It was discovered a few years ago that many networks in the real world exhibit self-similarity. A lot of researches on the structures and processes on real and artificial fractal complex networks have been done, drawing an analogy to critical phenomena. However, the non-Markovian dynamics on fractal networks has not been understood well yet. We here study the self-avoiding walk on complex fractal networks through the mapping of the self-avoiding walk to the n-vector model by a generating function formalism. First, we analytically calculate the critical exponent ν and the effective coordination number (the connective constant) by a renormalization-group analysis in various fractal dimensions. We find that the exponent ν is equal to the exponent of displacement, which describes the speed of diffusion in terms of the shortest distance. Second, by obtaining an exact solution, we present an example which supports the well-known conjecture that the universality class of the self-avoiding walk is not determined only by a fractal dimension. Our finding suggests that the scaling theory of polymers can be applied to graphs which lack the Euclidian distance as well. Furthermore, the self-avoiding walk has been exactly solved only on a few lattices embedded in the Euclidian space, but we show that consideration on general graphs can simplify analytic calculations and leads to a better understanding of critical phenomena. The scaling theory of the self-avoiding path will shed light on the relationship between path numeration problems in graph theory and statistical nature of paths.
We study the self-avoiding walk on complex fractal networks called the (u,v)-flower by mapping it to the N-vector model in a generating function formalism. First, we analytically calculate the critical exponent ν and the connective constant by a renormalization-group analysis in arbitrary fractal dimensions. We find that the exponent ν is equal to the displacement exponent, which describes the speed of diffusion in terms of the shortest distance. Second, by obtaining an exact solution for the (u,u)-flower, we provide an example which supports the conjecture that the universality class of the s…
Self-avoiding walks are studied on the 3-simplex fractal lattice as a model of linear polymer conformations in a dilute, non-homogeneous solution. A model is supplemented with bending energies and attractive-interaction energies between non-consecutively visited pairs of nearest-neighboring sites (contacts). It captures the main features of a semi-flexible polymer subjected to variable solvent conditions. Hierarchical structure of the fractal lattice enabled determination of the exact recurrence equations for the generating function, through which universal and local properties of the model we…
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 $< P_n(S)>, <W_n(S)>, <log P_n(S)>$ and $<log W_n(S)>$ over different positions of S. We find that the connectivi…
We have analyzed geometric and topological features of self-avoiding walks. We introduce a new kind of walk: the loop-deleted self-avoiding walk (LDSAW) motivated by the interaction of chromatin with the nuclear lamina. Its critical exponent is calculated and found to be different from that of the ordinary SAW. Taking the walks as point-clouds, the LDSAW is a subset of the SAW. We study the difference between the LDSAW and SAW by comparing their fractal dimensions and growth rates of the Betti number. In addition, the spatial distribution of the contacts inside a SAW, which is also a subset of…
We study the polymer system consisting of two polymer chains situated in a fractal container that belongs to the three--dimensional Sierpinski Gasket (3D SG) family of fractals. Each 3D SG fractal has four fractal impenetrable 2D surfaces, which are, in fact, 2D SG fractals. The two-polymer system is modelled by two interacting self-avoiding walks (SAWs), one of them representing a 3D floating polymer, while the other corresponds to a chain adhered to one of the four 2D SG boundaries. We assume that the studied system is immersed in a poor solvent inducing the intra-chain interactions. For the…
This is a pedagogical review of the subject of linear polymers on deterministic finitely ramified fractals. For these, one can determine the critical properties exactly by real-space renormalization group technique. We show how this is used to determine the critical exponents of self-avoiding walks on different fractals. The behavior of critical exponents for the $n$-simplex lattice in the limit of large $n$ is determined. We study self-avoiding walks when the fractal dimension of the underlying lattice is just below 2. We then consider the case of linear polymers with attractive interactions,…
Using data on the Berlin public transport network, the present study extends previous observations of fractality within public transport routes by showing that also the distribution of inter-station distances along routes displays non-trivial power law behaviour. This indicates that the routes may in part also be described as Lévy-flights. The latter property may result from the fact that the routes are planned to adapt to fluctuating demand densities throughout the served area. We also relate this to optimization properties of Lévy flights.
We calculate the fractal dimension $d_{\rm f}$ of critical curves in the $O(n)$ symmetric $(\vec ϕ^2)^2$-theory in $d=4-\varepsilon$ dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at $n=-2$, self-avoiding walks ($n=0$), Ising lines $(n=1)$, and XY lines ($n=2$), in agreement with numerical simulations. It can be compared to the fractal dimension $d_{\rm f}^{\rm tot}$ of all lines, i.e. backbone plus the surrounding loops, identical to $d_{\rm f}^{\rm tot} = 1/ν$. The combination $ϕ_{\rm c}= d_{\rm f}/d_{\rm f}^{\rm tot} = νd_{\rm f}$ is the crossover e…
We study the problem of critical adsorption of piecewise directed random walks on a boundary of fractal lattices that belong to the Sierpinski gasket family. By applying the exact real space renormalization group method, we calculate the crossover exponent $ϕ$, associated with the number of adsorbed steps, for the complete fractal family. We demonstrate that our results are very close to the results obtained for ordinary self-avoiding walk, and discuss the asymptotic behaviour of $ϕ$ at the fractal to Euclidean lattice crossover.
We refute the claims made by Riera and Chalub [Phys.Rev.E {\bf 58}, 4001 (1998)] by demonstrating that they have not provided enough data (requisite in their series expansion method) to draw reliable conclusions about criticality of self-avoiding walks on the Sierpinski gasket family of fractals.
Using self-avoiding walk model on three-dimensional Sierpinski fractals (3d SF) we have studied critical properties of self-interacting linear polymers in porous environment, via exact real-space renormalization group (RG) method. We have found that RG equations for 3d SF with base b=4 are much more complicated than for the previously studied b=2 and b=3 3d SFs. Numerical analysis of these equations shows that for all considered cases there are three fixed points, corresponding to the high-temperature extended polymer state, collapse transition, and the low-temperature state, which is compact …
We study self-avoiding walks on three-dimensional critical percolation clusters using a new exact enumeration method. It overcomes the exponential increase in computation time by exploiting the clusters' fractal nature. We enumerate walks of over $10^4$ steps, far more than has ever been possible. The scaling exponent $ν$ for the end-to-end distance turns out to be smaller than previously thought and appears to be the same on the backbones as on full clusters. We find strong evidence against the widely assumed scaling law for the number of conformations and propose an alternative, which perfec…
The fractal dimensions of polymer chains and high-temperature graphs in the Ising model both in three dimension are determined using the conformal bootstrap applied for the continuation of the $O(N)$ models from $N=1$ (Ising model) to $N=0$ (polymer). The unitarity bound below $N=1$ of the scaling dimension for the the $O(N)$-symmetric-tensor develops a kink as a function of the fundamental field as in the case of the energy operator dimension in the Ising model. Although this kink structure becomes less pronounced as $N$ tends to zero, an emerging asymmetric minimum in the current central cha…
We numerically test the correspondence between the scaling limit of self-avoiding walks (SAW) in the plane and Schramm-Loewner evolution (SLE) with k=8/3. We introduce a discrete-time process approximating SLE in the exterior of the unit disc and compare the distribution functions for an internal point in the SAW and a point at a fixed fractal variation on the SLE, finding good agreement. This provides numerical evidence in favor of a conjecture by Lawler, Schramm and Werner. The algorithm turns out to be an efficient way of computing the position of an internal point in the SAW.
We study the adsorption problem of linear polymers, when the container of the polymer--solvent system is taken to be a member of the three dimensional Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integer $b$ ($2\le b\le\infty$), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the critical exponents $γ_1, γ_{11}$, and $γ_s$ which, within the self--avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends grafted on the adsorbing impen…
The self-avoiding walk on the square site-diluted correlated percolation lattice is considered. The Ising model is employed to realize the spatial correlations of the metric space. As a well-accepted result, the (generalized) Flory's mean field relation is tested to measure the effect of correlation. After exploring a perturbative Fokker-Planck-like equation, we apply an enriched Rosenbluth Monte Carlo method to study the problem. To be more precise, the winding angel analysis is also performed from which the diffusivity parameter of Schramm-Loewner evolution (SLE) theory ($κ$) is extracted. W…
We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to 2-loo…
The critical behaviour of correlation functions near a boundary is modified from that in the bulk. When the boundary is smooth this is known to be characterised by the surface scaling dimension $\xt$. We consider the case when the boundary is a random fractal, specifically a self-avoiding walk or the frontier of a Brownian walk, in two dimensions, and show that the boundary scaling behaviour of the correlation function is characterised by a set of multifractal boundary exponents, given exactly by conformal invariance arguments to be $λ_n = 1/48 (\sqrt{1+24n\xt}+11)(\sqrt{1+24n\xt}-1)$. This re…
We investigate asymptotical behavior of numbers of long Hamiltonian walks (HWs), i.e. self-avoiding random walks that visit every site of a lattice, on various fractal lattices. By applying an exact recursive technique we obtain scaling forms for open HWs on 3-simplex lattice, Sierpinski gasket, and their generalizations: Given-Mandelbrot (GM), modified Sierpinski gasket (MSG) and n-simplex fractal families. For GM, MSG and n-simplex lattices with odd values of n, number of open HWs $Z_N$, for the lattice with $N\gg 1$ sites, varies as $ω^N N^γ$. We explicitly calculate exponent $γ$ for severa…
We solve the O(n) model, defined in terms of self- and mutually avoiding loops coexisting with voids, on a 3-simplex fractal lattice, using an exact real space renormalization group technique. As the density of voids is decreased, the model shows a critical point, and for even lower densities of voids, there is a dense phase showing power-law correlations, with critical exponents that depend on n, but are independent of density. At n=-2 on the dilute branch, a trivalent vertex defect acts as a marginal perturbation. We define a model of biconnected clusters which allows for a finite density of…
The Shcramm-Loewner evolution (SLE) is a correlated exploration process, in which for the chordal set up, the tip of the trace evolves in a self-avoiding manner towards the infinity. The resulting curves are named SLE$_κ$, emphasizing that the process is controlled by one parameter $κ$ which classifies the conformal invariant random curves. This process when experiences some environmental imperfections, or equivalently some scattering random points (which can be absorbing or repelling) results to some other effective scale-invariant curves, which are described by the other effective fractal di…
This paper analyzes a new self-avoiding (SA) meshwork model using the canonical Monte Carlo simulation technique on lattices that consist of connection-fixed triangles. The Hamiltonian of this model includes a self-avoiding potential and a pressure term. The model identifies a crumpled-to-tubule (CT) transition between the crumpled and tubular phases. This is a second-order transition, which occurs when the pressure difference between the inner and outer sides of the surface is close to zero. We obtain the Flory swelling exponents $ν_{{\rm R}^2}(=\!D_f/2)$ and $\barν_{\rm v}$ corresponding to …
The harmonic measure (or diffusion field or electrostatic potential) near a percolation cluster in two dimensions is considered. Its moments, summed over the accessible external hull, exhibit a multifractal spectrum, which I calculate exactly. The generalized dimensions D(n) as well as the MF function f(alpha) are derived from generalized conformal invariance, and are shown to be identical to those of the harmonic measure on 2D random walks or self-avoiding walks. An exact application to the anomalous impedance of a rough percolative electrode is given. The numerical checks are excellent. Anot…
The existing estimation of the upper critical dimension of the Abelian Sandpile Model is based on a qualitative consideration of avalanches as self-avoiding branching processes. We find an exact representation of an avalanche as a sequence of spanning sub-trees of two-component spanning trees. Using equivalence between chemical paths on the spanning tree and loop-erased random walks, we reduce the problem to determination of the fractal dimension of spanning sub-trees. Then, the upper critical dimension $d_u=4$ follows from Lawler's theorems for intersection probabilities of random walks and l…
A microscopic model of the effect of unbinding in diffusion limited aggregation based on a cellular automata approach is presented. The geometry resembles electrochemical deposition - ``ions'' diffuse at random from the top of a container until encountering a cluster in contact with the bottom, to which they stick. The model exhibits dendritic (fractal) growth in the diffusion limited case. The addition of a field eliminates the fractal nature but the density remains low. The addition of molecules which unbind atoms from the aggregate transforms the deposit to a 100% dense one (in 3D). The mol…
The finite-frequency dynamics of the $^4$He superfluid phase transition can be formulated in terms of the response of thermally excited vortex loops to an oscillating flow field. The key parameter is the Hausdorff fractal dimension $d_H$ of the loops, which affects the dynamics because the frictional force on a loop is proportional to the total perimeter $P$ of the loop, which varies as $P \sim a^{d_H}$ where $a$ is the loop diameter. Solving the 3D Fokker-Planck equation for the loop response at frequency $ω$ yields a superfluid density which varies at $T_λ$ as $ω^{1/(d_H -1)}$. This power-la…
We formulate the angular structure of Lacunarity in fractals, in terms of a symmetry reduction of the three point correlation function. This provides a rich probe of universality, and first measurements yield new evidence in support of the equivalence between self-avoiding walks and percolation perimeters in two dimensions. We argue the Lacunarity reveals much of the Renormalisation Group in real space. This is supported by exact calculations for Random Walks and measured data for percolation clusters and SAW's. Relationships follow between exponents governing inwards and outwards propagating …
Driven periodic elastic systems such as charge-density waves (CDWs) pinned by impurities show a non-trivial, glassy dynamical critical behavior. Their proper theoretical description requires the functional renormalization group. We show that their critical behavior close to the depinning transition is related to a much simpler model, $O(n)$-symmetric $ϕ^4$ theory in the unusual limit of $n\to -2$. We demonstrate that both theories yield identical results to 4-loop order and give both a perturbative and a non-perturbative proof of their equivalence. As we show, both theories can be used to desc…
