Random walks in asymmetric random environments

We consider random walks on Z ^d with transition ratesp(x, y) given by a random matrix. Ifp is a small random perturbation of the simple random walk, we show that the walk remains diffusive for almost all environmentsp ifd>2. The result also holds for a continuous time Markov process with a random drift. The corresponding path space measures converge weakly, in the scaling limit, to the Wiener process, for almost everyp.Dedicated to Joel Lebowitz on his 60th birthdaySupported by NSF-grant DMS-8903041     

Spatial structure and stability based on random walks

A general expression for a recursion formula which describes a random walk with coupled modes is given. In this system, the random walker is specified by the jumping probabilities P⁺ and P^– which depend on the modes. The transition probability between the modes is expressed by a jumping probabilityR ^(ij) (orr _ij). With the aid of this recursion formula, spatial structures of the steady state of a coupled random walk are studied. By introducing a Liapunov function and entropy, it is shown that the stability condition for the present system can be expressed as the principle of the extremum entropy production.On leave of absence from Tohoku University, Department of Applied Science, Faculty of Engineering, Sendai, 980 Japan.     

A renormalization result for the intersection local time of lattice random walks ind≧3 dimensions

In this paper we derive some general conditions on stable walks inZ ^d , under which the central limit theorem holds for their normalized intersection local time. In particular, we prove that the process given by the normalized intersection local time of the simple random walk inZ ^d , withd3, is weakly convergent to the standard Brownian motion.BiBoS; SFB 237 Bochum-Essen-Düsseldorf; CERFIM, Locarno.     

Current noise and long time tails in biased disordered random walks

We calculate the mean velocity and the velocity correlation function for a random walk with a uniform bias on a disordered chain. We find a new long time tail in the velocity correlation function due to the combined effects of the bias and of the disorder in the site variables. This long time tail persists to a low-frequency cutoff inversely proportional to the square of the bias. By associating the velocity correlation function with the spectrum of current fluctuations, we calculate the excess low-frequency current noise associated with this long time tail. The spectrum of current fluctuations goes as(I ²/N)f ^–1/2, whereI is the DC current,N is the number of charge carriers, andf is the frequency. The possible connection to 1/f noise is discussed. The calculation is done by a perturbation expansion in the strength of the disorder, but is shown to be exact to all orders for weak enough bias.Supported by a fellowship of the German Academic Exchange Service (DAAD).Supported by the National Science Foundation through Grant No. DMR-8108328 and through the Cornell Materials Science Center.     

Multiple trapping of random walkers on periodic lattices

Formulas are obtained for the mean absorption time of a set ofk independent random walkers on periodic space lattices containingq traps. We consider both discrete (here we assume simultaneous stepping) and continuous-time random walks, and find that the mean lifetime of the set of walkers can be obtained, via a convolution-type recursion formula, from the generating function for one walker on the perfect lattice. An analytical solution is given for symmetric walks with nearest neighbor transitions onN-site rings containing one trap (orq equally spaced traps), for both discrete and exponential distribution of stepping times. It is shown that, asN , the lifetime of the walkers is of the form Ta_kN², whereT is the average time between steps. Values ofa _k, 2 Sk 6, are provided.     

Random Time-Dependent Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on the lattice \mathbb Z^d{\mathbb {Z}^d} performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in \mathbb Z^d{\mathbb {Z}^d} when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. A generalization to unitary updates distributed according to a Markov process is also provided.     

Spectral Dimension and Random Walks on the Two Dimensional Uniform Spanning Tree

We study the simple random walk on the uniform spanning tree on \mathbb Z²{\mathbb {Z}^2} . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on \mathbb Z²{\mathbb {Z}^2} is 16/13 almost surely.     

线性高分子体系中高分子链间排除体积效应的模型化

提出了线性高分子体系中高分子链间排除体积效应的一个它回避模型，并且针对具体的四个模型系统进行了计算机模拟. 计算结果表明：1)线团的均方末端距〈R²〉与行走步数(N)仍然保持着与无规线团模型一样的线性关系；2)但与无规线团相比，线团的空间尺寸被压缩；3)与两侧方向的回避相比，在行走的前进方向的回避而导致的压缩效应更加明显. 关键词： 排除体积效应 回避行走 无规线团 线性高分子     

Polyelectrolyte chains in poor solvent. A variational description of necklace formation

We study the properties of polyelectrolyte chains under different solvent conditions, using a variational technique. The free energy and the conformational properties of a polyelectrolyte chain are studied by minimizing the free energy F_N, depending on N(N - 1)/2 trial probabilities that characterize the conformation of the chain. The Gaussian approximation is considered for a ring of length 2⁴ < N < 2⁸ and for an open chain of length 50 < N < 200 in poor- and theta-solvent conditions, including a Coulomb repulsion between the monomers. In theta-solvent conditions the blob size is measured and found in agreement with scaling theory, including charge depletion effects, expected for the case of an open chain. In poor-solvent conditions, a globule instability, driven by electrostatic repulsion, is observed. We notice also inhomogeneous behavior of the monomer-monomer correlation function, reminiscence of necklace formation in poor-solvent polyelectrolyte solutions. A global phase diagram in terms of solvent quality and inverse Bjerrum length is presented. Received 7 June 2001 and Received in final form 17 October 2001     

Dynamical Triangulation Models with Matter:¶High Temperature Region

We consider a canonical ensemble with a fixed number N of triangles for planar dynamical triangulation models with compact spin in the high temperature region. We find the asymptotics of the partition function Z(N) and reveal the analytic properties of the generating function U(x)=∑: Z(N)x ^N . New cluster expansion techniques are developed for this case. For fixed triangulation it would be quite standard but for random triangulations one has to deal with the non-zero entropy of the space between clusters. It is a multiscale expansion, where the role of scale is played by a topological parameter – the maximal length of chains of imbedded not simply connected clusters. Received: 19 June 2001 / Accepted: 12 October 2001     

Are random walks random?

Random walks have been created using the pseudo-random generators in different computer language compilers (BASIC, PASCAL, FORTRAN, C++) using a Pentium processor. All the obtained paths have apparently a random behavior for short walks (2¹⁴ steps). From long random walks (2³³ steps) different periods have been found, the shortest being 2¹⁸ for PASCAL and the longest 2³¹ for FORTRAN and C++, while BASIC had a 2²⁴ steps period. The BASIC, PASCAL and FORTRAN long walks had even (2 or 4) symmetries. The C++ walk systematically roams away from the origin. Using deviations from the mean-distance rule for random walks, d² ∝ N, a more severe criterion is found, e.g. random walks generated by a PASCAL compiler fulfills this criterion to N < 10 000.     

Use of the momentum approximation in classical scattering calculations

Abstract

The validity range of the momentum approximation in binary collisions : s investigated. It is shown that at high energies, E ? 10 · Z ₁ Z ₂ e ²/a _TF, the approximation is even applicable to scattering angles ? as large as 60 degrees. An empirical formula for the evaluation of the momentum approximation employing the Thomas-Fermi-Moliére potential is given. It reduces the computer time to calculate ? to ～⊥₁₀ of the time consumed in using the standard algebraic functions appearing in the momentum approximation.     

Random walk on a disordered directed Bethe lattice

The random walk of a particle on a directed Bethe lattice of constant coordinanceZ is examined in the case of random hopping rates. As a result, the higher the coordinance, the narrower the regions of anomalous drift and diffusion. The annealed and quenched mean square dispersions are calculated in all dynamical phases. In opposition to the one-dimensional (Z=2) case, the annealed and quenched mean quadratic dispersions are shown to be identical in all phases.We shall employ indifferently the expressions Bethe lattice or infinite Cayley tree to denote an infinite ramified lattice of constant coordinanceZ.^{(4, 5)}     

Forward on the <Emphasis Type="Italic">N</Emphasis> = <Emphasis Type="Italic">Z</Emphasis> line with GASP

The experimental study of the proton-rich nuclei close to the N = Z line is a constant challenge for nuclear spectroscopy, mainly due to the difficulty to produce them with the currently available beam/target combinations. Significant advances on this direction were obtained from experiments performed with the GASP array during the last two years: the yrast line of ⁸⁴Mo was extended up to 10 ⁺ , ⁸⁸Ru observed for the first time, and the N = Z + 1 line was mapped from ⁸¹Zr to ⁹⁵Ag. These new results allow us to have a more complete image of the transition from the well-deformed shell closure at N,Z = 40 to the spherical-shell closure at N,Z = 50, and highlights some particular effects that can be observed only in the vicinity of the N = Z line.Received: 10 January 2003, Published online: 23 March 2004PACS: 21.10.-k Properties of nuclei; nuclear energy levels - 21.10.Pc Single-particle levels and strength functions - 21.10.Re Collective levels - 23.20.-g Electromagnetic transitions     

Atoms in Strong Magnetic Fields:¶The High Field Limit at Fixed Nuclear Charge

Let E(B,Z,N) denote the ground state energy of an atom with N electrons and nuclear charge Z in a homogeneous magnetic field B. We study the asymptotics of E(B,Z,N) as B→∞ with N and Z fixed but arbitrary. It is shown that the leading term has the form (ln B)² e(Z,N), where e(Z,N) is the ground state energy of a system of N bosons with delta interactions in one dimension. This extends and refines previously known results for N= 1 on the one hand, and N,Z→∞ with B/Z ³→∞ on the other hand. Received: 9 December 1999 / Accepted: 15 February 200     

Mass Measurements of Exotic Nuclei around N=Z=40 with CSS2

Mass measurements of the N=Z nuclei ⁸⁰Zr, ⁷⁶Sr, ⁶⁸Se were performed for the first time and a new measurement was obtained for ⁸⁰Y, using the second cyclotron CSS2 of GANIL as a high-resolution spectrometer. Ions around N=Z were produced by fusion-evaporation in the inverse ⁵⁸Ni (4.32MeVA) + ²⁴Mg and ¹²C reactions. New masses were measured by a time-of-flight method, with a precision of 2⋅10⁻⁶, by using well-known masses as references. Study of the double binding energy difference δV _np is then performed leading to a strong N=Z Wigner effect around N=Z=40. Knowledge of new masses in this region also plays a crucial role in the modelling of the astrophysical rp process. This revised version was published online in July 2006 with corrections to the Cover Date.     

Two-dimensional random walk on the ground state energy surface: The N-Z distributions from nuclear heavy-ion collisions

The nucleon transfer process in deep inelastic collision is formulated as a two-dimensional random walk process on the N-Z plane. The probability of each step is assumed to be proportional to the available final state density which is determined by the ground state energy surface of the colliding dinuclear system. Here a model problem is considered, where the long and narrow valley in the energy surface is replaced by a slot with a flat bottom. Even in this crude model, good agreement is obtained between the calculated widths of the N-Z distribution and the experimental values, indicating that the ground state energy surface plays an important role in the mass and charge transfer phenomena in deep inelastic processes.     

Entropic repulsion of the lattice free field,II. The 0-boundary case

This paper is a continuation of [5]. We consider the Euclidean massless free field on a boxV _N of volumeN ^d with O-boundary condition; that is the centered Gaussian field with covariances given by the Green function of the simple random walk on ^d ,d3, killed as it exitsV _N . We show that the probability, that all the spins are positive in the boxV _N decays exponentially at a surface rateN ^d–1 . This is in contrast with the rateN ^d–2 logN for the infinite field of [5].     

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.