We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition of the Schramm-Loewner evolutions SLE, we define these objects, study its various properties, show how to compute (probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.
We consider shape, size and regularity of the hulls of the chordal Schramm-Loewner evolution driven by a symmetric alpha-stable process. We obtain derivative estimates, show that the complements of the hulls are Hoelder domains, prove that the hulls have Hausdorff dimension 1, and show that the trace is right-continuous with left limits almost surely.
We discuss the partition function point of view for chordal Schramm-Loewner evolutions and their relationship with correlation functions in conformal field theory. Both are closely related to crossing probabilities and interfaces in critical models in two-dimensional statistical mechanics. We gather and supplement previous results with different perspectives, point out remaining difficulties, and suggest directions for future studies.
We establish a large deviation principle for chordal SLE$_κ$ parametrized by capacity, as the parameter $κ\to 0+$, 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 E. Peltola and Y. Wang who obtained the analogous statement using the Hausdorff topology.
The Schramm-Loewner evolution (SLE) can be simulated by dividing the time interval into N subintervals and approximating the random conformal map of the SLE by the composition of N random, but relatively simple, conformal maps. In the usual implementation the time required to compute a single point on the SLE curve is O(N). We give an algorithm for which the time to compute a single point is O(N^p) with p<1. Simulations with kappa=8/3 and kappa=6 both give a value of p of approximately 0.4.
Schramm-Loewner evolution appears as the scaling limit of interfaces in lattice models at critical point. Critical behavior of these models can be described by minimal models of conformal field theory. Certain CFT correlation functions are martingales with respect to SLE. We generalize Schramm-Loewner evolution with additional Brownian motion on Lie group $G$ to the case of factor space $G/A$. We then study connection between SLE description of critical behavior with coset models of conformal field theory. In order to be consistent such construction should give minimal models for certain choic…
We prove the existence of the Green's function for radial SLE(k) for k<8. Unlike the chordal case where an explicit formula for the Green's function is known for all values of k<8, we give an explicit formula only for k=4. For other values of k, we give a formula in terms of an expectation with respect to SLE conditioned to go through a point.
We conformally weld (via "quantum zipping") two boundary arcs of a Liouville quantum gravity random surface to generate a random curve called the Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under welding via SLE martingales. As an application, we construct the natural quantum length and boundary intersection measures on the SLE curve itself.
In this note, we prove a version of the conjectured duality of Schramm-Loewner Evolutions, by establishing exact identities in distribution between some boundary arcs of chordal $\SLE_κ$, $κ>4$, and appropriate versions of $\SLE_{\hatκ}$, $\hatκ=16/κ$.
We consider the measure on multiple chordal Schramm-Loewner evolution ($SLE_κ$) curves. We establish a derivative estimate and use it to give a direct proof that the partition function is $C^2$ if $κ<4$.
We present a mathematical proof of theoretical predictions made by Arguin and Saint-Aubin, as well as by Bauer, Bernard, and Kytola, 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 SLE(3) 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 SLE(k) path, k in (0,4], and a Brownian motion excursion do not intersect.
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 integratio…
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 $\widehat{\mathfrak{osp}(1|2)}$. We also demonstrate computation of local martingales associated with the solution from a representation …
It is known that a backward Schramm--Loewner evolution (SLE) is coupled with a free boundary Gaussian free field (GFF) with boundary perturbation to give conformal welding of quantum surfaces. Motivated by a generalization of conformal welding for quantum surfaces with multiple marked boundary points, we propose a notion of multiple backward SLE. To this aim, we investigate the commutation relation between two backward Loewner chains, and consequently, we find that the driving process of each backward Loewner chain has to have a drift term given by logarithmic derivative of a partition functio…
The Rohde--Schramm theorem states that Schramm--Loewner Evolution with parameter $κ$ (or SLE$_κ$ for short) exists as a random curve, almost surely, if $κ\neq 8$. 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 obt…
We prove that the geodesics associated with any metric generated from Liouville quantum gravity (LQG) which satisfies certain natural hypotheses are necessarily singular with respect to the law of any type of SLE$_κ$. These hypotheses are satisfied by the LQG metric for $γ=\sqrt{8/3}$ constructed by the first author and Sheffield, and subsequent work by Gwynne and the first author has shown that there is a unique metric which satisfies these hypotheses for each $γ\in (0,2)$. As a consequence of our analysis, we also establish certain regularity properties of LQG geodesics which imply, among ot…
The smart kinetic self-avoiding walk (SKSAW) is a random walk which never intersects itself and grows forever when run in the full-plane. At each time step the walk chooses the next step uniformly from among the allowable nearest neighbors of the current endpoint of the walk. In the full-plane a nearest neighbor is allowable if it has not been visited before and there is a path from the nearest neighbor to infinity through sites that have not been visited before. It is well known that on the hexagonal lattice the SKSAW in a bounded domain between two boundary points is equivalent to an interfa…
We prove the existence and nontriviality of the $d$-dimensional 4 Minkowski content for the Schramm-Loewner evolution ($\mathrm {SLE}_κ$) with $κ<8$ and $d=1+\fracκ{8}$. We show that this is a multiple of the natural parameterization.
We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime that the curve is self-intersecting but not space-filling. We let ${\mathcal K}$ be the set of $κ\in (4,8)$ for which the adjacency graph of connected components of the complement of an SLE$_κ$ is a.s. connected, meaning that for every pair of complementary components $U, V$ there exist complementary components $U_1,\ldots,U_n$ with $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. We show that the range of an SLE$_κ$ for $κ\in {\mathcal K}$ is a.s…
We use the interpretation of the Schramm-Loewner evolution as a limit of path measures tilted by a loop term in order to motivate the definition of $n$-radial SLE going to a particular point. In order to justify the definition we prove that the measure obtained by an appropriately normalized loop term on $n$-tuples of paths has a limit. The limit measure can be described as $n$ paths moving by the Loewner equation with a driving term of Dyson Brownian motion. While the limit process has been considered before, this paper shows why it naturally arises as a limit of configurational measures obta…
Conformally-invariant curves that appear at critical points in two-dimensional statistical mechanics systems, and their fractal geometry have received a lot of attention in recent years. On the one hand, Schramm has invented a new rigorous as well as practical calculational approach to critical curves, based on a beautiful unification of conformal maps and stochastic processes, and by now known as Schramm-Loewner evolution (SLE). On the other hand, Duplantier has applied boundary quantum gravity methods to calculate exact multifractal exponents associated with critical curves. In the first par…
In the paper we prove that, for $κ\in(0,8)$, the $n$-point boundary Green's function of exponent $\frac8κ-1$ for chordal SLE$_κ$ exists. We also prove that the convergence is uniform over compact sets and the Green's function is continuous. We also give up-to-constant bounds for the Green's function.
The natural paramterization or length for the Schramm-Loewner evolution (SLEκ) is the candidate for the scaling limit of the length of discrete curves for κ< 8. We improve the proof of the existence of the parametrization and use this to establish some new results. In particular, we show that the natural parametrization is independent of domain and it is Hölder continuous with respect to the capacity parametrization. We also give up-to-constants bounds for the two-point Green's function. Although we do not prove the conjecture that the natural length is given by the appropriate Minkowski conte…
The Schramm-Loewner evolution (SLE_κ) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When κ< 8, an instance of SLE_κis a random planar curve with almost sure Hausdorff dimension d = 1 + κ/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume. For κ< 8, we use a Doob-Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE_κthat transforms like a d-dimensional volume measure under conformal maps. We prove that this param…
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 {\it et al.} [Phys. Rev. Lett. {\bf 95}, 251601 (2005)] and Alekseev {\it et al.} [Lett. Math. Phys. {\bf 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 $\mathfrak{sl}_{2}$. Our formulation enables us to find sever…
Schramm-Loewner evolution (SLE) is a random process that gives a useful description of fractal curves. After its introduction, many works concerning the connection between SLE and conformal field theory (CFT) have been carried out. In this paper, we develop a new method of coupling SLE with a Wess-Zumino-Witten (WZW) model for $SU(2)$, an example of CFT, relying on a coset construction of Virasoro minimal models. Generalizations of SLE that correspond to WZW models were proposed by previous works [Bettelheim {\it et al.}, Phys. Rev. Lett. {\bf 95}, 251601 (2005)] and [Alekseev {\it et al.}, Le…
In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy-Smirnov formula. This theorem, together with the introduction of Schramm-Loewner Evolution and techniques developed over the years in percolation, allow precise descriptions of the critical and near-critical regimes of the model. This survey aims to describe the different steps leading to the proof that the infinite-cluster density $θ(p)$ for site percolation on the triangular lattice behaves like $(p-p_c)^{5/36+o(1)}$ as $p\sear…
Square ice is a statistical mechanics model for two-dimensional ice, widely believed to have a conformally invariant scaling limit. We associate a Peano (space filling) curve to a square ice configuration, and more generally to a so-called 6-vertex model configuration, and argue that its scaling limit is a space-filling version of the random fractal curve SLE$_κ$, Schramm--Loewner evolution with parameter $κ$, where $4<κ\leq 12+8\sqrt{2}$. For square ice, $κ=12$. At the "free-fermion point" of the 6-vertex model, $κ=8+4\sqrt{3}$. These unusual values lie outside the classical interval $2\le κ\…
In this article, we study multiple $SLE_κ$, for $κ\in(0,4]$, driven by Dyson Brownian motion. This model was introduced in the unit disk by Cardy in connection with the Calogero-Sutherland model. We prove the Carathéodory convergence of perturbed Loewner chains under different initial conditions and under different diffusivity $κ\in (0,4]$ 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…
We provide multiple Schramm-Loewner evolutions (SLEs) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models is described by Wess-Zumino-Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct the multiple SLE with additional Lie algebra symmetries. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE trac…
Studying SLE$_κ$ on $S^2$ provides a new and interesting perspective for the conformality of some 2-dimensional physical models. We prove the existence and some basic properties of the spherical Minkowski content of SLE$_κ$, which is finite and can be viewed as the natural parametrization of SLE$_κ$ on $S^2$.
Simulations of the self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's Stochastic Loewner Evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance would imply that if we map the walks in the cut-plane into the half plane using the conformal map z -> sqrt(z), then the resulting walks …
A quantum surface (QS) is an equivalence class of pairs $(D,H)$ of simply connected domains $D\subsetneq\mathbb{C}$ and random distributions $H$ on $D$ induced by the conformal equivalence for random metric spaces. This distribution-valued random field is extended to a QS with $N+1$ marked boundary points (MBPs) with $N\in\mathbb{Z}_{\ge 0}$. We propose the conformal welding problem for it in the case of $N\in\mathbb{Z}_{\ge 1}$. If $N=1$, it is reduced to the problem introduced by Sheffield, who solved it by coupling the QS with the Schramm--Loewner evolution (SLE). When $N \ge 3$, there natu…
Bessel process is defined as the radial part of the Brownian motion (BM) in the $D$-dimensional space, and is considered as a one-parameter family of one-dimensional diffusion processes indexed by $D$, BES$^{(D)}$. It is well-known that $D_{\rm c}=2$ is the critical dimension. Bessel flow is a notion such that we regard BES$^{(D)}$ with a fixed $D$ as a one-parameter family of initial value. There is another critical dimension $\bar{D}_{\rm c}=3/2$ and, in the intermediate values of $D$, $\bar{D}_{\rm c} < D < D_{\rm c}$, behavior of Bessel flow is highly nontrivial. The dimension D=3 is speci…
We consider the Peano curve separating a spanning tree from its dual spanning tree on an embedded planar graph, where the tree and dual tree are weighted by $y$ to the number of active edges, and "active" is in the sense of the Tutte polynomial. When the graph is a portion of the square grid approximating a simply connected domain, it is known ($y=1$ and $y=1+\sqrt{2}$) or believed ($1<y<3$) that the Peano curve converges to a space-filling SLE$_κ$ loop, where $y=1-2\cos(4π/κ)$, corresponding to $4<κ\leq 8$. We argue that the same should hold for $0\le y<1$, which corresponds to $8<κ\leq 12$.
When studying stochastic processes, it is often fruitful to have an understanding of several different notions of regularity. One such notion is the optimal Hölder exponent obtainable under reparametrization. In this paper, we show that the chordal SLE_kappa path in the unit disk for kappa less than or equal to 4 can be reparametrized to be Hölder continuous of any order up to 1/(1+kappa/8). From this result, we obtain that the Young integral is well defined along such SLE paths with probability one, and hence that SLE admits a path-wise notion of integration. This allows for us to consider th…
Levy-Loewner evolution (LLE) is a generalization of the Schramm-Loewner evolution (SLE) where the branching is possible in a course of growth process. We consider a class of radial Levy-Loewner evolutions for which sets of points of the average means beta-spectrum can be found exactly. In this paper we show how to overcome difficulties arised in previous works on multi-fractal analysis of SLE/LLE.
We show that, for $κ\in(0,8)$, the integral of the laws of two-sided radial SLE$_κ$ curves through different interior points against a measure with SLE$_κ$ Green function density is the law of a chordal SLE$_κ$ curve, biased by the path's natural length. We also show that, for $κ>0$, the integral of the laws of extended SLE$_κ(-8)$ curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLE$_κ$ curve, biased by the path's capacity length restricted in that set. Another result is that, for $κ\in(4,8)$, if one i…
This is the first expository set of notes on SLE I have written since publishing a book two years ago [45]. That book covers material from a year-long class, so I cannot cover everything there. However, these notes are not just a subset of those notes, because there is a slight change of perspective. The main differences are: o I have defined SLE as a finite measure on paths that is not necessarily a probability measure. This seems more natural from the perspective of limits of lattice systems and seems to be more useful when extending SLE to non-simply connected domains. (However, I do not di…
Schramm Loewner Evolutions (SLE) are random increasing hulls defined through the Loewner equation driven by Brownian motion. It is known that the increasing hulls are generated by continuous curves. When the driving process is of the form \sqrtκ B+θ^{1/α}S for a Brownian motion B and a symmetric α-stable process S with κnot equal to 4 and 8, we prove that the corresponding increasing hulls are generated by Cadlag curves.
We show how to relate Schramm-Loewner Evolutions (SLE) to highest-weight representations of infinite dimensional Lie Algebras using the conformal restriction properties studied by Lawler, Schramm and Werner in the paper arXiv:math.PR/0209343. This confirms the prediction from theoretical physics and conformal field theory that two-dimensional critical systems are related to such degenerate representations.
Some possible definitions for the natural parametrization of SLE (Schramm-Loewner evolution) paths are proposed in terms of various limits. One of the definitions is used to give a new proof of the Hausdorff dimension of SLE paths.
In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally-invariant scaling limit seems sufficient to deduce the required condition. Our paper serves as an important step in establishin…
We simulate several models of random curves in the half plane and numerically compute their stochastic driving process (as given by the Loewner equation). Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion. We find that just testing the normality of the process at a fixed time is not effective at determining if the process is Brownian motion. Tests that involve the independence of the increments of Brownian motion are much more effective. We also study the …
Building on the identification of the scaling limit of the critical percolation exploration process as a Schramm-Loewner Evolution, we derive a PDE characterization for the crossing probability of an annulus.
We provide an order of convergence for a version of the Carathéodory convergence for the multiple SLE model with a Dyson Brownian motion driver towards its hydrodynamic limit, for $β=1$ and $β=2$. The result is obtained by combining techniques from the field of Schramm-Loewner Evolutions with modern techniques from random matrices. Our approach shows how one can apply modern tools used in the proof of universality in random matrix theory, in the field of Schramm-Loewner Evolutions.
We show how to combine Kesten's scaling relations, the determination of critical exponents associated to the stochastic Loewner evolution process by Lawler, Schramm, and Werner, and Smirnov's proof of Cardy's formula, in order to determine the existence and value of critical exponents associated to percolation on the triangular lattice.
This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Loewner Evolution (SLE) by Oded Schramm. This article opens with a discussion of Loewner's method, explaining how this method can be used to describe families of random curves. Then we define SLE and discuss some of its properties. We also explain how the connection can be made between SLE and the discrete models whose scaling limits it describes, or is believed to describe. Finally, we have inc…
We relate the formulas giving Brownian (and other) intersection exponents to the absolute continuity relations between Bessel process of different dimensions, via the two-parameter family of Schramm-Loewner Evolution processes SLE(kappa,rho) introduced in arXiv:math.PR/0209343. This allows also to compute the value of some new exponents (``hiding exponents'') related to SLEs and planar Brownian motions.
In this note, we show how to relate the Schramm-Loewner Evolution processes (SLE) to highest-weight representations of the Virasoro Algebra. The conformal restriction properties of SLE that have been recently studied in the paper arXiv:math.PR/0209343 by G. Lawler, O. Schramm and the second author play an instrumental role. In this setup, various considerations from conformal field theory can be interpreted and reformulated via SLE. This enables to make a concrete link between the two-dimensional discrete critical systems from statistical physics and conformal field theory.
We show how to combine our earlier results to deduce strong convergence of the interfaces in the planar critical Ising model and its random-cluster representation to Schramm's SLE curves with parameter $κ=3$ and $κ=16/3$ respectively.
