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Register a new userThe Rohde--Schramm theorem, via the Gaussian free field
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Added by
Nathanael Berestycki
Publication date
2017
fields
Physics
and research's language is
English
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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.
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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 on recent joint papers with O. Louidor and with J. Ding and S. Goswami. Still, new proofs of the tightness and distributional convergence of the centered DGFF maximum are presented that by-pass the use of the modified Branching Random Walk. The text contains a wealth of instructive exercises and a list of open questions and conjectures for future research.
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 point process on $Dtimesmathbb R$, the scaled spatial distribution of these near-extremal level sets in $D_N$ and the field values (in units of $sqrt{log N}$ below the absolute maximum) tends in law as $Ntoinfty$ to the product of the critical Liouville Quantum Gravity (cLQG) $Z^D$ and the Rayleigh law. The convergence holds jointly with the extremal process, for which $Z^D$ enters as the intensity measure of the limiting Poisson point process, and that of the DGFF itself; the cLQG defined by the limit field then coincides with $Z^D$. While the limit near-extremal process is measurable with respect to the limit continuum GFF, the limit extremal process is not. Our results explain why the various ways to norm the lattice cLQG measure lead to the same limit object, modulo overall normalization.
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 sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices like $mathbb{Z}^d$, for $d geqslant 3$, but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman in arXiv:0704.2560, and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from arXiv:1010.1490.
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 open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on $mathbb{Z}^d$, for all $dgeqslant 3$. Among other things, our construction of an infinite cluster above small, but positive level $h$ involves random interlacements at level $u>0$, a random subset of $mathbb{Z}^d$ with desirable percolative properties, introduced in arXiv:0704.2560 in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field, see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two objects, lifted to a continuous metric graph structure over $mathbb{Z}^d$.
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