N-sided radial Schramm–Loewner evolution Healey, Vivian Olsiewski; Lawler, Gregory F.
Probability theory and related fields,
11/2021, Volume:
181, Issue:
1-3
Journal Article
Peer reviewed
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 obtained from loop measures.
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We prove the existence and nontriviality of the d-dimensional 4 Minkowski content for the Schramm–Loewner evolution (SLEκ) with κ < 8 and d = 1 + κ/8. We show that this is a multiple of the natural ...parameterization.
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The Schramm-Loewner evolution (SLE) is a probability measure on random fractal curves that arise as scaling limits of two-dimensional statistical physics systems. In this paper we survey some results ...about the Hausdorff dimension and Minkowski content of {\rm SLE}_\kappa paths and then extend the recent work on Minkowski content to the intersection of an SLE path with the real line.
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
.
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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 κ is 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 κ that transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is nontrivial (i.e., the curve is not entirely traversed in zero time) for κ < 4(7 − √33) = 5.021...
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Developing the theory of two-sided radial and chordal SLE, we prove that the natural parametrization on SLE κ curves is well defined for all κ < 8. Our proof uses a two-interior-point local ...martingale.
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Compressed self-avoiding walks, bridges and polygons Beaton, Nicholas R; Guttmann, Anthony J; Jensen, Iwan ...
Journal of physics. A, Mathematical and theoretical,
11/2015, Volume:
48, Issue:
45
Journal Article
Peer reviewed
We study various self-avoiding walks (SAWs) which are constrained to lie in the upper half-plane and are subjected to a compressive force. This force is applied to the vertex or vertices of the walk ...located at the maximum distance above the boundary of the half-space. In the case of bridges, this is the unique end-point. In the case of SAWs or self-avoiding polygons, this corresponds to all vertices of maximal height. We first use the conjectured relation with the Schramm-Loewner evolution to predict the form of the partition function including the values of the exponents, and then we use series analysis to test these predictions.
We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green’s function is the probability that ...the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the
SLE
2
Green’s function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a “spinor” observable for random walk started from one of the vertices of the marked edge.
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