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Derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc. In particular, the author proves the relations p(dpij/dp)=( lambda ij) where p is the bond density, pij is the pair connectedness function and ( lambda ij) is the average number of cutting bonds between i and j. From this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi , diverges as mod p-pc mod -1. The remaining (multiply connected) bonds in the percolating backbone, which lump together in 'blobs', diverge with a dimensionality-dependent exponent. He also shows that in the cell renormalisation group of Reynolds et al. (1978, 1980) the 'thermal' eigenvalue is simply related to the average number of cutting bonds in the spanning cluster. He discusses a percolation model in which the 'blobs' can be controlled by varying a parameter, and study the influence on the critical exponents.

The free energies of directed lattice animals in good and theta -solvents, and the free energy of directed percolation are found by the use of a simple Flory-type approximation, which accounts for the inherent anisotropy of these systems. From these free energies, the authors obtain the upper critical dimension below which mean-field theory breaks down. They also calculate closed-form, dimension-dependent expressions for the parallel and perpendicular correlation length exponents, which characterise the asymptotic cluster shapes. These exponents are in excellent agreement with existing numerical data.

We present Monte Carlo experiments on nucleation theory in the nearest-neighbor three-dimensional Ising model and in Ising models with long-range interactions. For the nearest-neighbor model, our results are compatible with the classical nucleation theory (CNT) for low temperatures, while for the long-range model a breakdown of the CNT was observed near the mean-field spinodal. A new droplet model and a zeroth-order theory of droplet growth are also presented.

Rigorous inequalities are proved, which relate percolation probability, mean cluster size and pair connectedness respectively with magnetization, susceptibility and pair correlation function in ferromagnetic Ising models. In two dimensions the critical point is shown to be a percolation point, while in three dimensions this is not true.

We have studied the voltage distribution for a two-component random mixture of conductances σa and σb. A scaling theory is developed for the moments of the distribution, which predicts, for small values ofh=σa/σb, an infinite number of crossover exponents, one for each moment, for Euclidean dimensiond >2, and only one crossover exponent ford=2. Monte Carlo results on the square lattice confirm this prediction.

A general theory is given for the quenched dilute $s$-state Potts and $n$-vector models in any dimension $d$. It is shown that for $T\ensuremath{\rightarrow}0$ at the percolation threshold the Potts thermal exponent ${\ensuremath{\nu}}_{T}$ equals the percolation exponent ${\ensuremath{\nu}}_{p}$, implying a crossover exponent $\ensuremath{\varphi}=1$, for any $s$ and $d$. For the $n$-vector model ($ng1$), ${\ensuremath{\nu}}_{T}=\frac{{\ensuremath{\nu}}_{p}}{{\ensuremath{\zeta}}_{R}}$, where ${\ensuremath{\zeta}}_{R}$ is a resistivity critical exponent. Agreement with recent experiments for two-dimensional dilute Ising and Heisenberg systems is excellent.

The authors study the droplet size distribution of the correlated site-bond percolation model introduced by Coniglio and Klein (1980), and also the usual clusters of two-dimensional Ising models near the critical point. Equilibrium configurations of the Ising model with nearest-neighbour interaction and also one with nearest- and next-nearest-neighbour interactions are generated through a Monte Carlo simulation, and then a cluster analysis is performed. The exponents beta and gamma for the Coniglio and Klein droplet distribution are found to agree, for both the nearest-neighbour and the next-nearest-neighbour model, with the corresponding exponents of the Ising model. The usual Ising clusters diverge only at Tc in the Ising model with nearest-neighbour interaction but not for the model with next-nearest-neighbour interaction. The Potts model formulation is used to predict the behaviour of the droplet for general further-neighbour interactions.

A Potts model formulation of the statistics of branched polymers or lattice animals in a solvent is given. The Migdal-Kadanoff renormalisation group is employed to study the critical behaviour or fractal dimension of the branched polymer. Four different critical behaviours are found, corresponding to random animal, collapse or theta point, percolation and compact cluster. The theta point behaviour is described by a tricritical point while percolation corresponds to a higher-order critical point, where the effect of the solvent on the branched polymer is the same as the screening effect of the other clusters in percolation.

A general formalism is developed to obtain series expansions of the average number of physical clusters of particles in the framework of Mayer's theory. The special case of lattice systems is investigated in more detail and some preliminary results are given on the relation between percolation (namely the formation of an infinite cluster) and condensation in fluid systems in the lowest approximation (summation of chain diagrams).

Various phenomena on a given fractal object have different critical behavior. This is related to the underlying multifractal structure of the aggregate. Here we discuss the problem with particular emphasis on a scaling approach which leads to a simple real space renormalization group. A mechanism which generates multifractality, based on a multiplicative process, is also illustrated. The multifractal structure of the incipient infinite cluster in percolation and diffusion limited aggregation is discussed in detail.

We calculate the critical temperature as a function of doping using a BCS formalism and a mean field approach to the annealed diluted quasi-two-dimensional antiferromagnet. We find reasonable agreement with the experimental data of Torrance et al. on La 2− x Sr x CuO 4 .

We show that for all Euclidean dimensions $d \stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}={\overline{d}}_{w}\ensuremath{-}{\overline{d}}_{f}$, where ${L}_{R}\ensuremath{\sim}{\ensuremath{\xi}}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}}$ is the effective resistance between two points separated by a distance comparable with the correlation length $\ensuremath{\xi},{\overline{d}}_{f}$ is the fractal dimension of the backbone, and ${\overline{d}}_{w}$ is the fractal dimension of a random walk on the same backbone. We also find a relation between the backbone and the full percolation cluster, ${\overline{d}}_{w}\ensuremath{-}{\overline{d}}_{f}={d}_{w}\ensuremath{-}{d}_{f}$. Thus the Alexander-Orbach conjecture ($\frac{{d}_{f}}{{d}_{w}}=\frac{2}{3}$ for $dg~2$) fails numerically for the backbone.

We show that a proper description of the temperature dependence of the spin-lattice relaxation rate of low-spin hemoproteins and ferredoxin requires that both the fractal structure of the protein backbone (polypeptide chain) and the cross connections (H bridges) between segments of the folded chain be taken into account. Within this picture the fracton dimensionality ${d}_{\mathrm{fr}}$ (recently introduced by Alexander and Orbach), the fractal dimensionality ${d}_{f}$, and the experimental noninteger exponent $n$ of Stapleton et al. (spin-lattice relaxation rate $\frac{1}{{T}_{1}}\ensuremath{\propto}{T}^{n}$) become satisfactorily consistent.

A definition of clusters of particles and holes with antiferromagnetic order is given for a lattice gas with coupling constant K < 0. In two dimensions it is shown that the Ising antiferromagnetic critical line is also a percolation line if Pb = 1 - exp(-|K|/2). Along this line these clusters called “droplets” diverge with Ising exponents.

A growth model can be characterized by the set of probabilities { p i } i ∈ Γ that each site at a given time on the external perimeter Γ becomes part of the aggregate. Equations for the set of pi are given for DLA and other growth models using the electrostatic analogy of the dielectric breakdown model. A scaling approach is developed for the probability distribution and is compared with the voltage distribution in a random resistor and random superconducting network at the percolation threshold. An infinite set of exponents is necessary to fully characterize the moments of the distribution which are related to the surface structure of the aggregate.

We exploit the relationship between the limit of the n → 0 of the n-vector model and self-avoiding walks (SAW) to relate the number of closed polygons, N of N + 1 links to the radius of gyration RN of SAW’s of N steps, i.e., N(N + 1) ~ R N −d where d is the dimensionality of the space. The relationship also holds at the Theta point: N ω(N + 1) ~ R θ −d where Rθ is the radius of gyration of the interacting SAW’s at the θ-temperature and Nω is the appropriately weighted polygon number. We show that a walk on the hull of the percolation clusters at the critical threshold Pc of the triangular lattice is identical to an interacting SAW and the critical properties of this walk are the θ-point critical properties.

By analyzing the voltage distribution of a random resistor network, we show that the backbone of the percolating cluster can be partitioned into an infinity of subsets, each one characterized by a fixed value of x\ensuremath{\equiv}lnV/ln${V}_{\mathrm{max}}$, where V is the voltage across each bond and ${V}_{\mathrm{max}}$ is its maximum value. Each subset is characterized by a distinct value of the fractal dimension \ensuremath{\varphi}(x), and as a consequence an infinite set of order parameters is required to describe the backbone structure. A new scaling approach and a real-space renormalization-group treatment are presented to treat the novel aspects of this problem. The mechanism for multifractality based on an underlying multiplicative process is illustrated on a hierarchical model.

We develop a new approach for studying the random resistor network by focusing on the distribution of voltage drops across each bond We introduce a simple model which provides a useful description of the percolating backbone, and which shows that the voltage distribution is log normal, with an infinite set of exponents required to describe the voltage moments. This latter prediction is verified by simulations of a resistor network on the square lattice.

We present an argument that the statistics of polymer rings at the theta point in two dimensions is exactly given by the statistics of the external perimeter (``hull'') of a percolation cluster. As a consequence, the fractal dimension ${d}_{f}$(theta) of a polymer chain at the theta point coincides with that of the hull of the percolating cluster, ${d}_{f}$(theta)=${d}_{H}$. Here theta' is the coil-globule transition point for a special interaction parameter. We also discuss conditions under which the theta' point may be related to the conventional theta point.

The opening talk of an interdisciplinary meeting should ideally start at “square one”. In the present case this means I should assume no previous background in the field of equilibrium phase transitions. Although everyone in the audience has some background in this field, the background of no two people is identical. Hence I begin with a brief introduction to phase transitions. Accordingly, I shall organize this talk around three simple questions: (i) “What happens?” That is to say, “What are the basic phenomena under consideration?” (ii) “Why do we care?” (iii) “What do we actually do?”

The authors explore the connection between the higher moments of the current (or voltage) distribution in a random linear resistor network, and the resistance of a nonlinear random resistor network. They find that the two problems are very similar, and that an infinite set of exponents are required to fully characterise each problem. These exponent sets are shown to be identical on a particular hierarchical lattice, a simple model which accurately describes the geometrical properties of the backbone of the infinite cluster at the percolation threshold and also the voltage distribution on this structure.

Fractal objects strongly screen external fields; only a small ``surface'' portion of the object is exposed appreciably to the field. We have studied this exposed surface of several random fractals as measured by random walkers and by ballistic particles launched from outside and absorbed by the fractal. The number of absorbing sites weighted by their rate of absorption shows an apparent power-law scaling with fractal mass. For diffusion-limited aggregates, ballistically generated aggregates, and screened-growth clusters in two dimensions, this power-law relationship is for the most part in accord with mean-field predictions of previous work. This accord is poorest for the objects of lowest fractal dimensionality. We have confirmed that this scaling is different from that of the old-growth--new-growth interface studied previously. We also find that a ``hierarchy'' of fractal dimensions describes the external surface of diffusion-limited aggregates.

The authors extend the definition of droplets in Ising and Potts models to the case of an external field different from zero. They also find exact relations between thermal properties and connectivity properties which show why, in mean field, the mean cluster size does not diverge as the susceptibility when the critical temperature is approached from below.

A growth model can be characterized by the set of probabilities {pi}iɛг that each site at a given time on the external perimeter Г becomes part of the aggregate. All quantities of interest both static and dynamic can be expressed in terms of the pi. Equations for the set of pi are given for DLA and other growth models using the electrostatic analogy of the dielectric breakdown model. Due to this electrostatic analogy the scaling properties of the probability distribution are related to those of the voltage distribution in a random resistor and random superconducting network at the percolation threshold. An infinite set of exponents is necessary to fully characterize the moments of the distribution which are related to the surface structure of the aggregate.

We show that the Interacting Self-Avoiding Walk/polymer chain at the theta temperature may be mapped onto the Indefinitely Growing Self-Avoiding Walk/Smart Kinetic Walk. Extremely accurate statistics for these walks exist which together with our independent numerical results support v = 0.57 (-4/7) where v is the exponent describing the dependence of the polymerization index N on the end-to-end distance RN.

Percolation theory is reviewed. Intuitive arguments are given to derive scaling and hyperscaling relations. Above six dimensions the breakdown of hyperscaling is related to the interpenetration of the critical large clusters, and to the appearence at p c of an infinite number of infinite clusters of zero density with fractal dimension d f = 4. The structure of the percolating cluster made of links and blobs is characterized by an infinite set of exponents related to the anomalous voltage distribution in a random resistor network at p c . The surface structure of critical clusters below pc, which is relevant to the study of random superconducting networks, is also discussed. In particular, an exact result is presented which shows that in any dimension the interface of two critical clusters diverge as (p c -p)-1 as the percolation threshold is approached.

It is shown that previously defined clusters, which give a geometrical description of the fluctuations in the q-state Potts model, at criticality have a fractal structure made of links and blobs as in percolation. Using the mapping from the Potts model to the Coulomb gas it is found that the fractal dimension of the links or red bonds ${D}_{R}$ is given by 5/4, 3/4, 13/24, 7/20, 0, while the fractal dimension of the external hull ${\mathit{D}}_{\mathit{h}}$ is given by 2, 7/4, 5/3, 8/5, 3/2, for q=0,1,2,3,4. A model originally introduced by Mandelbrot and Given for percolation clusters is found to correctly describe the fractal structure of the Potts clusters.

To describe the irreversible growth of linear polymers, we introduce a new type of perturbed random walk, related to the zero initiator concentration limit of the kinetic gelation model. Our model simulates real polymer growth by permitting the initiator (walker) to form the next bond with an unsaturated monomer at one of the neighbouring sites of its present location. A heuristic kinetic self-consistent field argument along the lines introduced by Pietronero suggests a fractal dimensionality, df = (d + 1)/2, in agreement with our Monte Carlo and series expansion results (including the usually expected logarithmic correction at the upper critical dimension dc = 3.

We report on a variety of novel features for the distribution of voltage drops across the bonds of a random resistor network. To describe this distribution analytically, we introduce a simple geometrical model, with a hierarchical structure of links and blobs, which appears to capture the basic features of random networks near the percolation threshold. On this model, we find that the voltage distribution is a log binomial, and that an infinite hierarchy of exponents is required to characterize the moments of this distribution. On general grounds, we argue that this exponent hierarchy emerges naturally from an underlying distribution which, at the percolation threshold, can be written in the form, ${L}^{\ensuremath{\varphi}(\mathrm{l}\mathrm{n}\mathit{V}/\mathrm{l}\mathrm{n}{V}_{\mathrm{max})}}$, where L is the linear size of the system, V is the voltage drop, and ${V}_{\mathrm{max}}$ is the maximum value of this voltage drop. The nonconstancy of \ensuremath{\varphi}(y) as a function of y is an unconventional feature in the context of a scaling approach, and a variety of novel properties result. These are tested by numerical simulations of the voltage distribution for square-lattice networks at the percolation threshold. In particular, the moments of this distribution are found to scale independently, with the exponents of the positive moments in excellent agreement with those of the hierarchical model. We also discuss some intriguing properties associated with the voltage distribution above the percolation threshold, most notably, that the higher moments of the distribution are nonmonotonic functions of the bond concentration. Finally, we exploit duality arguments to investigate the voltage distribution of a random superconducting network.

A Potts correlated polychromatic percolation is studied. The clusters are made of sites corresponding to a given value of the q-state Potts variables, connected by bonds being active with probability pB. To treat this problem an s-state Potts Hamiltonian diluted with q-state Potts variables (instead of lattice gas variables) is introduced to which the the Migdal-Kadanoff renormalisation group is applied. It is found for a particular choice of pB=1-e-K (where K is the Potts coupling constant divided by the Boltzmann factor) that these clusters, called droplets diverge at the Potts critical point with Potts exponents.

A new relation is given in percolation theory from which follows that the backbone of the incipient infinite cluster is made of singly-connected links and “blobs”, the number of links diverge with an universal exponent 1. It is also shown that this exponent characterizes the crossover exponent of the dilute s-state Potts model while for the n-vector model is given by the low density resistivity exponent.

We give a description of the mechanism of phase transitions in the Ising model, pointing out the connection between the spontaneous magnetization and the existence of infinite clusters of “up” and “down” spins. The picture is more complete in the two-dimensional Ising model, where we can also use a generalized version of a result by Miyamoto.

We review some properties of frustrated models which reproduce the logaritmic relaxation in granular compaction.

A general scheme for devising efficient cluster dynamics proposed in a previous letter [Phys.Rev.Lett. 72, 1541 (1994)] is extensively discussed. In particular the strong connection among equilibrium properties of clusters and dynamic properties as the correlation time for magnetization is emphasized. The general scheme is applied to a number of frustrated spin model and the results discussed., 17 pages LaTeX + 16 figures; will appear in Phys. Rev. E

We present a detailed Monte Carlo evaluation of the equilibrium distribution of local spin-flip rates and local energies in the paramagnetic phase of the $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ and $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3\ifmmode\pm\else\textpm\fi{}J$ Ising spin glass. Both quantities are spatially heterogeneous, and we find that the shapes of the distributions change dramatically with decreasing temperature. In particular as temperature decreases we find that for an increasing fraction of spins the local spin-flip rate and local energy increase as the glass transition is approached.

Despite the large use of granular materials in the industry, and the large number of natural phenomena where granular materials are involved, a comprehensive theoretical framework of their physics is still missing. An important perspective was proposed almost 20 years ago by S.Edwards, which suggested the possibility of a statistical mechanics description of granular materials at rest in their mechanically stable states. This article focuses on the theoretical foundations and current understanding of a statistical mechanics approach of granular materials under gravity. Experimental and numerical results discussing such an approach and clarifying its limit of validity are described as well.

The Invaded Cluster (IC) dynamics introduced by Machta et al. [Phys. Rev. Lett. 75 2792 (1995)] is extended to the fully frustrated Ising model on a square lattice. The properties of the dynamics which exhibits numerical evidence of self-organized criticality are studied. The fluctuations in the IC dynamics are shown to be intrinsic of the algorithm and the fluctuation-dissipation theorem is no more valid. The relaxation time is found very short and does not present critical size dependence., Comment: notes and refernences added, some minor changes in text and fig.3,5,7 16 pages, Latex, 8 EPS figures, submitted to Phys. Rev. E

How to characterize geometrically a fluctuation near a critical point is a longstanding problem1−8 that recently has received renewed attention, due to a novel experiment in which direct visual observation of critical fluctuations was possible9. Here I want to describe a static and dynamic approach to this problem. The static approach is based on a particular site bond correlated percolation model which was previously introduced as a model for sol gel transition10, For a particular value of the bond probability4, this approach can be related to a formalism developped by Kasteleyn and Fortuin3,11 and gives the correct definition of the Ising and Potts clusters for a geometrical description of the phase transition. I will first review the main results and show how the formalism can be extended to the case of non zero magnetic field. In Sect. III–V I will present exact results12 on the fractal structure of the Ising and Potts clusters based on a mapping13 from the Potts model to the Coulomb gas. In Sec. VI it will be shown that a model originally introduced by Mandelbrot and Given14 for percolation clusters is found extremely good to describe the fractal structure of the Potts clusters. Finally Sec. VII is devoted to a dynamical approach which has been recently15 introduced and is based on the propagation of “damage” in a spin system.

Many eukaryotic cell types share the ability to migrate directionally in response to external chemoattractant gradients. This ability is central in the development of complex organisms, and is the result of billion years of evolution. Cells exposed to shallow gradients in chemoattractant concentration respond with strongly asymmetric accumulation of several signaling factors, such as phosphoinositides and enzymes. This early symmetry-breaking stage is believed to trigger effector pathways leading to cell movement. Although many factors implied in directional sensing have been recently discovered, the physical mechanism of signal amplification is not yet well understood. We have proposed that directional sensing is the consequence of a phase ordering process mediated by phosphoinositide diffusion and driven by the distribution of chemotactic signal. By studying a realistic computational model that describes enzymatic activity, recruitment to the plasmamembrane, and diffusion of phosphoinositide products we have shown that the effective enzyme-enzyme interaction induced by catalysis and diffusion introduces an instability of the system towards phase separation for realistic values of physical parameters. In this framework, large reversible amplification of shallow chemotactic gradients, selective localization of chemical factors, macroscopic response timescales, and spontaneous polarization arise.

Experiments of in vitro formation of blood vessels show that cells randomly spread on a gel matrix autonomously organize to form a connected vascular network. We propose a simple model which reproduces many features of the biological system. We show that both the model and the real system exhibit a fractal behavior at small scales, due to the process of migration and dynamical aggregation, followed at large scale by a random percolation behavior due to the coalescence of aggregates. The results are in good agreement with the analysis performed on the experimental data., Comment: 4 pages, 11 eps figures