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.     

Application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media

We consider the application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media. It is found that answers to interesting physics questions can be expressed in terms of several new fractal dimensions (in addition to the fractal dimensiond _f ): (1)d _f ^BB , the fractal dimension of the backbone, arises in connection with electric current flow, (2)d _red, the fractal dimension of the singly connected bonds in the backbone, arises in connection with its equivalence to the thermal scaling power, (3)d _E, the fractal dimension of the of the elastic backbone, (4)d _u, the fractal dimension of the unscreened perimeter, arises in connection with the viscosity singularity at the gelation threshold, (5)d _min the fractal dimension of the minimum path (or chemical distance) between two sites, arises in co-nnection with the Aharony-Stauffer conjecture, (6)dw, the fractal dimension of a random walk, (7)d _G, the fractal dimension of growth sites that arise as a random walk creates a cluster. Relations among these fractal dimensions are discussed, some of which can be proved and others of which are conjectures whose validity has been established only in certain limiting cases.Supported in part at the Center for Polymer Studies by grants from ONR and NSF.     

Percolation on infinitely ramified fractals

We present a family of exact fractals with a wide range of fractal and fracton dimensionalities. This includes the case of the fracton dimensionality of 2, which is critical for diffusion. This is achieved by adjusting the scaling factor as well as an internal geometrical parameter of the fractal. These fractals include the cases of finite and infinite ramification characterized by a ramification exponentp. The infinite ramification makes the problem of percolation on these lattices a nontrivial one. We give numerical evidence for a percolation transition on these fractals. This transition is tudied by a real-space renormalization group technique on lattices with fractal dimensionality ¯d between 1 and 2. The critical exponents for percolation depend strongly on the geometry of the fractals.     

Restoration of universality for the rod-to-coil transition scaling in the infinite-dimensionality limit: Exact results for directed walks

We derive scaling forms for the thermodynamic and correlation quantities for the turn-weighted fully and partially directed self-avoiding walks on the hypercubic lattices ind2. In the grand canonical (fixed fugacity per step) ensemble, the conformational rod-to-coil transition sets up in the regimew¯N=O(1), wherew is the weight of each 90° turn and¯N is the (fugacity-dependent) average number of steps. Contrary to the conventional critical phenomena wisdom, the scaling functions for the two different walk models, directed and partially directed, become universal only in the limitd.     

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 dissipative network model with neighboring activation

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν ≈ 0.65 that corresponds to the fractal dimension of the network d ≈ 1.3. An independent computation of the fractal dimension by the cluster growing method, generalized for directed networks, gives a close value d ≈ 1.4. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension d < 2.     

Computational analysis of factors influencing thermal conductivity of nanofluids

Numerical investigations are conducted to study the effect of factors such as particle clustering and interfacial layer thickness on thermal conductivity of nanofluids. Based on this, parameters including Kapitza radius and fractal and chemical dimension which have received little attention by previous research are rigorously investigated. The degree of thermal enhancement is analyzed for increasing aggregate size, particle concentration, interfacial thermal resistance, and fractal and chemical dimensions. This analysis is conducted for water-based nanofluids of Alumina (Al₂O₃), CuO, and Titania (TiO₂) nanoparticles where the particle concentrations are varied up to 4 vol%. Results from the numerical work are validated using available experimental data. For the case of aggregate size, particle concentration, and interfacial thermal resistance, the aspect ratio (ratio of radius of gyration of aggregate to radius of primary particle, R _g/a) is varied from 2 to 60. It was found that the enhancement decreases with interfacial layer thickness. Also the rate of decrease is more significant after a given aggregate size. For a given interfacial resistance, the enhancement is mostly sensitive to R _g/a < 20 indicated by the steep gradients of data plots. Predicted and experimental data for thermal conductivity enhancement are in good agreement. On the influence of fractal and chemical dimensions (d _l and d _f) of Alumina–water nanofluid, the R _g/a was varied from 2 to 8, d _l from 1.2 to 1.8, and d _f from 1.75 to 2.5. For a given concentration, the enhancement increased with the reduction of d _l or d _f. It appears a distinctive sensitivity of the enhancement to d _f, in particular, in the range 2–2.25, for all values of R _g/a. However, the sensitivity of d _l was largely depended on the value of R _g/a. The information gathered from this study on the sensitivity of thermal conductivity enhancement to aggregate size, particle concentration, interfacial resistance, and fractal and chemical dimensions will be useful in manufacturing highly thermally conductive nanofluids. Further research on the refine cluster evolution dynamics as a function of particle-scale properties is underway.     

Loop-erased self-avoiding random walk in two and three dimensions

If(n) is the position of the self-avoiding random walk in ^d obtained by erasing loops from simple random walk, then it is proved that the mean square displacementE(¦(n)¦²) grows at least as fast as the Flory predictions for the usual SAW, i.e., at least as fast asn ^3/2 ford=2 andn ^6/5 ford=3. In particular, if the mean square displacement of the usual SAW grows liken ^1.18... ind=3, as expected, then the loop-erased process is in a different universality class.     

Transience, Recurrence and Critical Behavior¶for Long-Range Percolation

We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d= 1,2, where x and y are connected with probability . We show that if d<s<2d, then the walk is transient, and if s≥ 2d, then the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d≥ 1, if d>s>2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d≥& 2 and d>s>2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network. Received: 27 October 2000 / Accepted: 29 November 2001     

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).     

Structural properties of silver nanoparticle agglomerates based on transmission electron microscopy: relationship to particle mobility analysis

In this work, the structural properties of silver nanoparticle agglomerates generated using condensation and evaporation method in an electric tube furnace followed by a coagulation process are analyzed using Transmission Electron Microscopy (TEM). Agglomerates with mobility diameters of 80, 120, and 150 nm are sampled using the electrostatic method and then imaged by TEM. The primary particle diameter of silver agglomerates was 13.8 nm with a standard deviation of 2.5 nm. We obtained the relationship between the projected area equivalent diameter (d _pa) and the mobility diameter (d _m), i.e., d _pa = 0.92 ± 0.03 d _m for particles from 80 to 150 nm. We obtained fractal dimensions of silver agglomerates using three different methods: (1) D _f = 1.84 ± 0.03, 1.75 ± 0.06, and 1.74 ± 0.03 for d _m = 80, 120, and 150 nm, respectively from projected TEM images using a box counting algorithm; (2) fractal dimension (D _fL) = 1.47 based on maximum projected length from projected TEM images using an empirical equation proposed by Koylu et al. (1995) Combust Flame 100:621–633; and (3) mass fractal-like dimension (D _fm) = 1.71 theoretically derived from the mobility analysis proposed by Lall and Friedlander (2006) J Aerosol Sci 37:260–271. We also compared the number of primary particles in agglomerate and found that the number of primary particles obtained from the projected surface area using an empirical equation proposed by Koylu et al. (1995) Combust Flame 100:621–633 is larger than that from using the relationship, d _pa = 0.92 ± 0.03 d _m or from using the mobility analysis.     

A Pearson Random Walk with Steps of Uniform Orientation and Dirichlet Distributed Lengths

A constrained diffusive random walk of n steps in ℝ ^d and a random flight in ℝ ^d , which are equivalent, were investigated independently in recent papers (J. Stat. Phys. 127:813, 2007; J. Theor. Probab. 20:769, 2007, and J. Stat. Phys. 131:1039, 2008). The n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1,2,4. Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D.     

Some fractal properties of the percolating backbone in two dimensions

A new algorithm is presented, based on elements of artificial intelligence theory, to determine the fractal properties of the backbone of the incipient infinite cluster. It is found that the fractal dimensionality of the backbone isd _f ^BB =1.61±0.01, the chemical dimensionality isd _t=1.40±0.01, and the fractal dimension of the minimum pathd _min=1.15 ± 0.02 for the two-dimensional triangular lattice.     

Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝ^d, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.     

Dynamic behavior of the voter model on fractals: logarithmic-periodic oscillations as a signature of time discrete scale invariance

The understanding of the dynamic behavior of the voter model, in low-dimensional media, is a very interesting open topic. In fact, due to the absence of the interfacial tension, only the interfacial noise becomes relevant during the coarsening processes, bringing the possibility of studing a new physical process. In this way, it is known that below the upper critical dimension (d < 2) and starting from a disordered configuration, a critical coarsening process takes place, and the density of interfaces, ρ(t), decays as a power-law function of time. Recently published numerical studies performed on low-dimensional fractal substrates (d _F < 2) [Physica A 362, 338 (2006)] show the existence of logarithmic-periodic oscillations superimposed on the standard ρ(t) power-law behavior, but the origin of those oscillations remains unclear. In this work, we provide an explanation of these oscillations in terms of the interplay between the dynamics of the voter model and the discrete scale invariance of the underlying fractal substrate. Our arguments are verified by means of extensive numerical simulations carried out on different fractal substrates.     

An universal relation between fractal and Euclidean (topological) dimensions of random systems

It is shown that a dimension-invariant form for fractal dimension D of random systems (where d is Euclidean dimension of the embedding space) is in good agreement with results of numerical simulations performed by different authors for critical (p=p _c ) and subcritical (p<p _c ) percolation, for lattice animals, and for different aggregation processes. Received: 9 July 1998 / Revised and Accepted: 12 July 1998     

First passage time and escape time distributions for continuous time random walks

We consider an arbitrary continuous time random walk (ctrw)via unbiased nearest-neighbour jumps on a linear lattice. Solutions are presented for the distributions of the first passage time and the time of escape from a bounded region. A simple relation between the conditional probability function and the first passage time distribution is analysed. So is the structure of the relation between the characteristic functions of the first passage time and escape time distributions. The mean first passage time is shown to diverge for all (unbiased)ctrw’s. The divergence of the mean escape time is related to that of the mean time between jumps. A class ofctrw’s displaying a self-similar clustering behaviour in time is considered. The exponent characterising the divergence of the mean escape time is shown to be (1−H), whereH(0<H<1) is the fractal dimensionality of thectrw.     

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).      

Mott Law as Upper Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic simple point process on , d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective model for the phonon–induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8].