amplitude ratios in particular we make 4 an extremely sensitive test of the hyperscaling relation d In two 4 dimensions we conrm the predicted exp onent and the hyperscal 2 2 ing relation we estimate the universal ratios hR ihR i g e 2 2 hR ihR i and condence m e limits In three dimensions we estimate with a correction toscaling exp onent sub jective condence limits This 1 value for agrees excellently with the eldtheoretic renormalizationgroup prediction but there is some discrepancy for Earlier Monte Carlo esti 1 mates of which were are now seen to b e biased by corrections to 2 2 scaling We estimate the universal ratios hR ihR i and g e since hyperscaling holds The approach to is from ab ove contrary to the prediction of the twoparameter renormalization group theory We critically reexamine this theory and explain where the error lies Contents Introduction The Problem of Hyp erscaling Which Quantities are Universal Plan of this Paper Background and Notation The SelfAvoiding Walk SAW A Review The Pivot Algorithm A Review Algorithms for Counting Overlaps Generalities Notation Deterministic Algorithms Monte Carlo Algorithms Generalities HitorMiss Monte Carlo Algorithm Barrett Algorithm Theory KarpLuby Algorithm Theory Scaling Theory Barrett and KarpLuby Algorithms Numerical Results Numerical Results Two Dimensions Three Dimensions Discussion Comparison with Previous Numerical Studies Two Dimensions Three Dimensions The Sign of Approach to Prosp ects for Future Work A Some Geometrical Theorems A Theorems and Pro ofs A Application to SAWs B Adequacy of Thermalization in the Pivot Algorithm C Some Statistical Subtleties D Remarks on the FieldTheoretic Estimates of Universal Amplitude Ratios Introduction The selfavoiding walk SAW is a wellknown lattice mo del of a p olymer molecule in a go o d solvent Its equivalence to the n limit of the nvector mo del has also made it an imp ortant testcase in the theory of critical phe nomena In this pap er we rep ort the results of an extensive Monte Carlo study of two and threedimensional SAWs of length up to steps using the pivot algorithm 1 We make a highprecision determination of the critical exp onents and as well as several universal amplitude ratios In particular we make an 4 extremely sensitive test of the hyperscaling relation d which plays a 4 central role in the general theory of critical phenomena Section Our results have also led us to reexamine critically the conventional theory of p olymer molecules the socalled twoparameter renormalizationgroup theory Indeed such a reexamination is unavoidable as our Monte Carlo data are inconsistent with this theory as it has b een heretofore applied But this is b ecause as we explain in Section the theory has heretofore b een applied incorrectly 1 These computations were carried out over a year p erio d on a variety of RISC workstations The total CPU time was several years but we have by now lost track of exactly how many These p oints were rst made three years ago by Nickel in an imp ortant but apparently underappreciated pap er they have recently b een extended by one of us The Problem of Hyp erscaling One of the key unsolved problems in the theory of critical phenomena has b een the status of the socalled hyperscaling relations scaling laws in which the spatial dimension d app ears explicitly These relations have long b een known to rest on a much more tenuous physical basis than the other scaling laws Indeed it has b een understo o d since the early s that hyperscaling should not hold for systems ab ove their upp er critical dimension d for d d the critical ex u u p onents are exp ected to b e those of meaneld theory and these exp onents satisfy 2 the hyperscaling relations only at d d For mo dels in an nvector univer u sality class including the SAW d equals It has generally b een b elieved that u hyperscaling should hold in dimensions d d but in our opinion there is no partic u ularly comp elling justication for such a b elief although the claim itself is probably correct Hyp erscaling in the threedimensional Ising mo del is the sub ject of a con 3 troversy that has b een raging for years and which is still not completely settled We remark that hyperscaling is also of interest in quantum eld theory where it 4 is equivalent to the nontriviality of the continuum limit for a stronglycoupled eld theory Although the hyperscaling relations app ear naively to b e ineluctable conse quences of the renormalizationgroup approach to critical phenomena closer exami nation reveals a mechanism by which hyperscaling can fail the socalled dangerous 4 irrelevant variables But the much more dicult question of whether this viola tion actually occurs in a given mo del can b e resolved only by detailed calculation Unfortunately a direct analytical test of hyperscaling app ears to b e p ossible only at or in the immediate neighborho o d of a Gaussian xed p oint that is for asymp totically free theories or for small d d or large n u We note that the realspace RG and eldtheoretic RG frameworks as typically used in approximate calculations implicitly assume the hyperscaling relations so they cannot b e used to test hyperscaling It is therefore of interest to make an unbiased numerical test of hyperscaling working directly from rst principles One approach is series extrap olation which aords a direct test of universality and scaling laws including hyperscaling 2 This b elief has now b een conrmed by rigorous pro ofs of the failure of hyperscaling for the Ising mo del in dimension d the selfavoiding walk in dimension d and spreadout p ercolation in dimension d 3 See for seriesextrap olation work and for Monte Carlo work 4 This mechanism was prop osed indep endently by Fisher and Wegner and Riedel p and fo otnote in the early s For further discussion see also Ma Section VI I Amit and Peliti Fisher App endix D and van Enter Fernandezand Sokal Section It gives numerical results of apparently very high accuracy the claimed sub jec tive error bars on critical exp onents are on the order of which is comparable to the b est alternative calculational schemes However as is inherent in any extrap olation metho d the results obtained dep end critically on the assump tions made ab out the singularity structure of the exact function notably the nature of the conuent singularity if any Indeed estimates by dier ent metho ds from the same series sometimes dier among themselves by several times their claimed error bars This together with systematic dierences b etween lattices of the same dimension accounts for much of the controversy over hyper scaling Quite a few extra terms would b e needed to resolve these discrepancies in a convincing manner Unfortunately the computer time required to eval uate the series co ecients grows exp onentially with the number of terms desired while the extrap olation error is prop ortional to some inverse p ower of the number of terms the p ower dep ends on the details of the correctiontoscaling terms and the extrap olation metho d In a Monte Carlo study by contrast one aims to prob e directly the regime where the correlation length is For SAWs this corresp onds to a chain length N The metho d aords a direct test of universality and scaling laws including hyperscaling In practice however it has b een extremely dicult to obtain go o d data in the neighborho o d of the critical p oint There are two essential diculties nite system size and critical slowingdown For spin mo dels and lattice eld theories these two factors together imply that the CPU d+z d z time needed to obtain one eectively indep endent sample grows as L where d is the spatial dimension of the system and z is the dynamic critical exp onent 5 of the Monte Carlo algorithm This situation may b e alleviated somewhat by a new nitesizescaling