Loewner Curvature

The purpose of this paper is to interpret the phase transition in the Loewner theory as an analog of the hyperbolic variant of the Schur theorem about curves of bounded curvature. We define a family of curves that have a certain conformal self-similarity property. They are characterized by a deterministic version of the domain Markov property, and have constant Loewner curvature. We show that every sufficiently smooth curve in a simply connected plane domain has a best-approximating curve of constant Loewner curvature, establish a geometric comparison principle, and show that curves of Loewner curvature bounded by 8 are simple curves.Comment: 18 pages, 7 figure

Collisions and Spirals of Loewner Traces

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 \lambda(t) so that the hulls driven by K\lambda(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-ph/0309006Comment: 34 pages, 11 figure

Political Power and Collective Action: British and Swedish Labor Movements, 1900-1950

http://deepblue.lib.umich.edu/bitstream/2027.42/50877/1/100.pd

Support schemes for renewable electricity in the EU

This paper discusses the level and design of support schemes used to promote renewable electricity in Europe. A theoretical model is presented to determine optimal renewable energy policies. Policies that solely aim to address environmental externalities and energy security risks are unlikely to make renewable power technologies competitive. Learning effects and spillovers are necessary to justify the need for support schemes. The analysis suggests that feed-in premiums guaranteed in addition to the electricity market price should be preferred over feed-in tariffs, which provide the eligible power producer with a guaranteed price. The premiums should be time limited and frequently reviewed. Once the technology becomes competitive, tradable green certificates would be a more suitable support instrument. As regards wind energy, the available estimates of externalities suggest that levels are probably too high in many Member States. In addition, the current promotion of photovoltaics could possibly be more cost-efficient if it targeted technology development more directly.european union, eu, setzer, wolff, van den Noord, euro area, money, heterogeneity, money holdings

Convergence of the Probabilistic Interpretation of Modulus

Given a Jordan domain $\Omega\subset\mathbb{C}$ and two arcs $A, B$ on $\partial\Omega$, the modulus of the curve family connecting $A$ and $B$ in $\Omega$ is famously related, via the conformal map $\phi$ mapping $\Omega$ to a rectangle $R=[0,L]\times[0,1]$ so that $A$ and $B$ are sent to the vertical sides, to the corresponding modulus in $R$. Moreover, in the case of the rectangle the family of horizontal segments connecting the two sides has the same modulus as the entire connecting family. Pulling these segments back to $\Omega$ via $\phi$ yields a family of extremal curves (also known as horizontal trajectories) connecting $A$ to $B$ in $\Omega$. In this paper, we show that these extremal curves can be approximated by some discrete curves arising from an orthodiagonal approximation of $\Omega$. Moreover, we show that these curves carry a natural probability mass function (pmf) deriving from the theory of discrete modulus and that these pmf's converge to the uniform distribution on the set of extremal curves. The key ingredient is an algorithm that, for an embedded planar graph, takes the current flow between two sets of nodes $A$ and $B$, and produces a unique path decomposition with non-crossing paths. Moreover, some care was taken to adapt recent results for harmonic convergence on orthodiagonal maps, due to Gurel-Gurevich, Jerison, and Nachmias, to our context

The scaling limit of fair Peano curves

We study random Peano curves on planar square grids that arise from fair random spanning trees. These are trees that are sampled in such a way as to have the same (if possible) edge probabilities. In particular, we are interested in identifying the scaling limit as the mesh-size of the grid tends to zero. It is known \cite{lawler-schramm-werner2002} that if the trees are sampled uniformly, then the scaling limit exists and equals ${\rm SLE}_8$. We show that if we simply follow the same steps as in \cite{lawler-schramm-werner2002}, then fair Peano curves have a deterministic scaling limit