Fractal dimension of critical curves in the $O(n)$-symmetric $ϕ^4$-model and crossover exponent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, XY and Heisenberg models
Mikhail Kompaniets, Kay Joerg Wiese
备注：
17 pages, 18 figures, 7 tables. v2 with additional explanations
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 exponent, describing a system with mass anisotropy. Introducing a novel self-consistent resummation procedure, and combining it with analytic results in $d=2$ allows us to give improved estimates in $d=3$ for all relevant exponents at 6-loop order.
We study the distribution of sizes of erased loops for loop-erased random walks on regular and fractal lattices. We show that for arbitrary graphs the probability $P(l)$ of generating a loop of perimeter $l$ is expressible in terms of the probability $P_{st}(l)$ of forming a loop of perimeter $l$ when a bond is added to a random spanning tree on the same graph by the simple relation $P(l)=P_{st}(l)/l$. On $d$-dimensional hypercubical lattices, $P(l)$ varies as $l^{-σ}$ for large $l$, where $σ=1+2/z$ for $1<d<4$, where z is the fractal dimension of the loop-erased walks on the graph. On recursi…
We study loop erased random walk (LERW) on the percolation cluster, with occupation probability $p\geq p_c$, in two and three dimensions. We find that the fractal dimensions of LERW$_p$ is close to normal LERW in Euclidean lattice, for all $p>p_c$. However our results reveal that LERW on critical incipient percolation clusters is fractal with $d_{f}=1.217\pm0.0015$ for d = 2 and $1.44\pm0.03$ for d = 3, independent of the coordination number of the lattice. These values are consistent with the known values for optimal path exponents in strongly disordered media. We investigate how the behavior…
We measure the fractal dimension of loop-erased random walk (LERW) in 3 dimensions, and estimate that it is 1.62400 +- 0.00005. LERW is closely related to the uniform spanning tree and the abelian sandpile model. We simulated LERW on both the cubic and face-centered cubic lattices; the corrections to scaling are slightly smaller for the face-centered cubic lattice.
We compute the crossover exponents of all quadratic and cubic deformations of critical field theories with permutation symmetry $S_q$ in $d=6-ε$ (Landau-Potts field theories) and $d=4-ε$ (hypertetrahedral models) up to three loops.We use our results to determine the $ε$-expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests ($q\to0$), and bond percolations ($q\to1$). We also explicitly verify several expected degeneracies in the spectrum of relevant operators for natural values of $q$ upon analytic continuation, which are…
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 show that in the loop-erased random walk problem, the exponent characterizing probability distribution of areas of erased loops is superuniversal. In d-dimensions, the probability that the erased loop has an area A varies as A^{-2} for large A, independent of d, for 2 <= d <= 4. We estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d = 2 and 3 by large-scale Monte Carlo simulations. Our estimate of the fractal dimension z in two-dimensions is consistent with the known exact value 5/4. In three-dimensions, we get z = 1.6183 +- 0.0004. The expone…
We study the scaling limit of planar loop erased random walk (LERW) on the percolation cluster, with occupation probability $p\geq p_c$. We numerically demonstrate that the scaling limit of planar LERW$_p$ curves, for all $p>p_c$, can be described by Schramm-Loewner Evolution (SLE) with a single parameter $κ$ which is close to normal LERW in Euclidean lattice. However our results reveal that the LERW on critical incipient percolation clusters is compatible with SLE, but with another diffusivity coefficient $κ$. Several geometrical tests are applied to ascertain this. All calculations are consi…
We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d=4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d=4, which can be used as a further test of the conjecture.
The thermodynamics of a long self-repulsive chain is studied. In $D<4$ dimensions it shows two distinctly different regimes, corresponding to weak and strong correlations in the system. A simple microscopic analytical model is presented which successfully describes both the regimes. The self-consistent scheme is used, in which the center of mass of a chain is fixed explicitly. This allows to take correlations into account in an indirect manner.
We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a "near-loop" when it comes within one lattice spacing of itself. We define a new curve by erasing these near-loops as we traverse the interface. Our Monte Carlo simulations of this model lead us to conclude that the scaling limit of this loop-erased percolation interface is conformally invariant and has fractal dimension 4/3. However, it is not SLE_8/3. We also consider the process in…
In this paper, we construct a three-phase model (that is, a system consisting of three homogeneous regions with various scattering length densities), which illustrate the behavior of small-angle scattering (SAS) scattering curves. Here two phases are a deterministic fractal embedded in another deterministic mass fractal, and they altogether are further embedded in a third phase, which can be a solution or solid matrix. We calculate SAS intensities, derive expressions for the crossover position (that is, the point where the power-law scattering exponent changes) as a function of control paramet…
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…
Small-angle scattering (SAS) intensities observed experimentally are often characterized by the presence of successive power-law regimes with various scattering exponents whose values vary from -4 to -1. This usually indicates multiple fractal structures of the sample characterized by different size scales. The existing models explaining the crossover positions (that is, the points where the power-law scattering exponent changes) involve only one contrast parameter, which depends solely on the ratio of the fractal sizes. Here, a model that describes SAS from a multi-phase system with a few con…
We study phase-ordering dynamics of a ferromagnetic system with a scalar order-parameter on fractal graphs. We propose a scaling approach, inspired by renormalization group ideas, where a crossover between distinct dynamical behaviors is induced by the presence of a length $λ$ associated to the topological properties of the graph. The transition between the early and the asymptotic stage is observed when the typical size $L(t)$ of the growing ordered domains reaches the crossover length $λ$. We consider two classes of inhomogeneous substrates, with different activated processes, where the effe…
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…
Self-avoiding walks (SAWs) and loop-erased random walks (LERWs) are two ensembles of random paths with numerous applications in mathematics, statistical physics and quantum field theory. While SAWs are described by the $n \to 0$ limit of $ϕ^4$-theory with $O(n)$-symmetry, LERWs have no obvious field-theoretic description. We analyse two candidates for a field theory of LERWs, and discover a connection between the corresponding and a priori unrelated theories. The first such candidate is the $O(n)$-symmetric $ϕ^4$ theory at $n=-2$ whose link to LERWs was known in two dimensions due to conformal…
We use field emission scanning electron microscope (FE-SEM) to investigate the growth of palladium colloids over the surface of thin films of WO3/glass. The film is prepared by Pulsed Laser Deposition (PLD) at different temperatures. A PdCl2 (aq) droplet is injected on the surface and in the presence of steam hydrogen the droplet is dried through a reduction reaction process. Two distinct aggregation regimes of palladium colloids are observed over the substrates. We argue that the change in aggregation dynamics emerges when the measured water drop Contact Angel (CA) for the WO3/glass thin film…
The spatial distribution of unvisited/persistent sites in $d=1$ $A+A\to\emptyset$ model is studied numerically. Over length scales smaller than a cut-off $ξ(t)\sim t^{z}$, the set of unvisited sites is found to be a fractal. The fractal dimension $d_{f}$, dynamical exponent $z$ and persistence exponent $θ$ are related through $z(1-d_{f})=θ$. The observed values of $d_{f}$ and $z$ are found to be sensitive to the initial density of particles. We argue that this may be due to the existence of two competing length scales, and discuss the possibility of a crossover at late times.
Simulations of irreversible growth of extended (fractal and square) islands with critical island sizes i=1 and 2 are performed in broad ranges of coverage θand diffusion-to-deposition ratios R in order to investigate scaling of island size and capture zone area distributions (ISD, CZD). Large θand small R lead to a crossover from the CZD predicted by the theory of Pimpinelli and Einstein (PE), with Gaussian right tail, to CZD with simple exponential decays. The corresponding ISD also cross over from Gaussian or faster decays to simple exponential ones. For fractal islands, these features are e…
Polymer adsorption on fractally rough walls of varying dimensionality is studied by renormalization group methods on hierarchical lattices. Exact results are obtained for deterministic walls. The adsorption transition is found continuous for low dimension $d_w$ of the adsorbing wall and the corresponding crossover exponent $ϕ$ monotonically increases with $d_w$, eventually overcoming previously conjectured bounds. For $d_w$ exceeding a threshold value $d_w^*$, $ϕ$ becomes 1 and the transition turns first--order. $d_w^*>d_{saw}$, the fractal dimension of the polymer in the bulk. An accurate num…
The variance in the winding number of various random fractal curves, including the self-avoiding walk, the loop-erased random walk, contours of FK clusters, and stochastic Loewner evolution, have been studied by numerous researchers. Usually the focus has been on the winding at the endpoints. We measure the variance in winding number at typical points along the curve. More generally, we study the winding at points where k strands come together, and some adjacent strands may be conditioned not to hit each other. The measured values are consistent with an interesting conjecture.
Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China（中国科学技术大学物理科学学院）；Department of Modern Physics, University of Science and Technology of China（中国科学技术大学近代物理系）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）；Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China（中国科学技术大学上海量子科学研究中心及中国科学院量子信息与量子物理学前沿卓越中心）
The spectral dimension has been widely used to understand transport properties on regular and fractal lattices. Nevertheless, it has been little studied for complex networks such as scale-free and small world networks. Here we study the spectral dimension and the return-to-origin probability of random walks on hierarchical scale-free networks, which can be either fractals or non-fractals depending on the weight of shortcuts. Applying the renormalization group (RG) approach to the Gaussian model, we obtain the spectral dimension exactly. While the spectral dimension varies between $1$ and $2$ f…
We apply Kauffman's automata on small-world networks to study the crossover between the short-range and the infinite-range case. We perform accurate calculations on square lattices to obtain both critical exponents and fractal dimensions. Particularly, we find an increase of the damage propagation and a decrease in the fractal dimensions when adding long-range connections.
Voids are a prominent feature of fractal point distributions but there is no precise definition of what is a void (except in one dimension). Here we propose a definition of voids that uses methods of discrete stochastic geometry, in particular, Delaunay and Voronoi tessellations, and we construct a new algorithm to search for voids in a point set. We find and rank-order the voids of suitable examples of fractal point sets in one and two dimensions to test whether Zipf's power-law holds. We conclude affirmatively and, furthermore, that the rank-ordering of voids conveys similar information to t…
A human pathogen, \textit{Neisseria gonorrhoeae} (NG), moves on surfaces by attaching and retracting polymeric structures called Type IV pili. The \textit{tug-of-war} between the pili results in a two-dimensional stochastic motion called \textit{twitching motility}. In this paper, with the help of real time NG trajectories, we develop coarse-grained models for their description. The \textit{fractal properties} of these trajectories are determined and their influence on \textit{first passage time} and formation of bacterial microcolonies is studied. Our main observations are as follows: (i) NG …
We consider the survival probability $f(t)$ of a random walk with a constant hopping rate $w$ on a host lattice of fractal dimension $d$ and spectral dimension $d_s\le 2$, with spatially correlated traps. The traps form a sublattice with fractal dimension $d_a<d$ and are characterized by the absorption rate $w_a$ which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps ($w_a\ll w$), we find that $f(t)$ can be closely approximated by a stretched exponential function over the ini…
A generalization of the Renormalization Group, which describes order-parameter fluctuations in finite systems, is developed in the specific context of percolation. This ``Stochastic Renormalization Group'' (SRG) expresses statistical self-similarity through a non-stationary branching process. The SRG provides a theoretical basis for analytical or numerical approximations, both at and away from criticality, whenever the correlation length is much larger than the lattice spacing (regardless of the system size). For example, the SRG predicts order-parameter distributions and finite-size scaling f…
Extensive Monte Carlo study of two-dimensional Ising model is done to investigate the statistical behavior of spin clusters and interfaces as a function of temperature, $T$. We use a \emph{tie-breaking} rule to define interfaces of spin clusters on square lattice with strip geometry and show that such definition is consistent with conformal invariant properties of interfaces at critical temperature, $T_c$. The \emph{effective} fractal dimensions of spin clusters and interfaces ($d_c$ and $d_I$, respectively) are obtained as a function of temperature. We find that the effective fractal dimensio…
Many resources, such as oil, gas, or water, are extracted from porous soils and their exploration is often shared among different companies or nations. We show that the effective shares can be obtained by invading the porous medium simultaneously with various fluids. Partitioning a volume in two parts requires one division surface while the simultaneous boundary between three parts consists of lines. We identify and characterize these lines, showing that they form a fractal set consisting of a single thread spanning the medium and a surrounding cloud of loops. While the spanning thread has fra…
We show that the equilibrium interfaces in the disordered phase have critical percolation fractal dimension over a wide range of length scales. We confirm that the system falls out of equilibrium at a temperature that depends on the cooling rate as predicted by the Kibble-Zurek argument and we prove that the dynamic growing length once the cooling reaches the critical point satisfies the same scaling. We determine the dynamic scaling properties of the interface winding angle variance and we show that the crossover between critical Ising and critical percolation properties is determined by the …
University of Luxembourg（卢森堡大学）；Center for Theoretical Physics of Complex Systems, Institute for Basic Science(IBS)（韩国基础科学研究院复杂系统理论物理中心）；Universidad del Valle（德瓦列大学）；ICTP South American Institute for Fundamental Research（国际理论物理研究所南美基础研究中心）；Instituto de Física Teórica, UNESP - Univ. Estadual Paulista（圣保罗州立大学理论物理研究所）；Center for Trapped Ions Quantum Science, Institute for Basic Science（韩国基础科学研究院捕获离子量子科学中心）；Basic Science Program, Korea University of Science and Technology (UST)（韩国科学技术院基础科学项目）
A Unified Computational Framework for Two Dimensional Diffusion Limited Aggregation via Finite-Size Scaling, Multifractality, and Morphological Analysis
Fractional diffusion without disorder in two dimensions
Max-Planck-Institut für Physik komplexer Systeme（马克斯·普朗克复杂系统物理研究所）；TCM Group, Cavendish Laboratory, University of Cambridge（剑桥大学卡文迪许实验室 TCM 组）；Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg（奥格斯堡大学物理学院电子相关与磁性中心理论物理第三系）；University of Cambridge（剑桥大学）；University of Augsburg（奥格斯堡大学）
The structure of the three-dimensional random field Ising magnet is studied by ground state calculations. We investigate the percolation of the minority spin orientation in the paramagnetic phase above the bulk phase transition, located at [Delta/J]_c ~= 2.27, where Delta is the standard deviation of the Gaussian random fields (J=1). With an external field H there is a disorder strength dependent critical field +/- H_c(Delta) for the down (or up) spin spanning. The percolation transition is in the standard percolation universality class. H_c ~ (Delta - Delta_p)^{delta}, where Delta_p = 2.43 +/…
We review some results on the effect of a specific type of quenched disorder on well known O(m)-vector models in three dimension. Evidences of changes of criticality in both systems, when confined in aerogel pores, are briefly referenced. The 3DXY model (m=2) represents the universality class to which the λ-transition of bulk superfluid $^4$He belongs. Experiments report changes of critical exponents for this transition, when superfluid $^4$He is confined in aerogels. Numerical results of the 3DXY model, confined in aerogel-like structures, are in agreement with experiments. Both results seem …
