We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle. The homeomorphism is constructed using the exponential of$βX$, where X is the restriction of the 2-dimensional free field on the circle and the parameter$β$is in the “high temperature” regime $$ \beta < \sqrt {2} $$ . The welding problem is solved by studying a non-uniformly elliptic Beltrami equation with a random complex dilatation. For the existence a method of Lehto is used. This requires sharp probabilistic estimates to control conformal moduli of annuli and they are proven by decomposing the free field as a sum of independent fixed scale fields and controlling the correlations of the complex dilatation restricted to dyadic cells of various scales. For the uniqueness we invoke a result by Jones and Smirnov on conformal removability of Hölder curves. Our curves are closely related to SLE($ϰ$) for$ϰ$<4.

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: • We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu’s Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. • We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs—in contrast, in the case of the Gromov monster expanders are not even coarsely embedded. We present further applications.

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).

Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich’s elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure $E$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_H X$ with the Gromov metric is H\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0 < \lambda < 1$ on $(X,E)$. We show that the Martin boundary ${\mathcal M}$ can be identified with $\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain precise estimates of the Martin kernel and the Na\"im kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate $\Theta (\xi,\eta) \asymp |\xi-\eta|^{-(\alpha+\beta)}$, where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ depends on $\lambda$. For suitable $\beta$, the kernel defines a regular non-local Dirichlet form on $K$. This extends the results of Kigami concerning random walks on certain trees with Cantor-type sets as boundaries.

This paper is devoted to exploiting the restrictions of Riesz–Morrey potentials on either unbounded or bounded domains in Euclidean spaces.