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The Rohde--Schramm theorem states that Schramm--Loewner Evolution with parameter $kappa$ (or SLE$_kappa$ for short) exists as a random curve, almost surely, if $kappa eq 8$. Here we give a new and concise proof of the result, based on the Liouville quantum gravity coupling (or reverse coupling) with a Gaussian free field. This transforms the problem of estimating the derivative of the Loewner flow into estimating certain correlated Gaussian free fields. While the correlation between these fields is not easy to understand, a surprisingly simple argument allows us to recover a derivative exponent first obtained by Rohde and Schramm, subsequently shown to be optimal by Lawler and Viklund, which then implies the Rohde--Schramm theorem.
These lecture notes offer a gentle introduction to the two-dimensional Discrete Gaussian Free Field with particular attention paid to the scaling limits of the level sets at heights proportional to the absolute maximum. The bulk of the text is based
For the Discrete Gaussian Free Field (DGFF) in domains $D_Nsubseteqmathbb Z^2$ arising, via scaling by $N$, from nice domains $Dsubseteqmathbb R^2$, we study the statistics of the values order-$sqrt{log N}$ below the absolute maximum. Encoded as a po
For a large class of amenable transient weighted graphs $G$, we prove that the sign clusters of the Gaussian free field on $G$ fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite
We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattic
We investigate the percolation phase transition for level sets of the Gaussian free field on $mathbb{Z}^d$, with $dgeqslant 3$, and prove that the corresponding critical parameter $h_*(d)$ is strictly positive for all $dgeqslant3$, thus settling an o