A simple model of DNA denaturation and mutually avoiding walks statistics

Recently Garel, Monthus and Orland [Europhys. Lett. 55, 132 (2001)] considered a model of DNA denaturation in which excluded volume effects within each strand are neglected, while mutual avoidance is included. Using an approximate scheme they found a first order denaturation. We show that a first order transition for this model follows from exact results for the statistics of two mutually avoiding random walks, whose reunion exponent is c > 2, both in two and three dimensions. Analytical estimates of c due to the interactions with other denaturated loops, as well as numerical calculations, indicate that the transition is even sharper than in models where excluded volume effects are fully incorporated. The probability distribution of distances between homologous base pairs decays as a power law at the transition. Received 8 July 2002 / Received in final form 25 July 2002 Published online 17 September 2002     

Symmetry in stochasticity: Random walk models of large-scale structure

This paper describes the insights gained from the excursion set approach, in which various questions about the phenomenology of large-scale structure formation can be mapped to problems associated with the first crossing distribution of appropriately defined barriers by random walks. Much of this is summarized in R K Sheth, AIP Conf. Proc. 1132, 158 (2009). So only a summary is given here, and instead a few new excursion set related ideas and results which are not published elsewhere are presented. One is a generalization of the formation time distribution to the case in which formation corresponds to the time when half the mass was first assembled in pieces, each of which was at least 1/n times the final mass, and where n ≥ 2; another is an analysis of the first crossing distribution of the Ornstein–Uhlenbeck process. The first derives from the mirror-image symmetry argument for random walks which Chandrasekhar described so elegantly in 1943; the second corrects a misuse of this argument. Finally, some discussion of the correlated steps and correlated walks assumptions associated with the excursion set approach, and the relation between these and peaks theory are also included. These are problems in which Chandra’s mirror-image symmetry is broken.     

Regularity criteria for linear chains

The structural analysis of linear chains of arbitrary fixed shape is discussed in the context of a spectral approach. The shape of the chain is characterized by a set of scalar and pseudoscalar invariants, which remain constant under translations and rotations. The statistical properties of the set of invariants are compared with the analogous characteristics for a freely linked chain. The proposed criteria have the self-averaging property for relatively short (∼100–300 links) chains and can be used to discern possible latent periodicities and symmetries in a system. As examples, two applications of the theory are considered: the structural analysis of chains generated by random walks on a cubic lattice and protein C_α backbones. Zh. éksp. Teor. Fiz. 116, 620–635 (August 1999)     

The electrostatic persistence length of polymers beyond the OSF limit

We use large-scale Monte Carlo simulations to test scaling theories for the electrostatic persistence length l _e of isolated, uniformly charged polymers with Debye-Hückel intrachain interactions in the limit where the screening length κ^-1 exceeds the intrinsic persistence length of the chains. Our simulations cover a significantly larger part of the parameter space than previous studies. We observe no significant deviations from the prediction l _e∝κ^-2 by Khokhlov and Khachaturian which is based on applying the Odijk-Skolnick-Fixman theories of electrostatic bending rigidity and electrostatically excluded volume to the stretched de Gennes-Pincus-Velasco-Brochard polyelectrolyte blob chain. A linear or sublinear dependence of the persistence length on the screening length can be ruled out. We show that previous results pointing into this direction are due to a combination of excluded-volume and finite chain length effects. The paper emphasizes the role of scaling arguments in the development of useful representations for experimental and simulation data. Received 12 February 2002     

Interaction of Anderson impurities with high orbital angular momenta: Non-RKKY behavior and instability of Kondo lattice

The hybridization-induced interaction of Anderson impurities with orbital angular momentum l is revisited. At short distances R<R _c ∝(l+1)/k _F the interaction has antiferromagnetic sign and decays as (R _c /R)^4l . At larger distances R>R _c the RKKY-like oscillatory interaction sets in. As l increases, the system will sooner or later enter the “short-distance” domain, where the intersite magnetic interaction dominates over the screening processes. This means that, contrary to previous expectations, the nonmagnetic state of the Anderson lattice is unstable at l→∞. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 1, 76–81 (10 January 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries

The general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias. Continuous time walks are characterized by random pauses between jumps, with a common pausing time distribution(t). An analytic solution in the form of an inverse Laplace transform for P(l, t), the probability of a walker being atl at timet if it started atl _o att=0, is obtained in the presence of completely absorbing boundaries. Numerical results for P(l, t) are presented for characteristically different (t), including one which leads to a non-Gaussian behavior for P(l, t) even for larget. Asymptotic results are obtained for the number of surviving walkers and the mean l showing the effect of the absorption at the boundary.This study was partially supported by ARPA and monitored by ONR(N00014-17-0308).     

Simulated migration and reaction of correlated point defects

Monte Carlo techniques are applied to defect migration in the vicinity of a fixed reaction volume. Intended to simulate “stage I_D ” annealing of correlated interstitials and vacancies in fcc metals, large reaction spheres of up to 720 atomic volume are employed, with mobile defects in the (100) split configuration symmetrically diffusing from maximum distances of more than twice the capture radius r ₀ and for a maximum of 100 jumps. This discrete approach and continuum theory are judged to be equally valid. Random walk recovery probabilities are not a smooth function of initial distance r, but fQr large reaction volumes agree with continuum theory nearly as well for r ～ r ₀ as for r > 2r ₀. Absolute agreement is improved as the “dumbbell” separation of the split defect is increased. Recovery due to an extended distribution of defects is obtained by weighting individual walks from distances < 3r ₀: the resulting composite annealing curves disallow observation of structure and compare favorably with resistivity data. The number of symmetric random walk jumps N required to reach the maximum rate of correlated recovery-the “I_D peak”-is found for a wide range of initial distributions, the best parameter estimates for Cu giving N = 50 ± 15. This value and resistivity data give N for the uncorrelated recovery peak in agreement with theory. One dimensional migration is excluded by these results.     

Open Quantum Random Walks

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.     

Recoil growth algorithm for chain molecules with continuous interactions

The recoil growth (RG) scheme is a dynamic Monte Carlo algorithm that has been suggested as an improvement over the configurational bias Monte Carlo (CBMC) method (Consta, S., Wilding, N. B., Frenkel, D. and Alexandrowicz, Z., 1999, J. chem. Phys., 110, 3220). The RG method had originally been tested for hard core polymers on a lattice, and it was found that RG outperforms CBMC for dense systems and long chain molecules. In the present paper, the RG scheme is extended to the practically more relevant case of off-lattice chain molecules with continuous interactions. It is found that for longer chain molecules RG becomes over an order of magnitude more efficient than CBMC. However, other schemes are better suited to the computation of the excess chemical potential. Moreover, it is more difficult to parallelize RG than CBMC.     

Scaling and universality in models of step bunching: the “C<Superscript>+</Superscript>–C<Superscript>-</Superscript>” model

We recently introduced a novel model of step flow crystal growth – the so-called “C⁺–C^-” model [B. Ranguelov et al., C.R. Acad. Bulgare Sci. 60, 389 (2007)]. In this paper we aim to develop a complete picture of the model’s behaviour in the framework of the notion of universality classes. The basic assumption of the model is that the reference (“equilibrium”) densities used to compute the supersaturation might be different on either side of a step, so C_L/C_R ≠ 1 (L/R stands for left/right in a step train descending from left to right), and that this will eventually cause destabilization of the regular step train. Linear stability analysis considering perturbation of the whole step train shows that the vicinal is always unstable when the condition C_L /C_R >1 is fulfilled. Numerical integration of the equations of step motion combined with an original monitoring scheme(s) results in obtaining the exact size- and time- scaling of the step bunches in the limit of long times (including the numerical prefactors). Over a broad range of parameters the surface morphology is characterized by the appearance of the minimal interstep distance at the beginning of the bunches (at the trailing edge of the bunch) and may be described by a single universality class, different from those already generated by continuum theories [A. Pimpinelli et al., Phys. Rev. Lett. 88, 206103 (2002), J. Krug et al., Phys. Rev. B 71, 045412 (2005)]. In particular, the scaling of the minimal interstep distance l_min in the new universality class is shown to be l_min = (S_n /N)^1/(n+1), where N is the number of steps in the bunch, n is the exponent in the step-step repulsion law U ~ 1/d₀ ⁿ for two steps placed a distance d₀ apart and S_n is a combination of the model parameters. It is also shown that N scales with time with universal exponent 1/2 independent of n. For the regime of slow diffusion it is obtained for the first time that the time scaling depends only on the destabilization parameter C_L/C_R. The bunching outside the parameter region where the above scaling exists cannot be assigned to a specific universality class and thus should be considered non-universal.     

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.     

On Connected Diagrams and Cumulants of Erdős-Rényi Matrix Models

Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian we introduce two families of discrete matrix models constructed both with the help of the Erdős-Rényi ensemble of random graphs. Corresponding matrix sums represent the characteristic functions of the average number of walks and closed walks over the random graph. These sums can be considered as discrete analogues of the matrix integrals of random matrix theory. We study the diagram structure of the cumulant expansions of logarithms of these matrix sums and analyze the limiting expressions as n → ∞ in the cases of constant and vanishing edge probabilities.     

Propriétés d'intersection des marches aléatoires

We study intersection properties of multi-dimensional random walks. LetX andY be two independent random walks with values in ? ^d (d≦3), satisfying suitable moment assumptions, and letI _n denote the number of common points to the paths ofX andY up to timen. The sequence (I _n ), suitably normalized, is shown to converge in distribution towards the “intersection local time” of two independent Brownian motions. Results are applied to the proof of a central limit theorem for the range of a two-dimensional recurrent random walk, thus answering a question raised by N. C. Jain and W. E. Pruitt.     

Adaptive random walks on the class of Web graphs

We study random walk with adaptive move strategies on a class of directed graphs with variable wiring diagram. The graphs are grown from the evolution rules compatible with the dynamics of the world-wide Web [B. Tadić, Physica A 293, 273 (2001)], and are characterized by a pair of power-law distributions of out- and in-degree for each value of the parameter β, which measures the degree of rewiring in the graph. The walker adapts its move strategy according to locally available information both on out-degree of the visited node and in-degree of target node. A standard random walk, on the other hand, uses the out-degree only. We compute the distribution of connected subgraphs visited by an ensemble of walkers, the average access time and survival probability of the walks. We discuss these properties of the walk dynamics relative to the changes in the global graph structure when the control parameter β is varied. For β≥ 3, corresponding to the world-wide Web, the access time of the walk to a given level of hierarchy on the graph is much shorter compared to the standard random walk on the same graph. By reducing the amount of rewiring towards rigidity limit β↦β_c≲ 0.1, corresponding to the range of naturally occurring biochemical networks, the survival probability of adaptive and standard random walk become increasingly similar. The adaptive random walk can be used as an efficient message-passing algorithm on this class of graphs for large degree of rewiring.     

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)).     

Continuum description of anomalous diffusion on a comb structure

Anomalous diffusion on a comb structure consisting of a one-dimensional backbone and lateral branches (teeth) of random length is considered. A well-defined classification of the trajectories of random walks reduces the original problem to an analysis of classical diffusion on the backbone, where, however, the time of this process is a random quantity. Its distribution is dictated by the properties of the random walks of the diffusing particles on the teeth. The feasibility of applying mean-field theory in such a model is demonstrated, and the equation for the Green’s function with a partial derivative of fractional order is obtained. The characteristic features of the propagation of particles on a comb structure are analyzed. We obtain a model of an effective homogeneous medium in which diffusion is described by an equation with a fractional derivative with respect to time and an initial condition that is an integral of fractional order. Zh. éksp. Teor. Fiz. 114, 1284–1312 (October 1998)     

Scale-Free Static and Dynamical Correlations in Melts of Monodisperse and Flory-Distributed Homopolymers

It has been assumed until very recently that all long-range correlations are screened in three-dimensional melts of linear homopolymers on distances beyond the correlation length ?? characterizing the decay of the density fluctuations. Summarizing simulation results obtained by means of a variant of the bond-fluctuation model with finite monomer excluded volume interactions and topology violating local and global Monte Carlo moves, we show that due to an interplay of the chain connectivity and the incompressibility constraint, both static and dynamical correlations arise on distances r???. These correlations are scale-free and, surprisingly, do not depend explicitly on the compressibility of the solution. Both monodisperse and (essentially) Flory-distributed equilibrium polymers are considered.     

Disordered Fermions on Lattices and Their Spectral Properties

We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra some standard results concerning the spectral properties exhibited by temperature states of disordered quantum spin systems. We investigate the Arveson spectrum, known to physicists as the set of the Bohr frequencies. We also establish its connection with the Connes and Borchers spectra, and with the associated invariants for such W ^∗-dynamical systems which determine the type of von Neumann algebras generated by a temperature state. We prove that all such spectra are independent of the disorder. Such results cover infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice ℤ ^d , including models exhibiting some standard spin-glass-like behavior. As a natural application, we show that a temperature state can generate only a type III\mathop {\rm {III}} von Neumann algebra (with the type III₀\mathop {\rm {III_{0}}} component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type III_l\mathop {\rm {III_{\lambda }}} for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.     

Improved Lower Bounds for the Critical Probability of Oriented Bond Percolation in Two Dimensions

We present a coupled decreasing sequence of random walks on Z that dominate the edge process of oriented bond percolation in two dimensions. Using the concept of random walk in a strip, we describe an algorithm that generates an increasing sequence of lower bounds that converges to the critical probability of oriented percolation p_c. From the 7th term on, these lower bounds improve upon 0.6298, the best rigorous lower bound at present, establishing 0.63328 as a rigorous lower bound for p_c. Finally, a Monte Carlo simulation technique is presented; the use thereof establishes 0.64450 as a non-rigorous five-digit-precision (lower) estimate for p_c. Mathematics Subject Classification (1991): 60K35 Supported by CNPq (grant N.301637/91-1). Supported by a grant from CNPq.     

Polymers from C60 barrelenes

Dimer, trimer, and chain structures consisting of covalently joined barrel-shaped C₆₀ monomers (barrelenes) with different types of bonds are modeled. The computed distances between the centers of the molecules within a barrelene chain (0.70–0.74 nm) and between the symmetry axes of the chains (0.63–0.77 nm) are comparable to the analogous distances on the surface of fullerites synthesized by Davydov’s group [V. A. Davydov, L. S. Koshevarova, A. V. Rakhmanina et al., JETP Lett. 63, 818 (1996)] and subjected to high pressures (∼10 GPa) and heating (∼700 K). The formation of special two-dimensional structures (triangles, squares, chains with a kink, stars), which are observed experimentally, is explained on the basis of the model, and 3D polymer structures consisting of C₆₀ barrelenes with density close to that of samples obtained earlier are constructed. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 9, 678–683 (10 May 1998)