Schramm-Loewner Evolution Research Articles

Bipolar orientations on planar maps and $\mathrm{SLE}_{12}$

We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the “peano curve” surrounding the tree of left-most paths to the sink, converges in law with respect to the peanosphere topology to a $\sqrt{4/3}$-Liouville quantum gravity surface decorated by an independent Schramm–Loewner evolution with parameter $\kappa=12$ (i.e., $\mathrm{SLE}_{12}$). This result is universal in the sense that it holds for bipolar-oriented triangulations, quadrangulations, $k$-angulations and maps in which face sizes are mixed.

Note on Schramm-Loewner Evolution for Superconformal Algebras

We propose variants of Schramm-Loewner evolution (SLE) that are related to superconformal algebras following the group theoretical formulation of SLE, in which the relevant stochastic differential equation is derived from a random process on an infinite dimensional Lie group. In this paper, we consider random processes on certain kind of groups of superconformal transformations generated by exponentiated elements of the Grassmann envelop of superconformal algebras. We also provide a prescription of obtaining local martingales from a representation of the superconformal algebra after integration by Grassmann variables.

Global and Local Multiple SLEs for $${\kappa \leq 4}$$ κ ≤ 4 and Connection Probabilities for Level Lines of GFF

This article pertains to the classification of multiple Schramm-Loewner evolutions (SLE). We construct the pure partition functions of multiple SLE$(\kappa)$ with $\kappa \in (0,4]$ and relate them to certain extremal multiple SLE measures, thus verifying a conjecture from [BBK05, KP16]. We prove that the two approaches to construct multiple SLEs - the global, configurational construction of [KL07, Law09a] and the local, growth process construction of [BBK05, Dub07, Gra07, KP16] - agree. The pure partition functions are closely related to crossing probabilities in critical statistical mechanics models. With explicit formulas in the special case of $\kappa = 4$, we show that these functions give the connection probabilities for the level lines of the Gaussian free field (GFF) with alternating boundary data. We also show that certain functions, known as conformal blocks, give rise to multiple SLE(4) that can be naturally coupled with the GFF with appropriate boundary data.

From Spin Glasses to Negative-Weight Percolation.

Spin glasses are prototypical random systems modelling magnetic alloys. One important way to investigate spin glass models is to study domain walls. For two dimensions, this can be algorithmically understood as the calculation of a shortest path, which allows for negative distances or weights. This led to the creation of the negative weight percolation (NWP) model, which is presented here along with all necessary basics from spin glasses, graph theory and corresponding algorithms. The algorithmic approach involves a mapping to the classical matching problem for graphs. In addition, a summary of results is given, which were obtained during the past decade. This includes the study of percolation transitions in dimension from up to and beyond the upper critical dimension , also for random graphs. It is shown that NWP is in a different universality class than standard percolation. Furthermore, the question of whether NWP exhibits properties of Stochastic–Loewner Evolution is addressed and recent results for directed NWP are presented.

High-precision studies of domain-wall properties in the 2D Gaussian Ising spin glass

In two dimensions, short-range spin glasses order only at zero temperature, where efficient combinatorial optimization techniques can be used to study these systems with high precision. The use of large system sizes and high statistics in disorder averages allows for reliable finite-size extrapolations to the thermodynamic limit. Here, we use a recently introduced mapping of the Ising spin-glass ground-state problem to a minimum-weight perfect matching problem on a sparse auxiliary graph to study square-lattice samples of up to 10 000 × 10 000 spins. We propose a windowing technique that allows to extend this method, that is formally restricted to planar graphs, to the case of systems with fully periodic boundary conditions. These methods enable highly accurate estimates of the spin-stiffness exponent and domain-wall fractal dimension of the 2D Edwards-Anderson spin-glass with Gaussian couplings. Studying the compatibility of domain walls in this system with traces of stochastic Loewner evolution (SLE), we find a strong dependence on boundary conditions and compatibility with SLE only for one out of several setups.

Optimal Hölder continuity and dimension properties for SLE with Minkowski content parametrization

We make use of the fact that a two-sided whole-plane Schramm–Loewner evolution (SLE $$_\kappa $$ ) curve $$\gamma $$ for $$\kappa \in (0,8)$$ from $$\infty $$ to $$\infty $$ through 0 may be parametrized by its d-dimensional Minkowski content, where $$d=1+\frac{\kappa }{8}$$ , and become a self-similar process of index $$\frac{1}{d}$$ with stationary increments. We prove that such $$\gamma $$ is locally $$\alpha $$ -Holder continuous for any $$\alpha <\frac{1}{d}$$ . In the case $$\kappa \in (0,4]$$ , we show that $$\gamma $$ is not locally $$\frac{1}{d}$$ -Holder continuous. We also prove that, for any deterministic closed set $$A\subset \mathbb {R}$$ , the Hausdorff dimension of $$\gamma (A)$$ almost surely equals d times the Hausdorff dimension of A.

On the Smoothness of the Partition Function for Multiple Schramm–Loewner Evolutions

We consider the measure on multiple chordal Schramm-Loewner evolution ($SLE_\kappa$) curves. We establish a derivative estimate and use it to give a direct proof that the partition function is $C^2$ if $\kappa<4$.

Local martingales associated with Schramm-Loewner evolutions with internal symmetry

We consider Schramm-Loewner evolutions (SLEs) with internal degrees of freedom that are associated with representations of affine Lie algebras, following group theoretical formulation of SLEs. We reconstruct the SLEs considered by Bettelheim et al. [Phys. Rev. Lett. 95, 251601 (2005)] and Alekseev et al. [Lett. Math. Phys. 97, 243–261 (2011)] in correlation function formulation. We also explicitly formulate stochastic differential equations on internal degrees of freedom for Heisenberg algebras and the affine sl2. Our formulation enables us to find several local martingales associated with SLEs with internal degrees of freedom from computation on a representation of an affine Lie algebra. Indeed, we formulate local martingales associated with SLEs with internal degrees of freedom described by Heisenberg algebras and the affine sl2. We also find an affine sl2 symmetry of a space of SLE local martingales for the affine sl2.

The Rohde--Schramm theorem, via the Gaussian free field

The Rohde–Schramm theorem states that Schramm–Loewner Evolution with parameter κ (or SLEκ for short) exists as a random curve, almost surely, if κ ≠ 8. Here we give a new and concise proof of the result, based on the Liouville quantum gravity coupling (or reverse coupling) with a Gaussian free field. This transforms the problem of estimating the derivative of the Loewner flow into estimating certain correlated Gaussian free fields. While the correlation between these fields is not easy to understand, a surprisingly simple argument allows us to recover a derivative exponent first obtained by Rohde and Schramm [14], subsequently shown to be optimal by Lawler and Viklund [17], which then implies the Rohde–Schramm theorem.

Schramm–Loewner evolution with Lie superalgebra symmetry

We propose a generalization of Schramm–Loewner evolution (SLE) that has internal degrees of freedom described by an affine Lie superalgebra. We give a general formulation of SLE corresponding to representation theory of an affine Lie superalgebra whose underlying finite-dimensional Lie superalgebra is basic classical type, and write down stochastic differential equations on internal degrees of freedom in case that the corresponding affine Lie superalgebra is [Formula: see text]. We also demonstrate computation of local martingales associated with the solution from a representation of [Formula: see text].

Percolation on Triangulations, and a Bijective Path to Liouville Quantum Gravity

We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the \emph{mating-of-trees} framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to $\sqrt{8/3}$-LQG and SLE$_6$. In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of $\sqrt{8/3}$-LQG and SLE$_6$. For instance, we show that the exploration tree of the percolation converges to a branching SLE$_6$, and that the collection of percolation cycles converges to the conformal loop ensemble CLE$_6$. We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation.

The Loewner equation and Lipschitz graphs

The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm–Loewner evolution SLE, H¨older continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we adopt the former method to the deterministic setting and obtain a new and elementary proof that Holder-1/2 driving functions with norm less than 4 generate simple arcs. We also give a sufficient condition for driving functions to generate curves that are graphs of Lipschitz functions.

The Loewner equation and Lipschitz graphs

The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm–Loewner evolution SLE, H¨older continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we adopt the former method to the deterministic setting and obtain a new and elementary proof that Hölder-1/2 driving functions with norm less than 4 generate simple arcs. We also give a sufficient condition for driving functions to generate curves that are graphs of Lipschitz functions.

SLE as a Mating of Trees in Euclidean Geometry

The mating of trees approach to Schramm-Loewner evolution (SLE) in the random geometry of Liouville quantum gravity (LQG) has been recently developed by Duplantier-Miller-Sheffield (2014). In this paper we consider the mating of trees approach to SLE in Euclidean geometry. Let $\eta$ be a whole-plane space-filling SLE with parameter $\kappa>4$, parameterized by Lebesgue measure. The main observable in the mating of trees approach is the contour function, a two-dimensional continuous process describing the evolution of the Minkowski content of the left and right frontier of $\eta$. We prove regularity properties of the contour function and show that (as in the LQG case) it encodes all the information about the curve $\eta$. We also prove that the uniform spanning tree on $\mathbb Z^2$ converges to $\mathrm{SLE}_8$ in the natural topology associated with the mating of trees approach.

Self-avoiding walk on a square lattice with correlated vacancies.

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 angle analysis is also performed from which the diffusivity parameter of Schramm-Loewner evolution theory (κ) is extracted. We find that at the critical Ising (host) system, the exponents are in agreement with Flory's approximation. For the off-critical Ising system, we find also a behavior for the fractal dimension of the walker trace in terms of the correlation length of the Ising system ξ(T), i.e., D_{F}^{SAW}(T)-D_{F}^{SAW}(T_{c})∼1/sqrt[ξ(T)].

Almost sure multifractal spectrum of Schramm–Loewner evolution

Suppose that η is a Schramm–Loewner evolution (SLEκ) in a smoothly bounded simply connected domain D⊂C and that ϕ is a conformal map from D to a connected component of D∖η([0,t]) for some t>0. The multifractal spectrum of η is the function (−1,1)→[0,∞) which, for each s∈(−1,1), gives the Hausdorff dimension of the set of points x∈∂D such that |ϕ'((1−ϵ)x)|=ϵ−s+o(1) as ϵ→0. We rigorously compute the almost sure multifractal spectrum of SLE, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the almost sure bulk integral means spectrum of SLE, we obtain the optimal Hölder exponent for a conformal map which uniformizes the complement of an SLE curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the SLE curve for κ≤4. Our results also hold for the SLEκ(ρ̲) processes with general vectors of weight ρ̲.

Coupling of c = −2 and c and c = 0 conformal field theories: the geometrical point of view

The coupling of the c = −2, and c = 0 conformal field theories are numerically considered in this paper. As the prototypes of the couplings, and , we consider the Bak–Tang–Weisenfeld (BTW) model on the 2D square critical site-percolation and the BTW model on Ising-correlated percolation lattices respectively. Some geometrical techniques are used to characterize the presumable conformal symmetry of the resultant systems. Based on the numerical analysis of the diffusivity parameter (κ) in the Schramm–Loewner evolution (SLE) theory we propose that the algebra of the central charges of the coupled models is closed. This result is based on the analysis of the conformal loop ensemble (CLE) analysis. The diffusivity parameter in each case is obtained by calculating the fractal dimension of loops (and the corresponding exponent of mean-square root distance), the direct SLE mapping method, the left passage probability and the winding angle analysis. More precisely we numerically show that the coupling results to 2D self-avoiding walk (SAW) fixed point corresponding to c = 0 conformal field theory, whereas the coupling results to the 2D critical Ising fixed point corresponding to the conformal field theory.

Decomposition of Schramm–Loewner evolution along its curve

We show that, for κ∈(0,8), the integral of the laws of two-sided radial SLEκ curves through different interior points against a measure with SLEκ Green’s function density is the law of a chordal SLEκ curve, biased by the path’s natural length. We also show that, for κ>0, the integral of the laws of extended SLEκ(−8) curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLEκ curve, biased by the path’s capacity length restricted in that set. Another result is that, for κ∈(4,8), if one integrates the laws of two-sided chordal SLEκ curves through different force points on R against a measure with density on R, then one also gets a law that is absolutely continuous w.r.t. that of a chordal SLEκ curve. To obtain these results, we develop a framework to study stochastic processes with random lifetime, and improve the traditional Girsanov’s Theorem.

Schramm-Loewner evolution and perimeter of percolation clusters of correlated random landscapes

Motivated by the fact that many physical landscapes are characterized by long-range height-height correlations that are quantified by the Hurst exponent H, we investigate the statistical properties of the iso-height lines of correlated surfaces in the framework of Schramm-Loewner evolution (SLE). We show numerically that in the continuum limit the external perimeter of a percolating cluster of correlated surfaces with H ∈ [−1, 0] is statistically equivalent to SLE curves. Our results suggest that the external perimeter also retains the Markovian properties, confirmed by the absence of time correlations in the driving function and the fact that the latter is Gaussian distributed for any specific time. We also confirm that for all H the variance of the winding angle grows logarithmically with size.

Hydrodynamic Limit of Multiple SLE

Recently del Monaco and Schlei{\ss}inger addressed an interesting problem whether one can take the limit of multiple Schramm--Loewner evolution (SLE) as the number of slits $N$ goes to infinity. When the $N$ slits grow from points on the real line ${\mathbb{R}}$ in a simultaneous way and go to infinity within the upper half plane ${\mathbb{H}}$, an ordinary differential equation describing time evolution of the conformal map $g_t(z)$ was derived in the $N \to \infty$ limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation governs the hydrodynamic limit of the Dyson model defined on ${\mathbb{R}}$ studied in random matrix theory, and when all particles start from the origin, the solution of this Burgers equation is given by the Stieltjes transformation of the measure which follows a time-dependent version of Wigner's semicircle law. In the present paper, first we study the hydrodynamic limit of the multiple SLE in the case that all slits start from the origin. We show that the time-dependent version of Wigner's semicircle law determines the time evolution of the SLE hull, $K_t \subset {\mathbb{H}} \cup {\mathbb{R}}$ , in this hydrodynamic limit. Next we consider the situation such that a half number of the slits start from $a>0$ and another half of slits start from $-a < 0$, and determine the multiple SLE in the hydrodynamic limit. After reporting these exact solutions, we will discuss the universal long-term behavior of the multiple SLE and its hull $K_t$ in the hydrodynamic limit.