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Optimal paths and growth processes
Abstract
Interfaces in systems with strong quenched disorder are fractal and are thus in a different universality class than the self-affine interfaces found in systems with weak quenched disorder. The geometrical properties of strands arising in loopless invasion percolation clusters, in loopless Eden growth clusters, and in the ballistic growth process are studied and their universality classes are identified.
... In the case of strong disorder, we present the following theoretical arguments. Cieplak et al. [CMB99] showed that finding the optimal path between nodes A and B in the strong disorder limit is equivalent to the following procedure. First, we sort all M links of the network in descending order of their weights, so that the first link in this list has the largest weight. ...
1. Introduction; Part I. Random Network Models: 2. The Erdos-Renyi
models; 3. Observations in real-world networks; 4. Models for complex
networks; 5. Growing network models; Part II. Structure and Robustness
of Complex Networks: 6. Distances in scale-free networks - the ultra
small world; 7. Self-similarity in complex networks; 8. Distances in
geographically embedded networks; 9. The network's structure - the
generating function method; 10. Percolation on complex networks; 11.
Structure of random directed networks - the bow tie; 12. Introducing
weights - bandwidth allocation and multimedia broadcasting; Part III.
Network Function - Dynamics and Applications: 13. Optimization of the
network structure; 14. Epidemiological models; 15. Immunization; 16.
Thermodynamic models on networks; 17. Spectral properties, transport,
diffusion and dynamics; 18. Searching in networks; 19. Biological
networks and network motifs; Part IV. Appendices; References; Index.
... In the case of strong disorder, we present the following theoretical arguments. Cieplak et al. [CMB99] showed that finding the optimal path between nodes A and B in the strong disorder limit is equivalent to the following procedure. First, we sort all M links of the network in descending order of their weights, so that the first link in this list has the largest weight. ...
Examining important results and analytical techniques, this graduate-level textbook is a step-by-step presentation of the structure and function of complex networks. Using a range of examples, from the stability of the internet to efficient methods of immunizing populations, and from epidemic spreading to how one might efficiently search for individuals, this textbook explains the theoretical methods that can be used, and the experimental and analytical results obtained in the study and research of complex networks. Giving detailed derivations of many results in complex networks theory, this is an ideal text to be used by graduate students entering the field. End-of-chapter review questions help students monitor their own understanding of the materials presented.
For Gaussian Spin Glasses in low dimensions, we introduce a simple Strong
Disorder renormalization procedure at zero temperature. In each disordered
sample, the difference between the ground states associated to Periodic and
Anti-Periodic boundary conditions defines a system-size Domain Wall. The
numerical study in dimensions $d=2$ (up to sizes $2048^2$) and $d=3$ (up to
sizes $128^3$) yields fractal Domain Walls of dimensions $d_s(d=2) \simeq 1.27$
and $d_s(d=3) \simeq 2.55$ respectively.
We present analytical and numerical results suggesting that the optimal path in an energy landscape in the strong disorder limit is in the universality class of the shortest path in invasion percolation with trapping. Our results imply that, in contrast to common belief, invasion percolation with trapping and regular percolation in d = 3 are in different universality classes.
Dynamic renormalization-group methods are used to study the large-distance, long-time behavior of velocity correlations generated by the Navier-Stokes equations for a randomly stirred, incompressible fluid. Different models are defined, corresponding to a variety of Gaussian random forces. One of the models describes a fluid near thermal equilibrium, and gives rise to the usual long-time tail phenomena. Apart from simplifying the derivation of the latter, our methods clearly establish their universality, their connection with Galilean invariance, and their analytic form in two dimensions, ∼(logt)-1/2/t. Nontrivial behavior results when the model is formally continued below two dimensions. Although the physical interpretation of the Navier-Stokes equations below d=2 is unclear, the results apply to a forced Burger's equation in one dimension. A large class of models produces a spectral function E(k) which behaves as k2 in three dimensions, as expected on the basis of equipartition. However, nonlinear effects (which become significant below four dimensions) control the infrared properties of models which force the Navier-Stokes equations at zero wave number.
We study the dynamical exponent $z$ for the directed percolation depinning (DPD) class of models for surface roughening in the presence of quenched disorder. We argue that $z$ for $(d+1)$ dimensions is equal to the exponent $d_{\rm min}$ characterizing the shortest path between two sites in an isotropic percolation cluster in $d$ dimensions. To test the argument, we perform simulations and calculate $z$ for DPD, and $d_{\rm min}$ for percolation, from $d = 1$ to $d = 6$. Comment: RevTex manuscript 3 pages + 6 figures (obtained upon request via email sth@iris.bu.edu)
This book brings together two of the most exciting and widely studied subjects in modern physics: namely fractals and surfaces. To the community interested in the study of surfaces and interfaces, it brings the concept of fractals. To the community interested in the exciting field of fractals and their application, it demonstrates how these concepts may be used in the study of surfaces. The authors cover, in simple terms, the various methods and theories developed over the past ten years to study surface growth. They describe how one can use fractal concepts successfully to describe and predict the morphology resulting from various growth processes. Consequently, this book will appeal to physicists working in condensed matter physics and statistical mechanics, with an interest in fractals and their application. The first chapter of this important new text is available on the Cambridge Worldwide Web server: http://www.cup.cam.ac.uk/onlinepubs/Textbooks/textbookstop.html
A numerical method is developed to study the scaling of distribution of couplings of highly random antiferromagnetic Ising and quantum Heisenberg spin- 1/2 systems. The method shows how freezing into inert local singlets prevents ordering down to temperatures well below the median nearest-neighbor coupling or bare exchange percolation threshold in positionally disordered systems with Heisenberg exchange varying exponentially with distance (e.g., doped semiconductors, quasi one-dimensional salts). This is contrasted with the Ising system.
By considering a model in which charge is transported via phonon-induced tunneling of electrons between localized states which are randomly distributed in energy and position, Mott has obtained an electrical conductivity of the form σ∝exp[-(λα3/ρ0kT)1/4]. Here T is the temperature of the system, ρ0 is the density of states at the Fermi level, λ is a dimensionless constant, and α-1 is the distance for exponential decay of the wave functions. We rederive these results, relating λ to the critical density of a certain dimensionless percolation problem, and we estimate λ to be approximately 16. The applicability of the model to experimental observations on amorphous Ge, Si, and C is discussed.
The nearest-neighbor random-exchange Heisenberg antiferromagnetic Heisenberg chain (s=1/2) is studied at low temperatures via an approximate renormalization-group method. This entails renormalization of the random-exchange coupling constant J and of the probability law for J. P0(J). After n iterations we find that the renormalized probability function Pn(J(n)), a function of the renormalized coupling J(n), develops singular behavior for small J(n), independent of the initial form of P0(J). This happens both for P0(J) that diverge or go to zero as J→0. The singular form of Pn(J(n)) is such that it leads to a specific heat and susceptibility that behave like C∼T1-αC(T) and χ∼T-αχ(T). αC(T) and αχ(T) are exponents weakly dependent on temperature (T) that go to one as T→0 and satisfy αχ(T)>αC(T), for arbitrary initial probability laws. The classical n-vector models, and in particular the classical Heisenberg model are also studied using a renormalization-group approach and it is found that their behavior is different than that of the quantum model: singular behavior in the thermodynamic properties is only obtained if the starting P0(J) is singular. An explanation of these results in terms of the scaling theory of localization is suggested. The relevance of our results to the experimental findings on the magnetic properties of tetracyanoquinodimethanide complexes is also discussed.
We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.
Optimal paths in disordered systems are studied using two different models interpolating between weak and infinitely strong disorder. In one case, exact numerical methods are used to study the optimal path in a two-dimensional square lattice whereas a renormalization-group analysis is employed on hierarchical lattices in the other. The scaling behaviour is monitored as a function of parameters that tune the strength of the disorder. Two distinct scenarios are provided by the models: in the first, fractal behaviour occurs abruptly as soon as the disorder widens, while in the other it emerges as a limiting case of a self-affine regime.
On the basis of preserving required symmetries and capturing the essential ingredients, the scaling properties of the frontier of Eden model clusters (1961) have been suggested by Kardar et al (1986) to be governed in the continuum limit by a Burger's equation with noise. The authors show that such a correspondence is exact by mapping explicitly the Eden model onto the problem of the conformation of a directed polymer in a random medium at zero temperature. The latter model has been shown to be itself in correspondence with Burgers' equations. In addition, this mapping allows them to have access to the distribution of noise to be included in Burgers' equation to exactly reproduce the Eden growth model. Finally, this implies that the Eden model always shows the same universal behaviour, in contrast to a recently found breakdown of universality for the directed polymer problem in a 'very disordered' medium.
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Kinetic interfaces form the basis of a fascinating, interdisciplinary branch of statistical mechanics. Diverse stochastic growth processes can be unified via an intriguing nonlinear stochastic partial differential equation whose consequences and generalizations have mobilized a sizeable community of physicists concerned with a statistical description of kinetically roughened surfaces. Substantial analytical, experimental and numerical effort has already been expended. Despite impressive successes, however, there remain many open questions, with much richness and subtlety still to be revealed. In this review, we give an unorthodox account of this rapidly growing field, concentrating on two main lines — the interface growth equations themselves, and their directed polymer counterparts. We emphasize the intrinsic links among the topics discussed, as well as the relationships to other branches of natural science. Our aim is to persuade the reader that multidisciplinary statistical mechanics can be challenging, enjoyable pursuit of surprising depth.
Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques
We show that percolation concepts lead to the definition of a characteristic length for the permeability in random porous media. Application of the model to sandstone and carbonate rocks yields quantitative agreement between theory and experiment with no adjustable parameters.
The problem of determining the ground state of a d-dimensional interface embedded in a (d+1)-dimensional random medium is treated numerically. Using a minimum-cut algorithm, the exact ground states can be found for a number of problems for which other numerical methods are inexact and slow. In particular, results are presented for the roughness exponents and ground-state energy fluctuations in a random bond Ising model. It is found that the roughness exponent ζ=0.41±0.01,0.22±0.01, with the related energy exponent being θ=0.84±0.03,1.45±0.04, in d=2,3, respectively. These results are compared with previous analytical and numerical estimates.
Randomly placed impurities that alter the local exchange couplings, but do not generate random fields or destroy the long-range order, roughen domain walls in Ising systems for dimensionality $\frac{5}{3}<d<5$. They also pin (localize) the walls in energetically favorable positions. This drastically slows down the kinetics of ordering. The pinned domain wall is a new critical phenomenon governed by a zero-temperature fixed point. For $d=2$, the critical exponents for domain-wall pinning energies and roughness as a function of length scale are estimated from numerically generated ground states.
DOI:https://doi.org/10.1103/PhysRevLett.55.2924
A Comment on the Letter by D. A. Huse and C. L. Henley, Phys. Rev. Lett. 54, 2708 (1985).
A model is proposed for the evolution of the profile of a growing interface. The deterministic growth is solved exactly, and exhibits nontrivial relaxation patterns. The stochastic version is studied by dynamic renormalization-group techniques and by mappings to Burgers's equation and to a random directed-polymer problem. The exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations. Predictions are made for more dimensions.
Directed polymers subject to quenched external impurities (as in a polyelectrolyte in a gel matrix) are examined analytically, and numerically. Transverse fluctuations ||x|| scale with the length t of the polymer as ||x||~tv. In all dimensions, for sufficiently strong disorder, v can be different from the random-walk value of (1/2. Extensive numerical simulations in 2, 3, and 4 dimensions in fact suggest a superuniversal exponent of v=(2/3. The importance of the tree structure of the polymer ensemble is emphasized.
Optimal paths in weakly disordered systems are self-affine. In the strong disorder limit^1 the optimal paths form a new universality class: they are fractal and form backbones of invasion percolation clusters without loops. The strong disorder limit arises when the bond strengths are so widely distributed that the sum of several bond strengths may be well approximated by the value of the largest bond strength. In the strong disorder limit, pinned domain walls are in a new universality class, spin glasses and ferromagnets behave alike in many respects and novel consequences result in several contexts including polymers in random media, fractally rough surfaces, percolation, lattice animals, and the ground state of spin glasses. Our studies^2 also shed light on the protein folding problem -- within the framework of a lattice model, we have shown that one can design rapidly folding sequences by assigning the strongest attractive couplings to the contacts present in a target native state. ^1 M. Cieplak, A. Maritan and J. R. Banavar, Phys. Rev. Lett. 72, 2320 (1994). ^2 I. Shrivastava, S. Vishveshwara, M. Cieplak, A. Maritan and J. R. Banavar, Proc. Nat. Acad. Sci. 92, 9206 (1995).
We introduce a new Ising spin-glass model and use it to study the relation between quenched disorder, frustration, ground state mulitplicity, and dimension. The Hamiltonian is of Edwards-Anderson type in any finite volume but the coupling magnitudes scale nonlinearly with volume. We find a mapping between the ground state structure of this model and the global connectivity structure of invasion percolation in the same dimension. We argue that our model has a singular pair of ground states below eight dimensions and infinitely many above.
The mapping of optimal paths in the strong disorder limit to the strands of invasion percolation clusters is shown to lead to a new universal property of these clusters. We suggest that the corresponding strands arising in the annealed Eden growth process are in the same universality class as directed polymers in weak quenched disorder with an upper critical dimension
Invasion bond percolation (IBP) is mapped exactly into Prim{close_quote}s algorithm for finding the shortest spanning tree of a weighted random graph. Exploring this mapping, which is valid for arbitrary dimensions and lattices, we introduce a new IBP model that belongs to the same universality class as IBP and generates the minimal energy tree spanning the IBP cluster. {copyright} {ital 1996 The American Physical Society.}
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