The Green's function for the radial Schramm-Loewner evolution

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.Comment: v1: 16 pages, 0 figure

Distribution of sizes of erased loops for loop-erased random walks

We study the distribution of sizes of erased loops for loop-erased random walks on regular and fractal lattices. We show that for arbitrary graphs the probability $P(l)$ of generating a loop of perimeter $l$ is expressible in terms of the probability $P_{st}(l)$ of forming a loop of perimeter $l$ when a bond is added to a random spanning tree on the same graph by the simple relation $P(l)=P_{st}(l)/l$. On $d$-dimensional hypercubical lattices, $P(l)$ varies as $l^{-\sigma}$ for large $l$, where $\sigma=1+2/z$ for $1<d<4$, where z is the fractal dimension of the loop-erased walks on the graph. On recursively constructed fractals with $\tilde{d} < 2$ this relation is modified to $\sigma=1+2\bar{d}/{(\tilde{d}z)}$, where $\bar{d}$ is the hausdorff and $\tilde{d}$ is the spectral dimension of the fractal.Comment: 4 pages, RevTex, 3 figure

SLE-type growth processes and the Yang-Lee singularity

The recently introduced SLE growth processes are based on conformal maps from an open and simply-connected subset of the upper half-plane to the half-plane itself. We generalize this by considering a hierarchy of stochastic evolutions mapping open and simply-connected subsets of smaller and smaller fractions of the upper half-plane to these fractions themselves. The evolutions are all driven by one-dimensional Brownian motion. Ordinary SLE appears at grade one in the hierarchy. At grade two we find a direct correspondence to conformal field theory through the explicit construction of a level-four null vector in a highest-weight module of the Virasoro algebra. This conformal field theory has central charge c=-22/5 and is associated to the Yang-Lee singularity. Our construction may thus offer a novel description of this statistical model.Comment: 12 pages, LaTeX, v2: thorough revision with corrections, v3: version to be publishe

Diagnosis: Reasoning from first principles and experiential knowledge

Completeness, efficiency and autonomy are requirements for suture diagnostic reasoning systems. Methods for automating diagnostic reasoning systems include diagnosis from first principles (i.e., reasoning from a thorough description of structure and behavior) and diagnosis from experiential knowledge (i.e., reasoning from a set of examples obtained from experts). However, implementation of either as a single reasoning method fails to meet these requirements. The approach of combining reasoning from first principles and reasoning from experiential knowledge does address the requirements discussed above and can possibly ease some of the difficulties associated with knowledge acquisition by allowing developers to systematically enumerate a portion of the knowledge necessary to build the diagnosis program. The ability to enumerate knowledge systematically facilitates defining the program's scope, completeness, and competence and assists in bounding, controlling, and guiding the knowledge acquisition process

Harmonic Measure and Winding of Conformally Invariant Curves

The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff dimension of the points where the potential scales with distance $r$ as $H \sim r^{\alpha}$ while the curve logarithmically spirals with a rotation angle phi=lambda ln r. It obeys the scaling law f(\alpha,\lambda)=(1+\lambda^2) f(\bar \alpha)-b\lambda^2 with \bar \alpha=\alpha/(1+\lambda^2) and b=(25-c)/{12}$, and where f(\alpha)\equiv f(\alpha,0) is the pure harmonic measure spectrum, and c the conformal central charge. The results apply to O(N) and Potts models, as well as to {\rm SLE}_{\kappa}.Comment: 3 figure

The dimension of loop-erased random walk in 3D

We measure the fractal dimension of loop-erased random walk (LERW) in 3 dimensions, and estimate that it is 1.62400 +- 0.00005. LERW is closely related to the uniform spanning tree and the abelian sandpile model. We simulated LERW on both the cubic and face-centered cubic lattices; the corrections to scaling are slightly smaller for the face-centered cubic lattice.Comment: 4 pages, 4 figures. v2 has more data, minor additional change

Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all \b>0 there exists \rho (d,\b)N(d,\b), any $x \in \Z^d$, and any $A\subset \{1,..., n\}^d$ | \{y\in\Z^d\colon \nu_{A,x}(y) \geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where $\nu_{A,x} (y)$ denotes the probability that $y$ is the first entrance point of the simple random walk starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as \b \to \infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne

Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk

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 $\kappa=8/3$ 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$_{8/3}$.Comment: TeX file using APS REVTeX 4.0. 10 pages, 5 figures (encapsulated postscript

Note on SLE and logarithmic CFT

It is discussed how stochastic evolutions may be linked to logarithmic conformal field theory. This introduces an extension of the stochastic Loewner evolutions. Based on the existence of a logarithmic null vector in an indecomposable highest-weight module of the Virasoro algebra, the representation theory of the logarithmic conformal field theory is related to entities conserved in mean under the stochastic process.Comment: 10 pages, LaTeX, v2: version to be publishe

Cardy's Formula for Certain Models of the Bond-Triangular Type

We introduce and study a family of 2D percolation systems which are based on the bond percolation model of the triangular lattice. The system under study has local correlations, however, bonds separated by a few lattice spacings act independently of one another. By avoiding explicit use of microscopic paths, it is first established that the model possesses the typical attributes which are indicative of critical behavior in 2D percolation problems. Subsequently, the so called Cardy-Carleson functions are demonstrated to satisfy, in the continuum limit, Cardy's formula for crossing probabilities. This extends the results of S. Smirnov to a non-trivial class of critical 2D percolation systems.Comment: 49 pages, 7 figure