Monte Carlo tests of stochastic Loewner evolution predictions for the 2D self-avoiding walk

The conjecture that the scaling limit of the two-dimensional self-avoiding walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE) with kappa = 8/3 leads to explicit predictions about the SAW. A remarkable feature of these predictions is that they yield not just critical exponents but also probability distributions for certain random variables associated with the self-avoiding walk. We test two of these predictions with Monte Carlo simulations and find excellent agreement, thus providing numerical support to the conjecture that the scaling limit of the SAW is SLE(8/3).     

The red queen's walk

We study a new type of walk with memory which might serve as a toy model for the behavior one must adopt to avoid exhaustion of resources and attraction of parasites and predators. The walk takes place on a regular square lattice with periodic boundary conditions. Although the walk is completely deterministic, it mimics a “true” self-avoiding walk, i.e. a random walk with weak autocorrelation. This shows that the Red Queen effect can lead to aperiodic behavior. In addition to the case of single walkers in a flat landscape we also study the cases of hilly landscapes and of several walkers performing simultaneous walks.     

On Continuous-Time Self-Avoiding Random Walk in Dimension Four

We study the distribution of the end-to-end distance of continuous-time self-avoiding random walks (CTRW) in dimension four from two viewpoints. From a real-space renormalization-group map on probabilities, we conjecture the asymptotic behavior of the end-to-end distance of a weakly self-avoiding random walk (SARW) that penalizes two-body interactions of random walks in dimension four on a hierarchical lattice. Then we perform the Monte Carlo computer simulations of CTRW on the four-dimensional integer lattice, paying special attention to the difference in statistical behavior of the CTRW compared with the discrete-time random walks. In this framework, we verify the result already predicted by the renormalization-group method and provide new results related to enumeration of self-avoiding random walks and calculation of the mean square end-to-end distance and gyration radius of continous-time self-avoiding random walks.     

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.     

Walks on Apollonian networks

We carry out comparative studies of random walks on deterministic Apollonian networks (DANs) and random Apollonian networks (RANs). We perform computer simulations for the mean first-passage time, the average return time, the mean-square displacement, and the network coverage for the unrestricted random walk. The diffusions both on DANs and RANs are proved to be sublinear. The effects of the network structure on the dynamics and the search efficiencies of walks with various strategies are also discussed. Contrary to intuition, it is shown that the self-avoiding random walk, which has been verified as an optimal local search strategy in networks, is not the best strategy for the DANs in the large size limit.     

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.     

An age-dependent approach to random walks on disordered lattices

We derive a new approach for the stochastic transport in random systems, starting from a phenomenological master equation with random transition rates. Our method combines the effective medium approximation with age-dependent dynamics. Within the framework of our approximation, the static disorder may be described by means of a system of age-dependent master equations. For translationally invariant systems which obey certain separability conditions, the approach is equivalent with the continuous time random walk theory. Moreover, for self-avoiding random walks our effective medium approximation is exact. For non self-avoiding random walks, the approximation neglects the correlations between successive transitions leading to closed paths on the lattice.     

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.     

Random walk statistics on fractal structures

We consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds: range, renewal theory, mean first passage time, etc. Asymptotic behaviors are shown to be controlled by the fractal (¯d) and spectral (¯d) dimensionalities of the considered structure. A simple decimation procedure giving the value of (¯d) is outlined and illustrated in the case of the Sierpinski gaskets. Recent results for the trapping problem, the self-avoiding walk, and the true-self-avoiding walk are briefly reviewed. New numerical results for diffusion on percolation clusters are also presented.     

On the connective constants of two-dimensional self-avoiding walks

A recent proposal that a new critical exponent characterises the variation of the self-avoiding walk connective constant with lattice co-ordination number is shown to be invalid. Instead, a functional relationship similar to that which holds for the Ising model in two dimensions is found to represent the available data for two-dimensional self-avoiding walks rather well.     

Sharp Phase Boundaries for a Lattice Flux Line Model

We consider a model of nonintersecting flux lines in a rectangular region on the lattice ^d , where each flux line is a non-isotropic self-avoiding random walk constrained to begin and end on the boundary of the region. The thermodynamic limit is reached through an increasing sequence of such regions. We prove the existence of several distinct phases for this model, corresponding to different regimes for the flux line density—a phase with zero density, a collection of phases with maximal density, and at least one intermediate phase. The locations of the boundaries of these phases are determined exactly for a wide range of parameters. Our results interpolate continuously between previous results on oriented and standard nonoriented self-avoiding random walks.     

Random walks on a ( 2+1)-dimensional deformable medium

A model of random walks on a deformable medium is proposed in 2+1 dimensions. The behavior of the walk is characterized by the stability parameter beta and the stiffness exponent alpha. The average square end-to-end distance l approximately equals (2nu) and the average number of visited sites approximately equals (k) are calculated. As beta increases, for each alpha there exists a critical transition point beta(c) from purely random walks ( nu = 1/2 and k approximate to 1) to compact growth ( nu = 1/3 and k = 2/3). The relationship between beta(c) and alpha can be expressed as beta(c) = e(alpha). The landscape generated by a walk is also investigated by means of the visit-number distribution N(n)(beta). There exists a scaling relationship of the form N(n)(beta)approximately n(-2)f(n/beta(z)).     

Thermal properties of a two-dimensional intrinsically curved semiflexible biopolymer

We study the behaviors of mean end-to-end distance and specific heat of a two-dimensional intrinsically curved semiflexible biopolymer with a hard-core excluded volume interaction. We find the mean square end-to-end distance R~2_N∝ N~βat large N, with N being the number of monomers. Both β and proportional constant are dependent on the reduced bending rigidity κ and intrinsic curvature c. The larger the c, the smaller the proportional constant, and 1.5 ≥β≥ 1. Up to a moderate κ = κ_c, or down to a moderate temperature T = T_c, β = 1.5, the same as that of a self-avoiding random walk, and the larger the intrinsic curvature, the smaller the κ_c. However, at a large κ or a low temperature, β is close to 1,and the conformation of the biopolymer can be more compact than that of a random walk. There is an intermediate regime with 1.5 β 1 and the transition from β = 1.5 to β = 1 is smooth. The specific heat of the system increases smoothly with increasing κ or there is no peak in the specific heat. Therefore, a nonvanishing intrinsic curvature seriously affects the thermal properties of a semiflexible biopolymer, but there is no phase transition in the system.     

Oriented polymers: A transfer matrix calculation

Based on transfer matrix techniques and finite-size scaling, we study the oriented polymer (self-avoiding walk) with nearest neighbor interaction. In the repulsive regime, various critical exponents are computed and compared with exact values predicted recently. The polymer is also found to undergo a spiral transition for sufficiently strong attractive interaction. The fractal dimension of the polymer is computed in the repulsive and attractive regimes and at the spiral transition point. The later is found to be different from that at the collapse transition of the ordinary self-avoiding walk.     

Orbital effect of a magnetic field on two-leg Hubbard ladder

By setting up the relevant recursion relations and by doing exact and approximate calculations, we show that there is no critical dimension in a self-avoiding random walk on a simplex fractal. Received: 6 April 1998 / Revised: 4 August 1998 / Accepted: 26 August 1998     

Discretization dependence of criticality in model fluids: a hard-core electrolyte

Grand-canonical simulations at various levels, zeta=5-20, of fine-lattice discretization are reported for the near-critical 1:1 hard-core electrolyte or restricted primitive model (RPM). With the aid of finite-size scaling analyses, it is shown convincingly that, contrary to recent suggestions, the universal critical behavior is independent of zeta (> or approximately 4), thus the continuum (zeta--> infinity ) RPM exhibits Ising-type (as against classical, self-avoiding walk, XY, etc.) criticality. A general consideration of lattice discretization provides effective extrapolation of the intrinsically erratic zeta dependence, yielding (T*(c),rho*(c)) approximately equal to (0.0493(3),0.075) for the zeta=infinity RPM.     

Self-avoiding walk model for proteins

We study a model for the backbone of proteins on a square lattice which consists of the path traced out by a self-avoiding walk (SAW) on the lattice and bridges not belonging to sites on the SAW but connecting nearest neighbor sites of the SAW. We calculated the fractal dimensiond _w for random walk on this model and found thatd _w2.6, in disagreement with a recent suggestion thatd _w should be 2.     

A Variational Formula for the Free Energy of the Partially Directed Polymer Collapse

Long linear polymers in dilute solutions are known to undergo a collapse transition from a random coil (expand itself) to a compact ball (fold itself up) when the temperature is lowered, or the solvent quality deteriorates. A natural model for this phenomenon is a 1+1 dimensional self-interacting and partially directed self-avoiding walk. In this paper, we develop a new method to study the partition function of this model, from which we derive a variational formula for the free energy. This variational formula allows us to prove the existence of the collapse transition and to identify the critical temperature in a simple way. We also provide a probabilistic proof of the fact that the collapse transition is of second order with critical exponent 3/2.     

Alzheimer random walk

Using the Monte Carlo simulation, we investigate a memory-impaired self-avoiding walk on a square lattice in which a random walker marks each of sites visited with a given probability p and makes a random walk avoiding the marked sites. Namely, p = 0 and p = 1 correspond to the simple random walk and the self-avoiding walk, respectively. When p> 0, there is a finite probability that the walker is trapped. We show that the trap time distribution can well be fitted by Stacy’s Weibull distribution \(b{\left( {\tfrac{a}{b}} \right)^{\tfrac{{a + 1}}{b}}}{\left[ {\Gamma \left( {\tfrac{{a + 1}}{b}} \right)} \right]^{ - 1}}{x^a}\exp \left( { - \tfrac{a}{b}{x^b}} \right)\) where a and b are fitting parameters depending on p. We also find that the mean trap time diverges at p = 0 as ~p ^{? α} with α = 1.89. In order to produce sufficient number of long walks, we exploit the pivot algorithm and obtain the mean square displacement and its Flory exponent ν(p) as functions of p. We find that the exponent determined for 1000 step walks interpolates both limits ν(0) for the simple random walk and ν(1) for the self-avoiding walk as [ ν(p) ? ν(0) ] / [ ν(1) ? ν(0) ] = p ^β with β = 0.388 when p ? 0.1 and β = 0.0822 when p ? 0.1.     

Linking numbers for self-avoiding loops and percolation: application to the spin quantum hall transition

Nonlocal twist operators are introduced for the O(n) and Q-state Potts models in two dimensions which count the numbers of self-avoiding loops (respectively, percolation clusters) surrounding a given point. Their scaling dimensions are computed exactly. This yields many results: for example, the number of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. Its mean behaves as (1/3sqrt[3] pi) |ln( p(c)-p)| as p-->p(c)-. As an application we compute the exact value sqrt[3]/2 for the conductivity at the spin Hall transition, as well as the shape dependence of the mean conductance in an arbitrary simply connected geometry with two extended edge contacts.