A new discrete Edwards model and a new polymer measure in two dimensions

The new discrete Edwards models in this paper are defined in terms of the so-called restricted intersection local times of the lattice random walk in two dimensions. We study the asymptotic behaviours of these new discrete Edwards models in the superrenormalizable cases. In particular, by approximating these models we can construct new polymer measures in two dimensions which are different from the original polymer measures obtained by approximating the original discrete Edwards models. The new discrete Edwards models can be thought of as zero-component lattice ω⁴-fields with different cutoffs in the free and interacting parts.     

Force-induced desorption of self-avoiding walks on Sierpinski gasket fractals

In this work we investigate force-induced desorption of linear polymers in good solvents in non-homogeneous environment, by applying the model of self-avoiding walk on two- and three-dimensional fractal lattices, obtained as generalization of the Sierpinski gasket fractal. For each of these lattices one of its boundaries represents an adsorbing wall, whereas along one of the fractal edges, not lying in the adsorbing wall, an external force acts on the self-avoiding walk. The hierarchical nature of the lattices under study enables an exact real-space renormalization group treatment, which yields the phase diagram of polymer critical behavior. We show that for this model there is no low-temperature reentrance in the cases of two-dimensional lattices, whereas in all studied three-dimensional cases the force-temperature dependance is reentrant. We also find that in all cases the force-induced desorption transition is of first order.     

Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk

We prove \({|x|^{-2}}\) decay of the critical two-point function for the continuous-time weakly self-avoiding walk on \({\mathbb{Z}^{d}}\), in the upper critical dimension d = 4. This is a statement that the critical exponent \({\eta}\) exists and is equal to zero. Results of this nature have been proved previously for dimensions \({d \ge 5}\) using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.     

Instanton Calculus for the Self-Avoiding Manifold Model

We compute the normalisation factor for the large order asymptotics of perturbation theory for the self-avoiding manifold (SAM) model describing flexible tethered (D-dimensional) membranes in d-dimensional space, and the ε-expansion for this problem. For that purpose, we develop the methods inspired from instanton calculus, that we introduced in a previous publication (Nucl. Phys. B 534 (1998) 555), and we compute the functional determinant of the fluctuations around the instanton configuration. This determinant has UV divergences and we show that the renormalized action used to make perturbation theory finite also renders the contribution of the instanton UV-finite. To compute this determinant, we develop a systematic large-d expansion. For the renormalized theory, we point out problems in the interplay between the limits ε→ 0 and d→∞, as well as IR divergences when ε=0. We show that many cancellations between IR divergences occur, and argue that the remaining IR-singular term is associated to amenable non-analytic contributions in the large-d limit when ε=0. The consistency with the standard instanton-calculus results for the self-avoiding walk is checked for D=1.     

From dilute to dense self-avoiding walks on hypercubic lattices

A field-theoretic representation is presented to count the number of configurations of a single self-avoiding walk on a hypercubic lattice ind dimensions with periodic boundary conditions. We evaluate the connectivity constant as a function of the fractionf of sites occupied by the polymer chain. The meanfield approximation is exact in the limit of infinite dimensions, and corrections to it in powers ofd ^–1 can be systematically evaluated. The connectivity constant and the site entropy calculated throughout second order compare well with known results in two and three dimensions. We also find that the entropy per site develops a maximum atf1–(2d)^–1. Ford=2 (d=3), this maximum occurs atf~0.80 (f~0.86) and its value is about 50% (30%) higher than the entropy per site of a Hamiltonian walk (f=1).     

Logarithmic Correction for the Susceptibility of the 4-Dimensional Weakly Self-Avoiding Walk: A Renormalisation Group Analysis

We prove that the susceptibility of the continuous-time weakly self-avoiding walk on \({\mathbb{Z}^d}\), in the critical dimension d = 4, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent \({\frac{1}{4}}\) for the logarithm. The susceptibility has been well understood previously for dimensions d ≥ 5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof begins by rewriting the walk two-point function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.     

Random walks with short-range interaction and mean-field behavior

We introduce a model of self-repelling random walks where the short-range interaction between two elements of the chain decreases as a power of the difference in proper time. The model interpolates between the lattice Edwards model and the ordinary random walk. We show by means of Monte Carlo simulations in two dimensions that the exponentv _MF obtained through a mean-field approximation correctly describes the numerical data and is probably exact as long as it is smaller than the corresponding exponent for self-avoiding walks. We also compute the exponent and present a numerical study of the scaling functions.     

Dimensional Reduction Formulas for Branched Polymer Correlation Functions

In [BI01] we have proven that the generating function for self-avoiding branched polymers in D+2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity in D dimensions. This result explains why the critical behavior of branched polymers should be the same as that of the i ³ (or Yang–Lee edge) field theory in two fewer dimensions (as proposed by Parisi and Sourlas in 1981). In this article we review and generalize the results of [BI01]. We show that the generating functions for several branched polymers are proportional to correlation functions of the hard-core gas. We derive Ward identities for certain branched polymer correlations. We give reduction formulae for multi-species branched polymers and the corresponding repulsive gases. Finally, we derive the massive scaling limit for the 2-point function of the one-dimensional hard-core gas, and thereby obtain the scaling form of the 2-point function for branched polymers in three dimensions.     

Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤ ^d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.     

Mean-Field Behavior for Long- and Finite Range Ising Model,Percolation and Self-Avoiding Walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d>2(α ∧2) for self-avoiding walk and the Ising model, and d>3(α ∧2) for percolation, where d denotes the dimension and α the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (Ann. Probab. 33(5):1886–1944, 2005).      

Reduced Dirac-Kähler lattice fermion action

It is shown that the lattice Dirac-Kähler action is reducible under a chiral-like transformation. This provides a new lattice fermion action for spinors that have 2^d−1 components (instead of 2^d), with the property that, in the free case, each component satisfies the lattice euclidean Klein-Gordon equation. Reflection positivity is satisfied on the lattice, thus assuring a (positive) physical Hilbert space. In d = 4 dimensions the spinors have 8 components, and the correct physical chiral anomaly in the continuum limit. The action is suitable for QCD quarks which, in the continuum limit, are described by Dirac spinors that occur in flavor doublets.     

Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D

We introduce a geometric generalization of the O(N)-field theory that describes N-colored membranes with arbitrary dimension D. As the O(N)-model reduces in the limit N → 0 to self-avoiding polymers, the N-colored manifold model leads to self-avoiding tethered membranes. In the other limit, for inner dimension D → 1, the manifold model reduces to the O(N)-field theory. We analyze the scaling properties of the model at criticality by a one-loop perturbative renormalization group analysis around an upper critical line. The freedom to optimize with respect to the expansion point on this line allows us to obtain the exponent ν of standard field theory to much better precision that the usual 1-loop calculations. Some other field theoretical techniques, such as the large N limit and Hartree approximation, can also be applied to this model. By comparison of low- and high-temperature expansions, we arrive at a conjecture for the nature of droplets dominating the 3d Ising model at criticality, which is satisfied by our numerical results. We can also construct an appropriate generalization that describes cubic anisotropy, by adding an interaction between manifolds of the same color. The two parameter space includes a variety of new phases and fixed points, some with Ising criticality, enabling us to extract a remarkably precise value of 0.6315 for the exponent ν in d = 3. A particular limit of the model with cubic anisotropy corresponds to the random bond Ising problem; unlike the field theory formulation, we find a fixed point describing this system at 1-loop order.     

Lace Expansion for the Ising Model

The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-point function for the nearest-neighbor model with and for the spread-out model with d > 4 and , without assuming reflection positivity.     

Group theory,Markov chains,and excluded volume effect in polymers

A restricted walk of orderr on a lattice is defined as a random walk in which polygons withr vertices or less are excluded. A study of restricted walks for increasingr provides an understanding of how the transition in properties is effected from random to self-avoiding walks which is important in our understanding of the excluded volume effect in polymers and in the study of many other problems. Here the properties of restricted walks are studied by the transition matrix method based on the theory of Markov chains. A group theoretical method is used to reduce the transition matrix governing the walk in a systematic manner and to classify the eigenvalues of the transition matrix according to the various representations of the appropriate group. It is shown that only those eigenvalues corresponding to two particular representations of the group contribute to the correlations among the steps of the walk. The distributions of eigenvalues for walks of various ordersr on the two-dimensional triangular lattice and the three-dimensional face-centered cubic lattice are presented, and they are shown to have some remarkable features.     

Renormalization of the self-avoiding walk on a lattice

It is shown that the Niemeijer-Van Leeuwen renormalization method can be extended, via a general n-vector model, to the case n = 0, which describes the self-avoiding walk. Results of a second cumulant approximation are given.     

Trapping in the Random Conductance Model

We consider random walks on ? ^d among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time 2n. We show that in the situations when the heat kernel exhibits subdiffusive decay—which is known to occur in dimensions d≥4—the walk gets trapped for a time of order n in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.     

On the well-posedness of the stochastic Allen–Cahn equation in two dimensions

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical systems in space dimensions d = 1, 2, 3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d ? 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen–Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that shrinking the mesh size in simulations of the two-dimensional white noise-driven Allen–Cahn equation does not lead to the recovery of a physically meaningful limit.     

A Faster Implementation of the Pivot Algorithm for Self-Avoiding Walks

The pivot algorithm is a Markov Chain Monte Carlo algorithm for simulating the self-avoiding walk. At each iteration a pivot which produces a global change in the walk is proposed. If the resulting walk is self-avoiding, the new walk is accepted; otherwise, it is rejected. Past implementations of the algorithm required a time O(N) per accepted pivot, where N is the number of steps in the walk. We show how to implement the algorithm so that the time required per accepted pivot is O(N ^q ) with q<1. We estimate that q is less than 0.57 in two dimensions, and less than 0.85 in three dimensions. Corrections to the O(N ^q ) make an accurate estimate of q impossible. They also imply that the asymptotic behavior of O(N ^q ) cannot be seen for walk lengths which can be simulated. In simulations the effective q is around 0.7 in two dimensions and 0.9 in three dimensions. Comparisons with simulations that use the standard implementation of the pivot algorithm using a hash table indicate that our implementation is faster by as much as a factor of 80 in two dimensions and as much as a factor of 7 in three dimensions. Our method does not require the use of a hash table and should also be applicable to the pivot algorithm for off-lattice models.     

From random to self-avoiding walks

A brief review will be given of the current situation in the theory of self-avoiding walks (SAWs). The Domb-Joyce model first introduced in 1972 consists of a random walk on a lattice in which eachN step configuration has a weighting factor Π _i=0 ^N?2 Π_j=i+2/N(1?ωδ_ij). Herei andj are the lattice sites occupied by the ith and jth points of the walk. When ω=0 the model reduces to a standard random walk, and when ω=1 it is a self-avoiding walk. The universality hypothesis of critical phenomena will be used to conjecture the behavior of the model as a function ofω for largeN. The implications for the theory of dilute polymer solutions will be indicated.