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 zipper algorithm for numerically computing the driving function of a simple curve. We give an implementation of this algorithm which runs in a time O(N^1.35) rather than the usual O(N^2), where N is the number of points on the curve.
We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version applies in particular to random lattice paths that have chordal $\mathrm {SLE}_κ$ as a scaling limit, with $κ<8$ (nonspace-filling). Existing $\mathrm {SLE}_κ$ convergence proofs often begin by showing that the Loewner driving functions of these paths (viewed from $\infty$) converge to Brownian motion. Unfortunately, this is not sufficient, and additional argumen…
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 …
Loewner hulls are determined by their real-valued driving functions. We study the geometric effect on the Loewner hulls when the driving function is composed with a random time change, such as the inverse of an $α$-stable subordinator. In contrast to SLE, we show that for a large class of random time changes, the time-changed Brownian motion process does not generate a simple curve. Further we develop criteria which can be applied in many situations to determine whether the Loewner hull generated by a time-changed driving function is simple or non-simple. To aid our analysis of an example with…
We analyze Loewner traces driven by functions asymptotic to K\sqrt{1-t}. We prove a stability result when K is not 4 and show that K=4 can lead to non locally connected hulls. As a consequence, we obtain a driving term λ(t) so that the hulls driven by Kλ(t) are generated by a continuous curve for all K > 0 with K not equal to 4 but not when K = 4, so that the space of driving terms with continuous traces is not convex. As a byproduct, we obtain an explicit construction of the traces driven by K\sqrt{1-t} and a conceptual proof of the corresponding results of Kager, Nienhuis and Kadanoff, math-…
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…
First we introduce the two tau-functions which appeared either as the $τ$-function of the integrable hierarchy governing the Riemann mapping of Jordan curves or in conformal field theory and the universal Grassmannian. Then we discuss various aspects of their interrelation. Subsequently, we establish a novel connection between free probability, growth models and integrable systems, in particular for second order freeness, and summarise it in a dictionary. This extends the previous link between conformal maps and large $N$-matrix integrals to (higher) order free probability. Within this context…
We construct radial stochastic Loewner evolution in multiply connected domains, choosing the unit disk with concentric circular slits as a family of standard domains. The natural driving function or input is a diffusion on the associated Teichmüller space. The diffusion stops when it reaches the boundary of the Teichmüller space. We show that for this driving function the family of random growing compacts has a phase transition for $κ=4$ and $κ=8$, and that it satisfies locality for $κ=6$.
Similar to the well-known phases of SLE, the Loewner differential equation with Lip(1/2) driving terms is known to have a phase transition at norm 4, when traces change from simple to non-simple curves. We establish the deterministic analog of the second phase transition of SLE, where traces change to space-filling curves: There is a constant C>4 such that a Loewner driving term whose trace is space filling has Lip(1/2) norm at least C. We also provide a geometric criterion for traces to be driven by Lip(1/2) functions, and show that for instance the Hilbert space filling curve and the Sierpin…
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…
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 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 $\rm{HL}(0)$ model. We describe the asymptotic…
We investigate the relation between the Laplacian-$b$ motion and stochastic Komatu-Loewner evolution (SKLE) on multiply connected subdomains of the upper half-plane, both of which are analogues to SLE. In particular, we show that, if the driving function of an SKLE is given by a certain stochastic differential equation, then this SKLE is the same as a time-changed Laplacian-$b$ motion. As an application, we prove the finite time explosion of SKLE corresponding to Laplacian-$0$ motion, or $\mathrm{SLE_6}$, in the sense that the solution to the Komatu-Loewner equation for the slits blows up.
This paper studies the behavior of solutions near the explosion time to the chordal Komatu-Loewner equation for slits, motivated by the preceding studies by Bauer and Friedrich (2008) and by Chen and Fukushima (2018). The solution to this equation represents moving slits in the upper half-plane. We show that the distance between the slits and driving function converges to zero at its explosion time. We also prove a probabilistic version of this asymptotic behavior for stochastic Komatu-Loewner evolutions under some natural assumptions.
Let $λ:[0,+\infty)\mapsto\mathbb{R}$ be the driving function of a chordal Loewner process. In this paper we find new conditions on $λ$ which imply that the process is generated by a simple curve. This result improves former one by Lind ,Marshall and Rhode, and it particular gives new results about the case $λ(t)=cW_b(t)$, $W_b$ being a Hölder-$1/2$ Weierstrass function. In the second part we find new conditions on $λ$ implying that the process is generated by a curve. The main tool here is a duality relation between the real part and the imaginary part of the Loewner equation.
The Loewner equation provides a correspondence between continuous real-valued functions $λ_t$ and certain increasing families of half-plane hulls $K_t$. In this paper we study the deterministic relationship between specific analytic properties of $λ_t$ and geometric properties of $K_t$. Our motivation comes, however, from the stochastic Loewner equation (SLE$_κ$), where the associated function $λ_t$ is a scaled Brownian motion and the corresponding domains $\mathbb{H} \backslash K_t$ are Hölder domains. We prove that if the increasing family $K_t$ is generated by a simple curve and the final d…
We estimate convergence rates for curves generated by Loewner's differential equation under the basic assumption that a convergence rate for the driving terms is known. An important tool is what we call the tip structure modulus, a geometric measure of regularity for Loewner curves parameterized by capacity. It is analogous to Warschawski's boundary structure modulus and closely related to annuli crossings. The main application we have in mind is that of a random discrete-model curve approaching a Schramm-Loewner evolution (SLE) curve in the lattice size scaling limit. We carry out the approac…
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…
The development of Schramm--Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde \cite{MR05}, is to sample Brownian motion at discrete times, interpolate appropriately in between and solve explicitly the Loewner equation with this approximation. This algorithm always produces piecewise smooth non self-intersecting curves whereas SLE$_κ$ has been proven to be simple for $κ\in[0,4]$, self-touching for $κ\in(4,8)$ and space-filling for $κ\geq 8$. In this p…
We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE$_κ$ as $κ$ tends to 0 and the known reversibility of the SLE$_κ$ curves for small $κ$, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal ie. of the same curve but viewed as going from B to A in D.
We present new results for the complex generalized integral means spectrum for two kinds of whole-plane Loewner evolutions driven by Lévy processes: - Lévy processes with continuous trajectories, which correspond to Schramm-Loewner evolutions (SLE) with a drift term in the Brownian driving function. A natural path to access the standard integral means spectrum in the presence of drift goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity. -Symmetric Lévy processes for which we generalize recent results by…
We derive a rate of convergence of the Loewner driving function for planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). The proof uses a new estimate of the difference between the discrete and continuous Green's functions that is an improvement over existing results for the class of domains we consider. Using the rate for the driving process convergence along with additional information about SLE(2), we also obtain a rate of convergence for the paths with respect to the Hausdorff distance.
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 describe the Loewner chains of the real locus of a class of real rational functions whose critical points are on the real line. Our main result is that the poles of the rational function lead to explicit formulas for the dynamical system that governs the driving functions. Our formulas give a simple method for mapping the class of rational functions into solutions to a non-trivial system of quadratic equations, and for directly showing that the curves in the real locus satisfy geometric commutation and have the geodesic multichord property. These results are entirely self-contained and have…
Schramm-Loewner evolution (SLE$_κ$) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $\sqrtκ$ times Brownian motion. This yields a (half-plane) valued random field $γ= γ(t, κ; ω)$. (Hölder) regularity of in $γ(\cdot,κ;ω$), a.k.a. SLE trace, has been considered by many authors, starting with Rohde-Schramm (2005). Subsequently, Johansson Viklund, Rohde, and Wong (2014) showed a.s. Hölder continuity of this random field for $κ< 8(2-\sqrt{3})$. In this paper, we improve their result to joint Hölder continuity up to $κ< 8/3$. Moreover, we show that th…
Let A be a bounded, relatively closed subset of the upper half plane H whose complement C is simply connected. If B_t is a standard complex Brownian motion starting at iy and t_A = inf {t > 0: B_t not in C}, the half-plane capacity of A, hcap(A) is defined to be the limit as y goes to infinity of y E[Im(B_{t_A}]. This quantity arises naturally in the study of Schramm-Loewner Evolutions (SLE). In this note, we show that hcap(A) is comparable to a more geometric quantity hsiz(A) that we define to be the 2-dimensional Lebesgue measure of the union of all balls tangent to R whose centers belong to…
Stochastic Loewner Evolutions (SLE) with a multiple sqrt(kappa)B of Brownian motion B as driving process are random planar curves (if kappa<=4) or growing compact sets generated by a curve (if kappa>4). We consider here more general Levy processes as driving processes and obtain evolutions expected to look like random trees or compact sets generated by trees, respectively. We show that when the driving force is of the form sqrt(kappa)B+theta^(1/alpha)S for a symmetric alpha-stable Levy process S, the cluster has zero or positive Lebesgue measure according to whether kappa<=4 or kappa>4. We als…
This paper determines values of intersection exponents between packs of planar Brownian motions in the half-plane and in the plane that were not derived in our first two papers. For instance, it is proven that the exponent $ξ(3,3)$ describing the asymptotic decay of the probability of non-intersection between two packs of three independent planar Brownian motions each is $(73-2 \sqrt {73}) / 12$. More generally, the values of $ξ(w_1, >..., w_k)$ and $\tx (w_1', ..., w_k')$ are determined for all $ k \ge 2$, $w_1, w_2\ge 1$, $w_3, ...,w_k\in[0,\infty)$ and all $w_1',...,w_k'\in[0,\infty)$. The …
This paper proves conjectures originating in the physics literature regarding the intersection exponents of Brownian motion in a half-plane. For instance, suppose that B and B' are two independent planar Brownian motions started from distinct points in a half-plane H. Then, as $t \to \infty$, $$ P [ B[0,t] \cap B'[0,t] = \emptyset, B[0,t] \cup B'[0,t] \subset H ] = t^{-5/3 + o(1)}.$$ The proofs use ideas and tools developped by the authors in previous papers. We prove that one of the stochastic Loewner evolution processes (with parameter 6, that we will call SLE_6 and which has been conjecture…
SLE is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed $κ$. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions. The present paper attempts a first systematic study of SLE. It is proved that for all $κ\ne 8$ the SLE trace is a path; for $κ\in[0,4]$ it is a simple path; for $κ\in(4,8)$ it is a self-intersecting path; and for $κ>8$ it is …
