a monte carlo method for series expansions the authors describe a simple and general algorithm to calculate series expansions in enumeration problems to large orders approximately by a monte carlo method it can be used to generate unbiased samples in cluster studies e g linear or branched polymers random surfaces etc in any dimension they calculate the number of site animals of size n on the square lattice for n or 50 and their average size to better than 1 accuracy

position space renormalization group for saws on the l lattice the self avoiding walk problem on the l lattice is treated with a position space renormalization group approach using a centre rule weight function

renormalization theory of self avoiding walks which cross a square the renormalization group theory is used to calculate the critical behaviour of selfwoiding walks which cross a square this problem which has been proposed recently is especially well suited for a renormalization group analysis the fixed endpoint diagonal span trademark of the square crossing walks leads naturally to a well defined renormaliz ation scheme unlike other finite lattice renormalization schemes for self avoiding walks this square crossing renormalization is exact in the sense that the finite lattice square is uniquely defined the spanning rule is unambiguous and the end to end correlations are exactly preserved the results for the critical point are in excellent agreement with series analysis estimates and support a conjecture on its exact value

simple renormalisation group method for calculating thermal equations of states renormalisation group rg techniques are currently used to derive relevant quantities in the vicinity of the critical point the authors present a real space rg procedure which directly yields the order parameter for all values of the external parameters e g temperature it is as simple as a mean field calculation although it provides non trivial results which can be systematically improved the method is successfully illustrated on the square lattice potts ferromagnet the whole approach suggests that the order parameter on a hierarchical lattice is on every site proportional to its coordination number

persistency of two dimensional self avoiding walks the authors investigate the persistency of self avoiding walks on the square lattice by studying the distribution of endpoint displacements projected in the direction of the first step exact enumeration and the constant fugacity monte carlo data indicate that the mean displacement after n steps xn increases logarithmically with n while the higher odd moments x2k 1n vary as n2kv in n where v is the correlation length exponent this unusual behaviour for the moments is found to arise from a simple scaling behaviour of the asymmetric component of the displacement distribution which in the constant fugacity ensemble can be written in the form a x approximately n 2v in n exp bx nv where n is the average number of steps in the walk at a fixed value of the fugacity and b is a coefficient of order unity

self avoiding walks on a penrose lattice direct real space renormalisation group procedures are constructed to calculate the self avoiding walk exponent nu on a two dimensional penrose lattice this lattice is neither translationally invariant nor hierarchical but still self similar and therefore calls for a renormalisation group treatment within the accuracy of a small cell calculation the obtained values of nu are consistent with the universal value of nu in two dimensions which is believed to be close to nu 0 75

correction to scaling exponents and amplitudes for the correlation length of linear polymers in two dimensions we consider the scaling behaviour of the radius of gyration for a system of dilute linear polymers we focus on pn the mean square end to end distance of an n step self avoiding walk for which an additional two terms were recently calculated for the close packed triangular lattice combining several extrapolation methods we find that a consistent description of the scaling behaviour exists if and only if the correction to scaling exponent a is roughly half as large as commonly believed we conclude that all data ale consistent with the equation pn an2 1 b na c n where y a a 1 j2 abso 21 and c lo

adsorption of a self avoiding walk a real space renormalisation group study of d 2 3 the author presents a real space renormalisation group calculation for the adsorption of a single self avoiding walk in d dimensions at a d 1 dimensional impenetrable wall the case d 3 is used to check the method for d 2 the result for the crossover exponent phi gives phi 0 55 or 0 15 this in connection with enumerations of ishinabe 1982 clearly rules out de gennes 1972 1976 1977 conjecture phi 1 v also for the case d 2 for d 2 cells up to 6 6 are used for d 3 cells up to 3 3 3

lower dimensional defect transitions in self avoiding walks and percolation the author studies self avoiding walks and percolation on d dimensional lattices focusing on the existence of transitions associated with a defect of dimensionality d or 2 in contrast to percolation for which they are absent if d 1 an apparent contradiction with scaling arguments is resolved a simple renormalisation group approximation yields phase diagrams consistent with these results and provides numerical estimates for crossover exponents and the splitting between special and homogeneous critical points

cell renormalisation study of biased self avoiding walks the authors present a cell renormalisation approach for the biased self avoiding walks and show that a stiff to isotropic crossover exponent is exactly one in all dimensions and for all cell sizes they also show by use of renormalisation flow diagrams a substantial difference between two and three dimensions in the crossover from stiff limit to isotropic limit as the length of walk n to infinity for fixed gauche weight p in three dimensions a crossover seems to occur first to the random walk limit and then to the self avoiding walk limit while in two dimensions it seems to occur directly to the self avoiding walk limit in agreement with recent observations based on monte carlo simulations

self avoiding walks on random networks of resistors and diodes abstract we study the self avoiding walks saw on a square lattice whose various degrees of randomness encompasses many different random networks including the incipient clusters of the directed mixed and isotropic bond percolation we apply the position space renormalization group psrg method and demonstrate that within the framework of this method one is bound to find that the critical exponent v of the mean end to end distance of saw on various two dimensional random networks should be equal to the critical exponent of saw on the ordinary square lattice a detailed analysis of this finding and similar findings of other authors lead us to conclude that a debatable opposite finding which has been predicted on the basis of different approaches could be attained after a substantial refinement of the method applied

relation between the two dimensional self avoiding walk and the eden model abstract when the interface of a two dimensional lattice gas is grown by an algorithm producing self avoiding walks some features of the eden model appear the scaling behavior of the widths and lengths of these interfaces and of their accessible perimeters are examined in several regions of the phase diagram it is found that both the accessible perimeter and the interface behave like the eden model below a critical point above this point only the accessible perimeter behaves like the eden model the behavior in the critical region is suprisingly the same for the interface and its external perimeter but is different from the eden model

self avoiding walks in quenched random environments the self avoiding walk in a quenched random environment is studied using realspace and field theoretic renormalization and flory arguments these methods indicate that the system is described for d de by a strong disorder fixed point corresponding to a glass state in which the polymer is confined to the lowest energy path this fixed point is characterized by scaling laws for the size of the walk l n with n the number of steps and the fluctuations in the free energy af l the bound 1 co pore and suggests that this inequality holds for d 2 and 3 although pure cannot be excluded particularly for d 2 for d d c there is a transition between strong and weak disorder phases at which pure the strong disorder fixed point for saws on percolation clusters is discussed the analogy with directed walks is emphasized

logarithmic voltage anomalies in random resistor networks the authors investigate the behaviour of the maximum voltage drop across the bonds in a random resistor network above the percolation threshold on the basis of numerical simulations on randomly diluted l l square lattice networks we find that the average value of the maximum voltage vmax p l exhibits a peak as a function of the bond concentration p which is located above the percolation threshold this peak value appears to grow logarithmically with l while the location of the peak appears to approach the percolation threshold very slowly as l increases to help understand these results they introduce the bubble model a quasi one dimensional structure in which the system length varies exponentially in the system width this model is exactly soluble and a rather good description of percolation in systems of greater than one dimension can be obtained moreover within the bubble model the peak value of the maximum voltage increases as in l while the peak location approaches the percolation threshold as in in l in l in good agreement with our numerical results

self avoiding walks in four dimensions logarithmic corrections we present simulation results for long n or 4000 self avoiding walks in four dimensions we find definite indications of logarithmic corrections but the data are poorly described by the asymptotically leading terms detailed comparisons are presented with renormalization group flow equations derived by direct renormalization and with the results of a field theoretic calculation

the hardy rellich inequality for polyharmonic operators the hardy rellich inequality given here generalizes a hardy inequality of davies from the case of the dirichlet laplacian of a region n to that of the higher order polyharmonic operators with dirichlet boundary conditions the inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents

position space renormalisation group for isolated polymer chains we develop a cell position space renormalisation group psrg with which we study the scaling properties of isolated polymer chains we model a chain by a self avoiding walk constrained to a lattice for rescaling factors b s 6 we calculate recursion relations analytically on the square lattice with several different choices for the psrg weight function we also calculate implicit cell to cell transformations in which a cell of size b is rescaled to a cell of size b the results of these psrgs improve both as b increases and as b b 1 in addition we construct a true infinitesimal psrg transformation which appears to become exact as the dimensionality d approaches 1 we obtain the closed form expression for the correlation length exponent u d l d in d the flory formula deviates from this already at first order in d 1 we also develop a constant fugacity monte carlo method which enables us to simu late in an unbiased way within the grand canonical ensemble chains of up to lo3 bonds with this method we extend the psrg to larger cells b s 150 on the square lattice our numerical method provides high statistical accuracy for all cell sizes however in the range of b we study the asymptotic behaviour of our results appears to depend on the choice of weight function one weight function provides smooth behaviour as a function of b and with it we extrapolate to find u 0 756 0 004 further work is required to resolve the apparent anomalies in the results based on the other weight functions

self avoiding random walks on the hexagonal lattice we use the algorithm recently introduce by a berretti and a d sokal to compute numerically the critical exponents for the self avoiding random walk on the hexagonal lattice we find 1 3509 0 0057 0 0023v 0 7580 0 0049 0 0046 0 519 0 082 0 077 where the first error is the systematic one due to corrections to scaling and the second is the statistical error for the effective coordination number we find 1 84779 0 00006 0 0017 the results support the nienhuis conjecture 43 32 and provide a rough numerical check of the hyperscaling relationdv 2 an additional analysis taking the nienhuis value of 2 21 2 1 2 for granted gives 1 3459 0 0040 0 0008

self avoiding walks on fractals studied by the monte carlo renormalization group the authors have applied the monte carlo renormalization group mcrg method to study the problem of self avoiding walks on the sierpinski gasket family of fractals each member of the family is labelled by an integer b 2 or b or infinity and when b to infinity both the fractal df and spectral ds dimension approach their euclidean value 2 they have calculated the critical exponent nu associated with the mean square end to end distance up to b 80 their mcrg results deviate at most 0 03 from the available exact results for 2 or b or 9 the obtained data show clearly that nu monotonically decreases with b and crosses the euclidean value nu 3 4 at b approximately 27 that is before entering the fractal to euclidean crossover region that occurs in the limit b to infinity

chain polymers near an adsorbing surface we present improved estimates of critical exponents for 3d polymers near a plane adsorbing surface the improvements were possible by simulating rather long chains up to n 2000 by means of a new algorithm apart from minor but significant adjustments in the location of the tricritical point and in the various gamma exponents our main result is that the crossover exponent for critical adsorption is compatible with phi 1 2 this is much lower than previous estimates and suggests that this exponent might be superuniversal

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

criticality of the discrete n vector ferromagnet in planar self dual lattices the authors study an extended version of the discrete n vector or cubic ferromagnetic model within a real space renormalisation group approach which preserves the two spin correlation function the n evolution for real values of n of the wheatstone bridge hierarchical lattice phase diagram which presents paramagnetic intermediate nematic like and ferromagnetic phases as well as of the thermal nu and crossover phi critical exponents is presented the self avoiding walk problem is recovered in the n to o limit and the so called corner rule is re obtained in a larger context the ising n and 2n state potts ferromagnets are recovered as particular cases an interchange of stability occurs at n n approximately 6 9 in such a way that the 2n state potts special point where all three existing phases join is multicritical if n n but only critical if n n consistently phi n o for the cubic model nu n presents a maximum at n nmax approximately 1 5 the results are exact for all n for the wheatstone bridge hierarchical lattice and approximate for n or 2 for the square lattice last but not least they discuss the connection between the present approach and the phenomenological renormalisation group

sharp moser trudinger inequality on the heisenberg group at the critical case and applications let h cn r be the n dimensional heisenberg group q 2n 2 be the homogeneous dimension of h q qq 1 and z 4 t2 14 be the homogeneous norm of z t h then we prove the following sharp moser trudinger inequality on h theorem 1 6 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q 0 q 1 q there holds sup u 1 1 h1 exp u q q 1 k 0q 2 kk u kq q 1 c q the constant q 1 q is best possible in the sense that the supremum is infinite if q 1 q here is any positive number and u 1 h hu q h u q 1 q our result extends the sharp moser trudinger inequality by cohn and lu 2001 19 on domains of finite measure on h and sharpens the recent result of cohn et al 2012 18 where such an inequality was studied for the subcritical case q 1 q we carry out a completely different and much simpler argument than that in cohn et al 2012 18 to conclude the critical case our method avoids using the rearrangement argument which is not available in an optimal way on the heisenberg group and can be used in more general settings such as riemanian manifolds appropriate metric spaces etc as applications we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of q laplacian type on the heisenberg group theorems 1 8 1 9 1 10 and 1 11 divh hu q 2 hu v u q 2u f u eh with nonlinear terms f of maximal exponential growth exp u qq 1 as u in particular when e 0 the existence of a nontrivial solution is also given

polymer coil globule transition by real space renormalisation a simple real space renormalisation scheme is proposed for the statistics of the polymer coil globule transition in two dimensions this approach describes the coil theta point and globular states of the macromolecule within a single two parameter transformation results from a small cell calculation include the theta point size exponent vtheta 0 494 and crossover exponent phi 0 486

recursive sampling of random walks self avoiding walks in disordered media the author presents a new algorithm for simulating random walks which is simple versatile and efficient it uses recursive function calls and can be used to obtain unbiased samples with any given length distribution this makes it particularly useful in disordered geometries where the effective connectivity constant is not known a priori when applying it to self avoiding random walks on two dimensional media with quenched randomness he finds evidence for at least two new renormalization group fixed points

single scaling field approach for an isolated polymer chain we develop an approach for describing an isolated polymer chain in solution without appeal to the n 0 limit of the n vector model our approach treats the generating function for self avoiding walks as a grand partition function in this description the fugacity per monomer plays a fundamental role as the unique scaling field in the problem a physical picture of a polymer chain in both an ordered and disordered phase emerges naturally from the resulting thermodynamic relations this picture turns out to be similar to percolation and leads us to classify the phases of a single polymer chain according to their fractal dimensionality we identify the monomer density as the order parameter furthermore since we have only a single scaling field the fugacity per monomer we can relate all the critical exponents to a single exponent we find cy y p 1 cy du 1 we emphasise that this theory is not equivalent to the n 0 limit of the n vector model though the underlying generating function and hence the correlation functions are the same

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

nonergodicity of local length conserving monte carlo algorithms for the self avoiding walk it is proved that every dynamic monte carlo algorithm for the self avoiding walk based on a finite repertoire of local n conserving elementary moves is nonergodic heren is the number of bonds in the walk indeed for largen each ergodic class forms an exponentially small fraction of the whole space this invalidates at least in principle the use of the verdier stockmayer algorithm and its generalizations for high precision monte carlo studies of the self avoiding walk

universal crossing of the self avoiding walk critical exponent nu at the euclidean value 3 4 for several different fractal families we find that the end to end distance critical exponent nu is close to the euclidean value 3 4 for self avoiding walks saws on fractals that are composed of homogeneous parts of size bh approximately 26 more precisely we find that saws on fractals with smaller homogeneous parts b 3 4 while nu 27 we establish this result in the case of three quite different fractal families the sierpinski gasket sg family the plane filling pf family and the checkerboard cb family of fractals in the case of the first two families sg and pf the relevant values for nu were found previously by applying the monte carte renormalization group mcrg method but the specific value bh was not recognized on the other hand for the cb fractals only the exact renormalization group rg results were known previously for 3 or b or 9 where b is an odd integer that serves as the fractal enumerator in this paper we extend the sequence of the known results for the cb family up to b 81 by generalizing the mcrg method the new results have revealed the occurrence of the aforementioned crossing of the euclidean value 3 4 we discuss the significance of the crossing within the current knowledge of saws on fractals

eden model on the manhattan lattice the authors study the eden growth model on the square manhattan lattice sml for this purpose it has been necessary to extend the standard eden model in such a way that each act of growing is conceived as a directional process by applying the position space renormalisation group technique they find that the eden processes on the sml and on the ordinary square lattice are in different universality classes this finding and corresponding fractal dimensions are compared with results obtained for similar models of irreversible kinetic processes

monte carlo renormalisation group for the true self avoiding walk following the general ideas by swendsen 1982 a monte carlo renormalisation group mcrg estimation of the end to end exponent nu for the one dimensional true self avoiding walk tsaw has been carried out because the tsaw is known to reach its asymptotic behaviour only very slowly an approximate expression for the fixed point has been determined by monte carlo techniques this approximate fixed point has been used as a starting point for the mcrg the final result for the exponent nu is 0 665 or 0 004

statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three dimensional space abstract we study the adsorption problem of linear polymers immersed in a good solvent when the container of the polymer solvent system is taken to be a member of the sierpinski gasket sg family of fractals embedded in the three dimensional euclidean space members of the sg family are enumerated by an integer b 2 b and it is assumed that one side of each sg fractal is impenetrable adsorbing boundary we calculate the surface critical exponents 11 1 and s which within the self avoiding walk model saw of polymer chain are associated with the numbers of all possible saws with both one and no ends grafted to the adsorbing surface adsorbing boundary respectively by applying the exact renormalization group method for 2 b 4 we have obtained specific values for these exponents for various types of polymer conformations to extend the obtained sequences of exact values for surface critical exponents we have applied the monte carlo renormalization group method for fractals with 2 b 40 the obtained results show that all studied exponents are monotonically increasing functions of the parameter b for all possible polymer states we discuss mutual relations between the studied critical exponents and compare their values with those found for other types of lattices in order to attain a unified picture of the attacked problem

grand canonical simulations of the interacting self avoiding walk model we report the results obtained from simulations of the interacting self avoiding walk isaw model on the square and simple cubic lattices in particular we verify a new scaling form for the canonical partition function in the low temperature phase as proposed by owczarek et al we also obtain estimates of the critical exponent in two dimensions finally we obtain good agreement with previous studies on the value of a correction to scaling amplitude at the point on the simple cubic lattice which is in disagreement with a field theoretic prediction

join and cut algorithm for self avoiding walks with variable length and free endpoints we introduce a new monte carlo algorithm for generating self avoiding walks of variable length and free endpoints the algorithm works in the unorthodox ensemble consisting of all pairs of saws such that the total number of stepsntot in the two walks is fixed the elementary moves of the algorithm are fixed n e g pivot moves on the individual walks and a novel join and cut move that concatenates the two walks and then cuts them at a random location we analyze the dynamic critical behavior of the new algorithm using a combination of rigorous heuristic and numerical methods in two dimensions the autocorrelation time in cpu units grows as n 1 5 and the behavior improves in higher dimensions this algorithm allows high precision estimation of the critical exponent

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

improved hardy adams inequality on hyperbolic space of dimension four abstract in this paper we establish a singular hardy adams inequality on the hyperbolic space of dimension four as our main result we will prove that for any 0 0 4 there exists a constant c 0 such that for all u c 0 b 4 satisfying b 4 h 9 4 h u u d v 1 then the following inequality holds b 4 exp 32 2 1 4 u 2 1 32 2 1 4 u 2 x d v c where x log 1 x 1 x is the hyperbolic distance from x to the origin our results improve that of lu and yang 2017

determination of the exponent for saws on the two dimensional manhattan lattice we present a high statistics monte carlo determination of the exponent for self avoiding walks on a manhattan lattice in two dimensions a conservative estimate is in agreement with the universal value on regular lattices but in conflict with predictions from conformal field theory and with a recent estimate from exact enumerations we find strong corrections to scaling that seem to indicate the presence of a non analytic exponent if we assume we find where the error is purely statistical

branched polymers on the two dimensional square lattice with attractive surfaces using the renormalization group approach an analysis is given of the asymptotic properties of branched polymers situated on the two dimensional square lattice with attractive impenetrable surfaces we modeled branched polymers as site lattice animals with loops and site lattice animals without loops on the simple square lattice we found the gyration radius critical exponent 0 6511 0 0003 and 0 6513 0 0003 for branched polymers with and without loops respectively our results for the crossover exponent 0 502 0 003 for branched polymers with loops and 0 503 0 003 for branched polymers without loops satisfy the recent hyperuniversality conjecture frac 1 2 in addition we have studied partially directed site lattice animals

monte carlo simulation of 3d self avoiding walks presents simulation results for 3d self avoiding walks of moderate length n or 2000 using a recently introduced recursive sampling method it is shown that this method is indeed maximally efficient for determining the connectivity constant mu combining the data with previous results from exact enumerations and from simulations of very long walks using the pivot algorithm the author obtains very precise estimates of mu and of the critical exponents gamma and nu he also points out significant deviations from uniformity in a very popular random number generator which was considered safe until now

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

sharp subcritical moser trudinger inequalities on heisenberg groups and subelliptic pdes the aim of this paper is to prove a sharp subcritical moser trudinger inequality on the whole heisenberg group let h cn rh cn r be the n n dimensional heisenberg group q 2n 2q 2n 2 be the homogeneous dimension of hh q qq 1 and z 4 t2 14 be the homogeneous norm of z t h z t h then we establish the following inequality on hh theorem 1 1 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q0 q 0 q 1 q there exists a constant 0 c c 0 c c such that the following inequality holds sup hu lq h 11 u lq h q h1 exp u q q 1 k 0q 2 kk u kq q 1 c the above result is the best possible in the sense when q 1 q the integral is still finite for any u w1 q h u w1 q h but the supremum is infinite in contrast to the analogous inequality in euclidean spaces proved in adachi and tanaka 1999 6 using symmetrization our argument is completely different and avoids the symmetrization method which is not available on the heisenberg group in an optimal way moreover our restriction on the norm hu lq h 1 hu lq h 1 of the function uu is much weaker than hu lq h u lq h 1 which was assumed in lam and lu 2012 16 as a consequence our inequality fails at q 1 q in contrast to the one in 16 as an application of this inequality we will prove that the following nonlinear subelliptic equation of qq laplacian type without perturbation equation 0 1 qu v u q 2u f u in h has a nontrivial weak solution where the nonlinear term ff has the critical exponential growth e u qq 1 as u u but does not satisfy the ambrosetti rabinowitz condition

the winding angle of planar self avoiding walks the problem of understanding the asymptotic statistical behaviour of the winding angle on of a self avoiding walk of n steps on a planar lattice is posed exact series expansion data for the square lattice up to n 21 are reported these data and monte carlo estimates up to n g 170 steps are fitted well by a logarithmic growth law for 06 the ratio oh appears to approach a limit of 2 9 to 3 2 which is close to the gaussian value 3 heuristic scaling arguments are consistent with simple logarithmic growth and also illuminate the logarithmic behaviour known rigorously for free brownian motion

recursion equations for static and kinetic models of branched polymers a new renormalisation group treatment of bond lattice animals is presented which yields the critical bond fugacity and nu exponent for general dimension d good agreement is obtained with predictions based on different methods for all d in particular nu 1 epsilon o epsilon 2 for d 1 epsilon results are also given for a kinetic version of the bond lattice animal problem

universality of neighbour avoiding and self avoiding walks on the square lattice it is shown that within position space renormalisation group psrg theory neighbour avoiding walks naws and self avoiding walks saws on the square lattice obey the same end to end distance critical exponent nu this universality is shown by applying a recent finite lattice renormalisation transformation to ordinary and oriented naw and saw problems and then using an exact equivalence between saws on an oriented lattice and naws on its covering lattice the universality class includes several oriented walk problems

real space renormalisation group for directed systems a new real space renormalisation group method is developed to treat systems with directionally dependent critical behaviour here two effective lengths s sub b and sperpendicular to b are defined for a given cell of linear size b the renormalised lattice is constructed from these effective lengths and is deformed from the original lattice it is the anisotropic rescaling of these effective lengths which give the anisotropic exponents nu sub and nu perpendicular to this method gives the exact result for the case of fully directed saws on a square lattice in the b to infinity limit for directed bond and directed site lattice animals the results obtained here are also in good agreement with the other known values

different types of self avoiding walks on deterministic fractals normal and indefinitely growing ig self avoiding walks saws are exactly enumerated on several deterministic fractals the manderbrot given curve with and without dangling bonds and the 3 simplex on then th fractal generation of linear sizel the average number of steps behaves asymptotically as n aldsaw b in contrast to saws on regular lattices on these factals igsaws and normal saws have the same fractal dimensiondsaw however they have different amplitudes a and correction terms b

bilocal dynamics for self avoiding walks we introduce several bilocal algorithms for lattice self avoiding walks we discuss their ergodicity in different confined geometries for instance in strips and in slabs a short discussion of the dynamical properties in the absence of interactions is given

a new renormalisation group weight function for linear polymers the author proposes a true centre rule weight function for treating linear polymers with a cell position space renormalisation group psrg for a cell to cell psrg the preliminary results for small cells give the critical fugacity per monomer k 0 3886 and the critical exponent nu 0 7471 accurate results are also obtained for nu with a cell to bond psrg

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

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

central projective quaternionic representations we obtain all the irreducible not similar central quaternionic projective representations of a finite group g from the knowledge of the vector complex irreducible representations of suitable extensions of g and classify them according to a generalized frobenius schur classification finally a definition of equivalence of such representations is proposed

on the critical behaviour of a surface interacting linear polymer chain abstract a method based on a real space renormalization group transformation is developed to describe the critical behaviour of a surface interacting linear flexible polymer chain represented by a self avoiding walk it is shown that a lattice model based on a central rule in which the starting point of the walk and the surface are taken to be in the middle of one cell provides a suitable framework to study both the penetrable and impenetrable surfaces in contrast to this a method based on a corner rule in which the starting point of the walk and the surface are fixed to be one corner of a cell cannot describe the behaviour of a chain interacting with an impenetrable surface the value of crossover exponent found by us for a square lattice are in agreement with those expected to be exact for both the cases

self avoiding walk model for proteins a real space renormalisation group treatment diffusion on the model for proteins introduced by helman coniglio and tsallis 1984 hct is discussed using real space renormalisation group ideas the authors perform cell to cell transformations in two dimensions for both the hct model and the self avoiding walk the latter system is quasi linear and so they expect trivial diffusive behaviour given by dw 2df 8 3 diffusion on the hct model however is complicated by the possibility of jumps across nearest neighbour hydrogen bond bridges the authors results are not consistent with the prediction that dw 2 for the hct model rather they support the assertion that dwhct is close to but slightly greater than dwsaw

a monte carlo analysis of self avoiding walks in three dimensions the authors present estimates for critical exponents of self avoiding walks on a cubic lattice they treat a grand canonical ensemble of walks with free ends in a monte carlo approach and make use of real space renormalisation ideas their estimate for v is 0 59 or 0 01 the estimate for gamma does not share the same degree of accuracy however they are able to pinpoint the source of this discrepancy in addition they define a quantity chi l beta the probability that a walk starting at the origin will end outside or at the border of a cube of side l this quantity turns out to be quite suitable for a real space renormalisation group analysis

a hardy moser trudinger inequality abstract in this paper we obtain an inequality on the unit disk b in r 2 which improves the classical moser trudinger inequality and the classical hardy inequality at the same time namely there exists a constant c 0 0 such that b e 4 u 2 h u d x c 0 u c 0 b 0 where h u b u 2 d x b u 2 1 x 2 2 d x this inequality is a two dimensional analog of the hardy sobolev maz ya inequality in higher dimensions which has been intensively studied recently we also prove that the supremum is achieved in a suitable function space which is an analog of the celebrated result of carleson chang for the moser trudinger inequality

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

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

on the correction to scaling exponent of linear polymers in two dimensions the authors present new results which indicate that the leading correction to scaling exponent in the mean squared end to end distance in two dimensions is the analytic term a potential source of the various correction to scaling terms reported in the literature is pointed out

exact and monte carlo study of adsorption of a self interacting polymer chain for a family of three dimensional fractals we study the problem of adsorption of self interacting linear polymers situated in fractal containers that belong to the three dimensional 3d sierpinski gasket sg family of fractals each member of the 3d sg fractal family has a fractal impenetrable 2d adsorbing surface which is in fact 2d sg fractal and can be labelled by an integer b 2 b by applying the exact and monte carlo renormalization group mcrg method we calculate the critical exponents associated with the mean squared end to end distance of polymers and associated with the number of adsorbed monomers for a sequence of fractals with 2 b 4 exactly and 2 b 40 monte carlo we find that both and monotonically decrease with increasing b that is with increase of the container fractal dimension df and the interesting fact that both functions b and b cross the estimated euclidean values besides we establish the phase diagrams for fractals with b 3 and b 4 which reveal the existence of six different phases that merge together at a multi critical point whose nature depends on the value of the monomer energy in the layer adjacent to the adsorbing surface

two dimensional polymers universality and correction to scaling improved monte carlo methods have been used to study the asymptotic behaviour of self avoiding walks on two dimensional lattices the mean square end to end distance rn2 and radius of gyration sn2 are both found to scale as n2v with exponent values very close to the expected v 3 4 and an amplitude ratio that is universal small n deviations of rn2 from the asymptotic results indicate a correction to scaling exponent delta 1 a value that differs from previous estimates

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

sharp hardy adams inequalities for bi laplacian on hyperbolic space of dimension four abstract we establish sharp hardy adams inequalities on hyperbolic space b 4 of dimension four namely we will show that for any 0 there exists a constant c 0 such that b 4 e 32 2 u 2 1 32 2 u 2 d v 16 b 4 e 32 2 u 2 1 32 2 u 2 1 x 2 4 d x c for any u c 0 b 4 with b 4 h 9 4 h u u d v 1 as applications we obtain a sharpened adams inequality on hyperbolic space b 4 and an inequality which improves the classical adams inequality and the hardy inequality simultaneously the later inequality is in the spirit of the hardy trudinger moser inequality on a disk in dimension two given by wang and ye in 46 and on any convex planar domain by lu and yang in 33 the fourier analysis techniques on hyperbolic and symmetric spaces play an important role in our work

self avoiding walks on percolation clusters the author studies self avoiding walks saws on percolation clusters a scaling function representation for r the mean end to end distance is proposed which describes a crossover from ordinary saws to saws on fractals he distinguishes between saws on a single cluster for which r approximately nx where x nu 1 and saws on all clusters for which r approximately nz where z nu 2 and n is the number of monomers in the walk he estimates nu 1 d 2 approximately 1 285 and the correction to scaling exponent omega d 2 approximately 1 3 and nu 1 d 3 approximately 1 38 two plausible generalisations of the flory 1953 approximation for nu 1 are investigated and it is argued that none of them provides a satisfactory approximation for nu 1 at all dimensions

efficiency of the incomplete enumeration algorithm for monte carlo simulation of linear and branched polymers we study the efficiency of the incomplete enumeration algorithm for linear and branched polymers there is a qualitative difference in the efficiency in these two cases the average time to generate an independent sample of configuration of polymer with n monomers varies as n2 for linear polymers for large n but as exp cn for branched undirected and directed polymers where 0 1 on the binary tree our numerical studies for n of order 104 gives 0 333 0 005 we argue that 1 3 exactly in this case

statistical properties of biased sampling methods for long polymer chains the authors present a detailed statistical analysis of the rosenbluth method 1955 of generating self avoiding walks this method became one of the standard methods for simulating long polymers they show that this method although very successful in yielding large samples becomes exponentially poor with increasing chain length this has to be taken into account for simulations and was not done yet they describe a way to quantify the number of chains needed however when compared to direct simple sampling the method still carefully used yields better results especially in the vicinity of the theta point of polymers special care has to be taken for d 2 some extensions to improve the situation are also discussed

size and shape of directed lattice animals uses position space renormalisation and exact enumeration methods to study configurational properties of directed lattice animals clusters of directed bonds in which the tail of a new bond must be added to the head of existing bonds exponents are found for the divergence of correlation lengths parallel and perpendicular to the preferred axis in addition an exponent is found that characterises the singularity of the directed animal generating function and critical points are located