Loewner Evolution Research Articles

CONFORMAL INVARIANCE OF CONTOUR LINES ON THE (2 + 1)-DIMENSIONAL RESTRICTED SOLID-ON-SOLID SURFACE

The contour lines of the saturated surface of the (2 + 1)-dimensional restricted solid-on-solid (RSOS) growth model are investigated by numerical method. It is shown that the calculated contour lines are conformal invariant curves with fractal dimension df = 1.34, and they belong to the universality class at large-scale limit, called the Schramm–Loewner evolution with diffusivity κ = 4. This is identical to the value obtained from the inverse cascade of surface quasigeostrophic (SQG) turbulence [Phys. Rev. Lett.98 (2007) 024501]. We also found that the measured fractal dimensions of contours on the (2 + 1)-dimensional RSOS saturated surfaces do not coincide well with that of SLE4 df = 1 + κ/8.

Percolation and Schramm–Loewner evolution in the 2D random-field Ising model

The presence of random fields is well known to destroy ferromagnetic order in Ising systems in two dimensions. When the system is placed in a sufficiently strong external field, however, the size of clusters of like spins diverges. There is evidence that this percolation transition is in the universality class of standard site percolation. It has been claimed that, for small disorder, a similar percolation phenomenon also occurs in zero external field. Using exact algorithms, we study ground states of large samples and find little evidence for a transition at zero external field. Nevertheless, for sufficiently small random-field strengths, there is an extended region of the phase diagram, where finite samples are indistinguishable from a critical percolating system. In this regime we examine ground-state domain walls, finding strong evidence that they are conformally invariant and satisfy Schramm–Loewner evolution ( SLE κ ) with parameter κ = 6 . These results add support to the hope that at least some aspects of systems with quenched disorder might be ultimately studied with the techniques of SLE and conformal field theory.

Loewner equation for Laplacian growth: A Schwarz-Christoffel-transformation approach

The problem of Laplacian growth is considered within the Loewner-equation framework. A new method of deriving the Loewner equation for a large class of growth problems in the half-plane is presented. The method is based on the Schwarz-Christoffel transformation between the so-called 'mathematical planes' at two infinitesimally separated times. Our method not only reproduces the correct Loewner evolution for the case of slit-like fingers but also can be extended to treat more general growth problems. In particular, the Loewner equation for the case of a bubble growing into the half-plane is presented.

Numerical study of Schramm–Loewner evolution in the random 3-state Pottsmodel

We have numerically studied the properties of the interface induced in the ferromagnetic random-bondthree-state Potts model by symmetry-breaking boundary conditions. The fractal dimensiondf of the interface was determined. The corresponding SLE parameterκ was estimated to be , compatible with a previous estimate. On the other hand, we estimatedκ independently, from the probability of passage of the interface at the left of a given point.The numerical data are well reproduced by the Schramm theoretical prediction and the fitleads to , in agreement with the first estimate. This provides evidence that the geometric propertiesof spin interfaces in the random 3-state Potts model may be described by chordalSLEκ.

Geometry Behind Chordal Loewner Chains

Loewner Theory is a deep technique in Complex Analysis affording a basis for many further important developments such as the proof of famous Bieberbach’s conjecture and well-celebrated Schramm’s stochastic Loewner evolution. It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the radial and the chordal Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there has been introduced a new approach (Bracci F et al. in Evolution families and the Loewner equation I: the unit disk. Preprint 2008. Available on ArXiv 0807.1594; Bracci F et al. in Math Ann 344:947–962, 2009; Contreras MD et al. in Loewner chains in the unit disk. To appear in Revista Matemática Iberoamericana; preprint available at arXiv:0902.3116v1 [math.CV]) bringing together, and containing as quite special cases, radial and chordal variants of Loewner Theory. In the framework of this approach we address the question what kind of systems of simply connected domains can be described by means of Loewner chains of chordal type. As an answer to this question we establish a necessary and sufficient condition for a set of simply connected domains to be the range of a generalized Loewner chain of chordal type. We also provide an easy-to-check geometric sufficient condition for that. In addition, we obtain analogous results for the less general case of chordal Loewner evolution considered in (Aleksandrov IA et al. in Complex Analysis. PWN, Warsaw, pp 7–32, 1979; Bauer RO in J Math Anal Appl 302: 484–501, 2005; Goryainov VV and Ba I in Ukrainian Math J 44:1209–1217, 1992).

Poisson point processes, excursions and stable processes in two-dimensional structures

Itô’s contributions lie at the root of stochastic calculus and of the theory of excursions. These ideas are also very useful in the study of conformally invariant two-dimensional structures, via conformal loop ensembles, excursions of Schramm–Loewner evolutions and Poisson point processes of Brownian loops.

Exact computations test stochastic Loewner evolution and scaling in glassy systems

Commentary on `Domain walls and chaos in the disordered SOS model', by K Schwarz, A Karrenbauer, G Schehr and H Rieger, 2009J. Stat. Mech. P08022.

Domain walls and chaos in the disordered SOS model

Domain walls, optimal droplets and disorder chaos at zero temperature are studied numericallyfor the solid-on-solid model on a random substrate. It is shown that the ensemble ofrandom curves represented by the domain walls obeys Schramm’s left passage formula withκ = 4 whereas theirfractal dimension is ds = 1.25, and therefore their behavior cannot be described as showing ‘Schramm– (orstochastic) Loewner evolution’ (SLE). Optimal droplets with a lateral size betweenL and2L have the same fractal dimension as domain walls but an energy that saturates at a value oforder for such that arbitrarily large excitations exist which cost only a small amount of energy.Finally it is demonstrated that the sensitivity of the ground state to small changes of orderδ in the disorder is subtle: beyond a crossover length scaleLδ∼δ−1 the correlations of the perturbed ground state with the unperturbed ground state, rescaledusing the roughness, are suppressed and approach zero logarithmically.

Using the Schramm–Loewner evolution to explain certain non-local observables in the 2D critical Ising model

We present a mathematical proof of theoretical predictions made by Arguin and Saint-Aubin, as well as by Bauer, Bernard and Kytölä, about certain non-local observables for the two-dimensional Ising model at criticality by combining Smirnov's recent proof of the fact that the scaling limit of critical Ising interfaces can be described by chordal SLE3 with Kozdron and Lawler's configurational measure on mutually avoiding chordal SLE paths. As an extension of this result, we also compute the probability that an SLEκ path (with 0 < κ ⩽ 4) and a Brownian motion excursion do not intersect.

Scaling behavior of domain walls at theT=0ferromagnet to spin-glass transition

We study domain-wall excitations in two-dimensional random-bond Ising spin systems on a square lattice with side length L, subject to two different continuous disorder distributions. In both cases an adjustable parameter allows to tune the disorder so as to yield a transition from a spin-glass ordered ground state to a ferromagnetic groundstate. We formulate an auxiliary graph-theoretical problem in which domain walls are given by undirected shortest paths with possibly negative distances. Due to the details of the mapping, standard shortest-path algorithms cannot be applied. To solve such shortest-path problems we have to apply minimum-weight perfect-matching algorithms. We first locate the critical values of the disorder parameters, where the ferromagnet to spin-glass transition occurs for the two types of the disorder. For certain values of the disorder parameters close to the respective critical point, we investigate the system size dependence of the width of the the average domain-wall energy (~L^\theta) and the average domain-wall length (~L^df). Performing a finite-size scaling analysis for systems with a side length up to L=512, we find that both exponents remain constant in the spin-glass phase, i.e. \theta ~- 0.28 and df~1.275. This is consistent with conformal field theory, where it seems to be possible to relate the exponents from the analysis of Stochastic Loewner evolutions (SLEs) via df-1=3/[4(3+\theta)]. Finally, we characterize the transition in terms of ferromagnetic clusters of spins that form, as one proceeds from spin-glass ordered to ferromagnetic ground states.

Ising (conformal) fields and cluster area measures

We provide a representation for the scaling limit of the d = 2 critical Ising magnetization field as a (conformal) random field by using Schramm–Loewner Evolution clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical Fortuin–Kasteleyn clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d = 3, and to Potts models are also considered.

Conditioning Schramm–Loewner evolutions and loop erased random walks

We discuss properties of dipolar Schramm–Loewner evolution (SLEκ) under conditioning. We show that κ=2, which describes continuum limits of loop erased random walks, is characterized as being the only value of κ such that dipolar SLE conditioned to stop on an interval coincides with dipolar SLE on that interval. We illustrate this property by computing a new bulk passage probability for SLE2.

Direct evidence for conformal invariance of avalanche frontiers in sandpile models

Appreciation of stochastic Loewner evolution (SLE_{kappa}) , as a powerful tool to check for conformal invariant properties of geometrical features of critical systems has been rising. In this paper we use this method to check conformal invariance in sandpile models. Avalanche frontiers in Abelian sandpile model are numerically shown to be conformally invariant and can be described by SLE with diffusivity kappa=2 . This value is the same as value obtained for loop-erased random walks. The fractal dimension and Schramm's formula for left passage probability also suggest the same result. We also check the same properties for Zhang's sandpile model.

A generalization of Schramm’s formula forSLE2

The scaling limit of planar loop-erased random walks is describedby using a stochastic Loewner evolution with parameterκ = 2. In thispaper SLE2 in the upper half-plane minus a simply connected compact subset is studied. As a main result, the left-passage probability with respect to is explicitly determined.

Geometric exponents, SLE and logarithmic minimal models

In statistical mechanics, observables are usually related to local degrees of freedom such as theQ<4 distinctstates of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuumscaling limit, these models are described by rational conformal field theories, namely theminimal models for suitable p,p′. More generally, as in stochastic Loewner evolution(SLEκ), one can consider observables related to non-local degrees of freedom such as paths orboundaries of clusters. This leads to fractal dimensions or geometric exponents related tovalues of conformal dimensions not found among the finite sets of values allowed by therational minimal models. Working in the context of a loop gas with loop fugacityβ = −2cos(4π/κ), we use Monte Carlo simulations to measure the fractal dimensions of variousgeometric objects such as paths and the generalizations of cluster mass, cluster hull,external perimeter and red bonds. Specializing to the case where the SLE parameterκ = (4p′/p) is rationalwith p<p′, we argue that the geometric exponents are related to conformal dimensions found in theinfinitely extended Kac tables of the logarithmic minimal models . These theories describe lattice systems with non-local degrees of freedom. We presentresults for critical dense polymers , critical percolation , the logarithmic Ising model , the logarithmic tricritical Ising model as well as . Our results are compared with rigorous results fromSLEκ, with predictions from theoretical physics and with other numericalexperiments. Throughout, we emphasize the relationships betweenSLEκ, geometric exponents and the conformal dimensions of the underlying CFTs.

Rescaled Lévy–Loewner hulls and random growth

We consider radial Loewner evolution driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings–Levitov HL ( 0 ) model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1. Using a different type of compound Poisson process, where the Poisson kernel is replaced by the heat kernel, as driving function, we recover one case of the aforementioned model and SLE ( κ ) as limits.

Schramm–Loewner Equations Driven by Symmetric Stable Processes

We consider shape, size and regularity of the hulls K t of the chordal Schramm–Loewner evolution driven by a symmetric α-stable process. We obtain derivative estimates, show that the domains $${\mathbb{H}{\setminus}K_{t}}$$ are Hölder domains, prove that K t has Hausdorff dimension 1, and show that the trace is right-continuous with left limits almost surely.

Nodal domain statistics for quantum chaotic maps**Dedicated to the memory of Charles Schmit.

We study the statistical distribution of nodal domains in the eigenvectors of quantum chaotic maps in the semiclassical limit. For generic quantum maps, which are believed to behave statistically like random matrices, the nodal domains are described by an uncorrelated critical percolation model. This leads to predictions for the distribution of the number, size and fractal dimension of the nodal domains, which are tested numerically. Furthermore, we find that the corresponding nodal lines can be modelled by stochastic Loewner evolution (SLE) with parameter κ close to 6. Interestingly, the percolation model and SLE are also found to describe the statistical properties of the nodal domains for the quantum cat map, which is non-generic in that its spectral statistics do not fall into any of the random matrix universality classes.

Scaling limits of two‐dimensional percolation: an overview

We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to chordal stochastic Loewner evolution (SLE6), the full scaling limit of critical cluster boundaries, and near‐critical scaling limits.

The dimension of the SLE curves

Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ ≥ 0. We prove that, with probability one, the Haus-dorff dimension of γ is equal to Min(2, 1 + κ/8). Introduction. It has been conjectured by theoretical physicists that various lattice models in statistical physics (such as percolation, Potts model, Ising model, uniform spanning trees), taken at their critical point, have a continuous confor-mally invariant scaling limit when the mesh of the lattice tends to 0. Recently, Oded Schramm [15] introduced a family of random processes which he called Stochastic Loewner Evolutions (or SLE), that are the only possible conformally invariant scaling limits of random cluster interfaces (which are very closely related to all above-mentioned models). An SLE process is defined using the usual Loewner equation, where the driving function is a time-changed Brownian motion. More specifically, in the present paper we will be mainly concerned with SLE in the upper-half plane (sometimes called chordal SLE), defined by the following PDE: