The scaling limit of the two-dimensional self-avoiding walk (SAW) is believed to be given by the Schramm-Loewner evolution (SLE) with the parameter kappa equal to 8/3. The scaling limit of the SAW has a natural parameterization and SLE has a standard parameterization using the half-plane capacity. These two parameterizations do not correspond with one another. To make the scaling limit of the SAW and SLE agree as parameterized curves, we must reparameterize one of them. We present Monte Carlo results that show that if we reparameterize the SAW using the half-plane capacity, then it agrees well with SLE with its standard parameterization. We then consider how to reparameterize SLE to make it agree with the SAW with its natural parameterization. We argue using Monte Carlo results that the so-called p-variation of the SLE curve with p=1/nu=4/3 provides a parameterization that corresponds to the natural parameterization of the SAW.
We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE with kappa=8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and then apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo s…
The conjecture that the scaling limit of the two-dimensional self-avoiding walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE) with $κ=8/3$ leads to explicit predictions about the SAW. A remarkable feature of these predictions is that they yield not just critical exponents, but probability distributions for certain random variables associated with the self-avoiding walk. We test two of these predictions with Monte Carlo simulations and find excellent agreement, thus providing numerical support to the conjecture that the scaling limit of the SAW is SLE$_{8/3}$.
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 that a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk (SAW) converges in the scaling limit to the metric gluing of two independent Brownian half-planes identified along their positive boundary rays. Combined with other work of the authors, this implies the convergence of the SAW on a random quadrangulation to SLE$_{8/3}$ on a certain $\sqrt{8/3}$-Liouville quantum gravity surface. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the local Gromov-Hausdorff topology to curve-decorate…
We define multiple chordal SLEs in a simply connected domain by considering a natural configurational measure on paths. We show how to construct these measures so that they are conformally covariant and satisfy certain boundary perturbation and Markov properties, as well as a cascade relation. As an example of our construction, we derive the scaling limit of Fomin's identity in the case of two paths directly; that is, we prove that the probability that an SLE(2) and a Brownian excursion do not intersect can be given in terms of the determinant of the excursion hitting matrix. Finally, we defin…
We consider the two-dimensional self-avoiding walk (SAW) in a simply connected domain that contains the origin. The SAW starts at the origin and ends somewhere on the boundary. The distribution of the endpoint along the boundary is expected to differ from the SLE partition function prediction for this distribution because of lattice effects that persist in the scaling limit. We give a precise conjecture for how to compute this lattice effect correction and support our conjecture with simulations. We also give a precise conjecture for the lattice corrections that persist in the scaling limit of…
We prove that for almost every Brownian motion sample, the corresponding SLE(κ) curves parameterized by capacity exist and change continuously in the supremum norm when κvaries in the interval [0,κ_0), where κ_0=8(2-\sqrt{3})=2.143... We estimate the κ-dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with κ.
Suppose that $η$ is a whole-plane space-filling SLE$_κ$ for $κ\in (4,8)$ from $\infty$ to $\infty$ parameterized by Lebesgue measure and normalized so that $η(0) = 0$. For each $T > 0$ and $κ\in (4,8)$ we let $μ_{κ,T}$ denote the law of $η|_{[0,T]}$. We show for each $ν, T > 0$ that the family of laws $μ_{κ,T}$ for $κ\in [4+ν,8)$ is compact in the weak topology associated with the space of probability measures on continuous curves $[0,T] \to {\mathbf C}$ equipped with the uniform distance. As a direct byproduct of this tightness result (taking a limit as $κ\uparrow 8$), we obtain a new proof o…
Let D be a domain in the plane containing the origin. We are interested in the ensemble of self-avoiding walks (SAW's) in D which start at the origin and end on the boundary of the domain. We introduce an ensemble of SAW's that we expect to have the same scaling limit. The advantage of our ensemble is that it can be simulated using the pivot algorithm. Our ensemble makes it possible to accurately study SLE predictions for the SAW in bounded simply connected domains. One such prediction is the distribution along the boundary of the endpoint of the SAW. We use the pivot algorithm to simulate our…
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.
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 …
We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as $β\rightarrow β_c-$ of the probability measure on all finite length walks $ω$ with the probability of $ω$ proportional to $β_c^{|ω|}$ where $|ω|$ is the number of steps in $ω$. The self-avoiding walk in a strip $\{z : 0<\Im(z)<y\}$ is defined by considering all self-avoiding walks $ω$ in the strip which start at the origin and end somewhere on the top boundary with probability proportional to $β_c^{|ω|}$ We pro…
We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived non rigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8/3).
We derive the exact value of intersection exponents between planar Brownian motions or random walks, confirming predictions from theoretical physics by Duplantier and Kwon. Let B and B' be independent Brownian motions (or simple random walks) in the plane, started from distinct points. We prove that the probability that the paths B[0,t] and B'[0,t] do not intersect decays like t^{-5/8}. More precisely, there is a constant c>0 such that if |B(0) - B'(0)| =1, for all t \ge 1, c^{-1} t^{-5/8} \le ¶[ B[0,t] \cap B'[0,t] = \emptyset ] \le c t^{-5/8}. One consequence is that the set of cut-points of…
We revisit the convergence of loop-erased random walk, LERW, to SLE(2) when the curves are parametrized by capacity. We construct a coupling of the chordal version of LERW and chordal SLE(2) based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner "difference" equation. This coupling is different than the ones previously considered in this context. Our recent work (arXiv:1603.05203) on the convergence of LERW parametrized by length to SLE(2) parameterized by Minkowski content uses specific features of the coupling constructed here.
Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has its roots in planar map combinatorics from the 1960s together with recent scaling limit results. This article surveys a series of works with Sheffield in which it is shown that Liouville quantum gravity (LQG) with parameter $γ=\sqrt{8/3}$ is equivalent to the Brownian map. We also briefly describe a series of works with …
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…
Loop-erased random walk, abbreviated LERW, is one of the most well-studied critical lattice models. It is the self-avoiding random walk one gets after erasing the loops from a simple random walk in order or alternatively by considering the branches in a uniformly chosen spanning tree. This paper proves that planar LERW parametrized by renormalized length converges in the lattice size scaling limit to SLE(2) parametrized by 5/4-dimensional Minkowski content. In doing this we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theore…
