Loewner Evolution Research Articles

Interfaces of the two-dimensional voter model in the context of SLE

Abstract This paper investigates various geometrical properties of interfaces of the two-dimensional voter model. Despite its simplicity, the model exhibits dual characteristics, resembling both a critical system with long-range correlations, while also showing a tendency towards order similar to the Ising–Glauber model at zero temperature. This duality is reflected in the geometrical properties of its interfaces, which are examined here from the perspective of Schramm–Loewner evolution. Recent studies have delved into the geometrical properties of these interfaces within different lattice geometries and boundary conditions. We revisit these findings, focusing on a system within a box of linear size L with Dobrushin boundary conditions, where values of the spins are fixed to either +1 or −1 on two distinct halves of the boundary, in order to enforce the presence of a pinned interface with fixed endpoints (or chordal interface). We also expand the study to compare the geometrical properties of the interfaces of the voter model with those of the critical Ising model and other related models. Scaling arguments and numerical studies suggest that, while locally the chordal interface of the voter model has fractal dimension d f = 3 / 2 , corresponding to a parameter κ = 4, it becomes straight at large scales, confirming a conjecture made by Holmes et al (2015 J. Stat. Phys. 159 937–57), and ruling out the possibility of describing the chordal interface of the voter model by SLE κ , for any non-zero value of κ. This contrasts with the critical Ising model, which is described by SLE3, and whose interface fluctuations remain of order L, and more generally with related critical models, which are in the same universality class.

Coastlines violate the Schramm–Loewner Evolution

Mandelbrot’s empirical observation that the coast of Britain is fractal has been confirmed by many authors, but it can be described by the Schramm–Loewner Evolution? Since the self-affine surface of our planet has a positive Hurst exponent, one would not expect a priori any critical behavior. Here, we investigate numerically the roughness and fractal dimension of the isoheight lines of real and artificial landscapes. Using a novel algorithm to take into account overhangs, we find that the roughness exponent of isoheight lines is consistent with unity regardless of the Hurst exponent of the rough surface. Moreover, the effective fractal dimension of the iso-height lines decays linearly with the Hurst exponent of the surface. We perform several tests to verify if the complete and accessible perimeters would follow the Schramm–Loewner Evolution and find that the left passage probability test is clearly violated, implying that coastlines violate SLE.

Loewner Theory for Stochastic Neuron Model

The nonlinear response of the dynamics of the stochastic neuronal activity based on the leaky integrate-and-fire model was reformulated using the Loewner theory. We observed that the signal-to-noise ratio (SNR), which is an indicator of stochastic resonance (SR), and fluctuation-dissipation relation (FDR) for the neuronal dynamics were newly obtained in the theoretical framework of the Loewner evolution. Particularly, we focus on the role of the Loewner entropy [Formula: see text], defined as the entropy of the Loewner driving force, while showing the efficacy of Loewner time conversion to theorize the nonlinear characteristic of the neuronal dynamics. The present results indicate a possible approach to the novel formulation of biophysical models.

Growth-induced stability in modified SLE curve

In this study, the non-equilibrium free energy corresponding to the curve generated by a modified stochastic Loewner evolution (SLE), which is driven by the Langevin equation, is theoretically investigated. Under certain conditions, we prove that the time derivative of the (generalized) free energy expressed by Kullback-Leibler divergence between the probability distributions of the curve and driving function has a positive value, indicating the negativity of Gibbs entropy production. In addition, it was implied that, in a certain restriction, the free energy can be expressed as a function of a Lyapunov-type exponent of the driving function. These results show a dissipative nature of conformal dynamics, and indicate the growth-induced stability of the modified SLE curve.

The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

AbstractWe study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure has finite Loewner–Kufarev energy, defined by whenever is of the form , and set to otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner–Kufarev energy is reversible, that is, invariant under inversion and time reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner–Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil–Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner–Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and that we construct using Hadamard's variational formula expressed by means of the Loewner–Kufarev equation. Our results are related to ‐parameter duality and large deviations of Schramm–Loewner evolutions coupled with Gaussian random fields.

Nonequilibrium critical dynamics of the two-dimensional ±J Ising model.

The ±J Ising model is a simple frustrated spin model, where the exchange couplings independently take the discrete value -J with probability p and +J with probability 1-p. It is especially appealing due to its connection to quantum error correcting codes. Here, we investigate the nonequilibrium critical behavior of the two-dimensional ±J Ising model, after a quench from different initial conditions to a critical point T_{c}(p) on the paramagnetic-ferromagnetic (PF) transition line, especially above, below, and at the multicritical Nishimori point (NP). The dynamical critical exponent z_{c} seems to exhibit nonuniversal behavior for quenches above and below the NP, which is identified as a preasymptotic feature due to the repulsive fixed point at the NP, whereas for a quench directly to the NP, the dynamics reaches the asymptotic regime with z_{c}≃6.02(6). We also consider the geometrical spin clusters (of like spin signs) during the critical dynamics. Each universality class on the PF line is uniquely characterized by the stochastic Loewner evolution with corresponding parameter κ. Moreover, for the critical quenches from the paramagnetic phase, the model, irrespective of the frustration, exhibits an emergent critical percolation topology at the large length scales.

FZZ formula of boundary Liouville CFT via conformal welding

Liouville Conformal Field Theory (LCFT) on the disk describes the conformal factor of the quantum disk, which is the natural random surface in Liouville quantum gravity with disk topology. Fateev, Zamolodchikov and Zamolodchikov (2000) proposed an explicit expression, the so-called FZZ formula, for the one-point bulk structure constant for LCFT on the disk. In this paper we give a proof of the FZZ formula in the probabilistic framework of LCFT, which represents the first step towards rigorously solving boundary LCFT using conformal bootstrap. In contrast to previous works, our proof is based on conformal welding of quantum disks and the mating-of-trees theory for Liouville quantum gravity. As a byproduct of our proof, we also obtain the exact value of the variance for the Brownian motion in the mating-of-trees theory. Our paper is an essential part of an ongoing program proving integrability results for Schramm–Loewner evolutions, LCFT, and in the mating-of-trees theory.

Loewner evolution driven by complex Brownian motion

Loewner evolution driven by complex Brownian motion

Generalized parametric representation and extension operators

We consider conditions under which mappings of the open Euclidean unit ball of Cn with generalized parametric representation with respect to an appropriately defined locally integrable function A:[0,∞)→L(Cn), where L(Cn) is the algebra of linear operators from Cn into Cn, on an interval [s,∞), s≥0, are extended to mappings of the same type defined in a higher-dimensional ball (with respect to a related operator-valued function) using two classes of extension operators, namely perturbations of the Pfaltzgraff–Suffridge extension operator and modifications of the Roper–Suffridge extension operator introduced by the author. Extensions of Loewner chains of order p∈[1,∞], i.e., non-normalized Loewner chains that satisfy a locally uniform local Lp-continuity condition on their real parameter, using these operators have previously been considered by the author, but, unlike in the case of standard parametric representation, mappings with generalized parametric representation may not be as nicely compatible with Loewner chains. We will consider the ramifications of this. As part of our work, we will see that A-normalized Loewner chains and evolution families will have order p when ‖A(⋅)‖ is locally Lp. In addition, we will completely characterize generalized parametric representation in the open unit disk. This characterization involves families of bounded univalent functions and allows us to show that the modified Roper–Suffridge extension operators extend families of univalent functions bounded by some M>1 to biholomorphic mappings bounded by M under certain conditions as a consequence of our main extension results.

A large deviation principle for the Schramm–Loewner evolution in the uniform topology

We establish a large deviation principle for chordal SLE\(_\kappa\) parametrized by capacity, as the parameter \(\kappa \to 0+\), in the topology generated by uniform convergence on compact intervals of the positive real line. The rate function is shown to equal the Loewner energy of the curve. This strengthens the recent result of Peltola and Wang who obtained the analogous statement using the Hausdorff topology.

Complex Generalized Integral Means Spectrum of Drifted Whole-Plane SLE and LLE

We present new exact results for the complex generalized integral means spectrum (in the sense of Duplantier et al. (Commun Math Phys 359(3):823–868, 2018)) for two kinds of whole-plane Loewner evolutions driven by a Lévy process:

Statistics of Winding Angle and Loewner Driving Force of Neurite Curves

In this study, the winding angle and Loewner driving force of neurite curves of cultured human induced pluripotent stem cell-derived neurons were statistically analyzed. Our previous study showed that the Loewner driving force exhibits deterministic (chaotic) properties rather than the Stochastic Loewner Evolution (SLE). The present results also showed that the neurite curves are not described by a “simple” SLE, although they have some SLE-like properties, suggesting that the neurite curves are not conformally invariant. On the other hand, the estimated fractal dimension showed that the neurite curves are scale invariant. These results suggest that neurite curves may be a new type of random curves.

The skew Brownian permuton: A new universality class for random constrained permutations

AbstractWe construct a new family of random permutons, called skew Brownian permuton, which describes the limits of several models of random constrained permutations. This family is parameterized by two real parameters. For a specific choice of the parameters, the skew Brownian permuton coincides with the Baxter permuton, that is, the permuton limit of Baxter permutations. We prove that for another specific choice of the parameters, the skew Brownian permuton coincides with the biased Brownian separable permuton, a one‐parameter family of permutons previously studied in the literature as the limit of uniform permutations in substitution‐closed classes. This brings two different limiting objects under the same roof, identifying a new larger universality class. The skew Brownian permuton is constructed in terms of flows of solutions of certain stochastic differential equations (SDEs) driven by two‐dimensional correlated Brownian excursions in the nonnegative quadrant. We call these SDEs skew perturbed Tanaka equations because they are a mixture of the perturbed Tanaka equations and the equations encoding skew Brownian motions. We prove existence and uniqueness of (strong) solutions for these new SDEs. In addition, we show that some natural permutons arising from Liouville quantum gravity (LQG) spheres decorated with two Schramm–Loewner evolution (SLE) curves are skew Brownian permutons and such permutons cover almost the whole range of possible parameters. Some connections between constrained permutations and decorated planar maps have been investigated in the literature at the discrete level; this paper establishes this connection directly at the continuum level. Proving the latter result, we also give an SDE interpretation of some quantities related to SLE‐decorated LQG spheres.

Loewner chains and evolution families on parallel slit half-planes

In this paper, we define and study Loewner chains and evolution families on finitely multiply-connected domains in the complex plane. These chains and families consist of conformal mappings on parallel slit half-planes and have one and two “time” parameters, respectively. By analogy with the case of simply connected domains, we develop a general theory of Loewner chains and evolution families on multiply connected domains and, in particular, prove that they obey the chordal Komatu–Loewner differential equations driven by measure-valued processes. Our method involves Brownian motion with darning, as do some recent studies.

Multi-point passage probabilities and Green’s functions for SLE$${}_{8/3}$$

We consider a loop representation of the $$O(n)$$ model at the critical point. In the case $$n=0$$ , the model reduces to statistical ensembles of self-avoiding loops, which can be described by Schramm–Loewner evolution (SLE) with $$\kappa=8/3$$ . In this limit, the $$O(n=0)$$ model corresponds to a logarithmic conformal field theory (LCFT) with the central charge $$c=0$$ . We study the LCFT correlation functions in the upper half-plane containing several twist operators in the bulk and a pair of the $$\Phi_{1,2}$$ boundary operators. By using a Coulomb gas representation for the correlation functions, we obtain explicit results for the probabilities of the SLE $${}_{8/3}$$ trace to pass in various ways about $$N\geq 1$$ marked points. When the points approach each other pairwise, the probabilities reduce to multipoint SLE Green’s functions. We propose an explicit representation for the Green’s functions in terms of the correlation functions of the bulk $$\Phi_{3,1}$$ and boundary $$\Phi_{1,2}$$ operators.

Fluctuation-dissipation theorem with Loewner time

Although the fluctuation-dissipation theorem (FDT) is recognized as a general result of statistical physics, its applicability to the non-equilibrium and nonlinear system has not been completely clarified. In this study, we observe that introducing a new type of time coordinate determined by the stochastic Loewner evolution enables the conversion of a certain class of nonlinear Langevin equations into linear ones. The formulation using Loewner time yields a linear response function for the nonlinear systems under a small perturbation. The proposed approach is expected to provide an extension of the conventional FDT in a form applicable to a wide class of non-equilibrium dynamics.

Perturbations of multiple Schramm–Loewner evolution with two non-colliding Dyson Brownian motions

In this article we study multiple SLEκ, for κ∈(0,4], driven by Dyson Brownian motions. This model was introduced in the unit disk by Cardy, (2003) in connection with the Calogero–Sutherland model. We prove the Carathéodory convergence of perturbed Loewner chains under different initial conditions and under different diffusivity κ∈(0,4] for the case of N=2 driving forces. Our proofs use the analysis of Bessel processes and estimates on Loewner differential equation with multiple driving forces. In the last section, we estimate the Hausdorff distance of the hulls under perturbations of the driving forces, with assumptions on the modulus of the derivative of the multiple Loewner maps.

Fractional stochastic Loewner evolution and scaling curves

In two-dimensional statistical mechanics, the Stochastic Loewner equation, introduced by Oded Schramm, provides a powerful technique to study and categorize critical random curves and interfaces. For the first time, a new type of stochastic Loewner equation entitled fractional stochastic Loewner evolution (FSLE) has been proposed. We introduce a wide class of fractal curves using the fractional time series as the driving function of the Loewner equation and local fractional integrodifferential operators. We suggest that, in addition to fractal dimension estimations, the Hurst index of the driving function leads significant differences in the FSLE curves. This modification introduces a new approach to categorize different types of scaling curves based on the Hurst index of the driving function. Such formalizations appear to be appropriate for studying a wide range of two-dimensional curves seen in statistical mechanics and natural phenomena.

Permutation-Loewner entropy analysis for 2-dimensional Ising system interface

The structure of the 2-dimensional (2D) Ising system interface has been investigated in various theoretical statistical physics studies. Although the relationships between the interface geometry and physical properties of the system have already been indicated, in this study, we consider their alternative expressions using the chordal Loewner evolution and permutation entropy method. The Loewner driving forces corresponding to the Ising interfaces for below and at the critical temperature Tc were numerically calculated and analyzed by examining their pattern frequency, entropy, and distance to white noise in the theoretical scheme of the permutation entropy method. Using this permutation-Loewner entropy (PLE) method, we found an exponential function-type relation between the PLE and mean energy of the Ising system, which provides a connection between the theory of Loewner evolution and 2D statistical mechanical systems.

Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding

We endow the 8/3-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere S2. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric dQ on a countable dense subset of S2. Here, we show that dQ a.s. extends uniquely and continuously to a metric d‾Q on all of S2. Letting d denote the Euclidean metric on S2, we show that the identity map between (S2,d) and (S2,d‾Q) is a.s. Hölder continuous in both directions. We establish several other properties of (S2,d‾Q), culminating in the fact that (as a random metric measure space) it agrees in law with the Brownian map. We establish analogous results for the Brownian disk and plane. Our proofs involve new estimates on the size and shape of QLE balls and related quantum surfaces, as well as a careful analysis of (S2,d‾Q) geodesics.