Random Curves by Conformal Welding

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally related to SLE$(\kappa)$ for $\kappa<4$.Comment: 5 page

Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of 1/f1/f Noises generated by Gaussian Free Fields

We compute the distribution of the partition functions for a class of one-dimensional Random Energy Models (REM) with logarithmically correlated random potential, above and at the glass transition temperature. The random potential sequences represent various versions of the 1/f noise generated by sampling the two-dimensional Gaussian Free Field (2dGFF) along various planar curves. Our method extends the recent analysis of Fyodorov Bouchaud from the circular case to an interval and is based on an analytical continuation of the Selberg integral. In particular, we unveil a {\it duality relation} satisfied by the suitable generating function of free energy cumulants in the high-temperature phase. It reinforces the freezing scenario hypothesis for that generating function, from which we derive the distribution of extrema for the 2dGFF on the $[0,1]$ interval. We provide numerical checks of the circular and the interval case and discuss universality and various extensions. Relevance to the distribution of length of a segment in Liouville quantum gravity is noted.Comment: 25 pages, 12 figures Published version. Misprint corrected, references and note adde

Isomorphisms of Royden Type Algebras over mathbbS1mathbb S^1

Let $mathbb S^1$ and $mathbb D$ be the unit circle and the unit disc in the plane and let us denote by $mathcal A(mathbb S^1)$ the algebra of the complex valued continuous functions on $mathbb S^1$ which are traces of functions in the Sobolev class $mathscr W^{1,2}(mathbb D)$. On $mathcal A(mathbb S^1)$ we define the following norm $|u|= |u|_{L^{infty}(mathbb S^1)}+ left(int int_{mathbb D}| abla ilde{u}|^2 ight)^{1/2}$ where $ilde{u}$ is the harmonic extension of $u$ to $mathbb D$. We prove that every isomorphism of the functional algebra $mathcal A(mathbb S^1)$ is a quasisymmetric change of variables on $mathbb S^1$