Exponential convergence to equilibrium for a class of random-walk models

We prove exponential convergence to equilibrium (L ² geometric ergodicity) for a random walk with inward drift on a sub-Cayley rooted tree. This randomwalk model generalizes a Monte Carlo algorithm for the self-avoiding walk proposed by Berretti and Sokal. If the number of vertices of levelN in the tree grows asC _N ~ ^N N ^–1 , we prove that the autocorrelation time satisfies N² N¹⁺     

Marriage of exact enumeration and 1/d expansion methods: Lattice model of dilute polymers

We consider the properties of a self-avoiding polymer chain with nearestneighbor contact energy on ad-dimensional hypercubic lattice. General theoretical arguments enable us to prescribe the exact analytic form of then-segment chain partition functionC _n ,and unknown coefficients for chains of up to 11 segments are determined using exact enumeration data ind=2–6. This exact form provides the main ingredient to produce a large-n expansion ind ^–1of the chain free energy through fifth order with the full dependence on the contact energy retained. The -dependent chain connectivity constant and free energy amplitude are evaluated within thed ^–1expansion toO(d ^–5). Our general formulation includes for the first time self-avoiding walks, neighboravoiding walks, theta, and collapsed chains as particular limiting cases.     

The Lattice Model for Addressing Phenomenological Diffusion Problems Associated with Grain Boundaries

Mapping a phenomenological diffusion problem onto a lattice permits lattice-based random walk theory to address the problem. The solution is generally carried out using Monte Carlo methods. This paper reviews the substantial progress that has been made with this method for tracer diffusion associated with grain boundaries.     

High-precision Monte Carlo test of the conformai-invariance predictions for two-dimensional mutually avoiding walks

Let _l be the critical exponent associated with the probability thatl independentN-step ordinary random walks, starting at nearby points, are mutually avoiding. Using Monte Carlo methods combined with a maximum-likelihood data analysis, we find that in two dimensions ₂=0.6240±0.0005±0.0011 and ₃=1.4575±0.0030±0.0052, where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second error bar represents statistical error (classical 95% confidence limits). These results are in good agreement with the conformal-invariance predictions ₂=5/8 and ₃=35/24.     

Ballistic behavior in a 1D weakly self-avoiding walk with decaying energy penalty

We consider a weakly self-avoiding walk in one dimension in which the penalty for visiting a site twice decays as exp[–|t–s| ^–p ] wheret ands are the times at which the common site is visited andp is a parameter. We prove that ifp<1 and is sufficiently large, then the walk behaves ballistically, i.e., the distance to the end of the walk grows linearly with the number of steps in the walk. We also give a heuristic argument that ifp>3/2, then the walk should have diffusive behavior. The proof and the heuristic argument make use of a real-space renormalization group transformation.     

基于自规避随机游走的节点排序算法

评估复杂网络系统的节点重要性有助于提升其系统抗毁性和结构稳定性. 目前, 定量节点重要性的排序算法通常基于网络结构的中心性指标如度数、介数、紧密度、特征向量等. 然而, 这些算法需要以知晓网络结构的全局信息为前提, 很难在大规模网络中实际应用. 基于自规避随机游走的思想, 提出一种结合网络结构局域信息和标签扩散的节点排序算法. 该算法综合考虑了节点的直接邻居数量及与其他节点之间的拓扑关系, 能够表征其在复杂网络系统中的结构影响力和重要性. 基于三个典型的实际网络, 通过对极大连通系数、网络谱距离数、节点连边数和脆弱系数等评估指标的实验对比, 结果表明提出的算法显著优于现有的依据局域信息的节点排序算法.     

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.     

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.     

Discrete Green's Function for Lattice Random Walk

１　Ｉｎｔｒｏｄｕｃｔｉｏｎ　　Ｏｐｔｉｃａｌｔｏｍｏｇｒａｐｈｙｐｒｏｖｉｄｅｓａｎａｌｔｅｒｎａｔｉｖｅｔｅｃｈｎｏｌｏｇｙｔｏｐｒｏｂｅｂｒｅａｓｔｃａｎｃｅｒａｎｄｍｏｎｉｔｏｒｈｕｍａｎｔｉｓｓｕｅ’ｓｆｕｎｃｔｉｏｎａｌｐａｒａｍｅｔｅｒｎｏｎｉｎｖａｓｉｖｅｌｙ［１,２］．Ｐｈｏｔｏｎｍｉｇｒａｔｉｏｎｉｎｔｉｓｓｕｅｐｌａｙｓａｋｅｙｒｏｌｅｉｎｏｐｔｉｃａｌｔｏｍｏｇｒａｐｈｙ．Ｒｅｃｅｎｔｌｙ,ａｌａｔｔｉｃｅｒａｎｄｏｍｗａｌｋｍｏｄｅｌ［３,４］ｉｓｅｍｐｌｏｙｅｄｔｏｄ…     

Discrete Green's Function for Lattice Random Walk

A lattice random walk model based on particles scattering on discrete lattice of homogenous space is introduced. The discrete Green's function (DFG) for two-dimensional and three-dimensional lattice random walk of photon is found and proved by mathematical induction. The convolution theorem of photon lattice random walk is presented. They can be used with the method of images to calculate the photon density distribution in semi-infinite and finite slab homogenous turbid media such as tissue.     

Quenched Averages for Self-Avoiding Walks and Polygons on Deterministic Fractals

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W _n (S), and rooted self-avoiding polygons P _n (S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P _n (S), and W _n(S) for an arbitrary point S on the lattice. These are used to compute the averages ,, and over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent are the same for the annealed and quenched averages. However, , and , where the exponents and , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives and , to be compared with the known annealed values and .     

Extended Quark Potential Model from Random Phase Approximation

The quark potential model is extended to include the sea quark excitation using the random phase approx-imation. The effective quark interaction preserves the important QCD properties - chiral symmetry and confinementsimultaneously. A primary qualitative analysis shows that the π meson as a well-known typical Goldstone boson andthe other mesons made up of valence qq quark pair such as the ρ meson can also be described in this extended quarkpotential model.     

Asymptotic distributions for self-avoiding walks constrained to strips,cylinders, and tubes

A transfer matrix method for treating self-avoiding walks on a lattice is developed. Single walks confined to infinitely long strips, cylinders, or tubes are considered, particularly in the limit where the length of the walk becomes infinite compared to the transverse dimensions. In this case relevant distributions are demonstrated to be asymptotically Gaussian. Explicit numerical results are given for a few of the narrower systems. Similar results for self-avoiding cycles are indicated, too. Finally, the behavior of the various distributions as a function of strip width is discussed.Supported by the Robert A. Welch Foundation, Houston, Texas.     

Phase Diagram and Tricritical Behavior of a Spin-2 Transverse Ising Model in a Random Field

The phase diagrams of a spin-2 transverse Ising model with a random field on honeycomb, square, and simple-cubic lattices, respectively, are investigated within the framework of an effective-field theory with correlations.We find the behavior of the tricritical point and the reentrant phenomenon for the system with any coordination number z, when the applied random field is bimodal. The behavior of the tricritical point is also examined as a function of applied transverse field. The reentrant phenomenon comes from the competition between the transverse field and the random field.     

A Lattice Model for the Line Tension of a Sessile Drop

Within a semi-infinite three-dimensional lattice gas model describing the coexistence of two phases on a substrate, we study, by cluster expansion techniques, the free energy (line tension) associated with the contact line between the two phases and the substrate. We show that this line tension, is given at low temperature by a convergent series whose leading term is negative, and equals 0 at zero temperature.     

Velocity and Dispersion for a Two-Dimensional Random Walk

In the paper, we consider the transport of a two-dimensional random walk. The velocity and the dispersion of this two-dimensional random walk are derived. It mainly show that： （i） by controlling the values of the transition rates, the direction of the random walk can be reversed; （ii） for some suitably selected transition rates, our two-dimensional random walk can be efficient in comparison with the one-dimensional random walk. Our work is motivated in part by the challenge to explain the unidirectional transport of motor proteins. When the motor proteins move at the turn points of their tracks （i.e., the cytoskeleton filaments and the DNA molecular tubes）, some of our results in this paper can be used to deal with the problem.     

On multiple phase transitions for branching Markov chains

We consider branching Markov chains on a countable set. We give a necessary and sufficient condition in terms of the transition kernel of the underlying Markov chain to have two phase transitions. We compute the critical values. We apply this result to prove that asymmetric branching random walks onZ have two phase transitions.     

Parrondo Games as Lattice Gas Automata

Parrondo games are coin flipping games with the surprising property that alternating plays of two losing games can produce a winning game. We show that this phenomenon can be modelled by probabilistic lattice gas automata. Furthermore, motivated by the recent introduction of quantum coin flipping games, we show that quantum lattice gas automata provide an interesting definition for quantum Parrondo games.     

Phase Diagram of a Two-Species Lattice Model with a Linear Instability

We discuss the properties of a one-dimensional lattice model of a driven system with two species of particles in which the mobility of one species depends on the density of the other. This model was introduced by Lahiri and Ramaswamy ( Phys . Rev. Lett ., 79 , 1150 (1997)) in the context of sedimenting colloidal crystals, and its continuum version was shown to exhibit an instability arising from linear gradient couplings. In this paper we review recent progress in understanding the full phase diagram of the model. There are three phases. In the first, the steady state can be determined exactly along a representative locus using the condition of detailed balance. The system shows phase separation of an exceptionally robust sort, termed strong phase separation, which survives at all temperatures. The second phase arises in the threshold case where the first species evolves independently of the second, but the fluctuations of the first influence the evolution of the second, as in the passive scalar problem. The second species then shows phase separation of a delicate sort, in which long-range order coexists with fluctuations which do not damp down in the large-size limit. This fluctuation-dominated phase ordering is associated with power law decays in cluster size distributions and a breakdown of the Porod law. The third phase is one with a uniform overall density, and along a representative locus the steady state is shown to have product measure form. Density fluctuations are transported by two kinematic waves, each involving both species and coupled at the nonlinear level. Their dissipation properties are governed by the symmetries of these couplings, which depend on the overall densities. In the most interesting case, the dissipation of the two modes is characterized by different critical exponents, despite the nonlinear coupling.