The conformally invariant measure on self-avoiding loops. (English) Zbl 1130.60016
Let \(S\) be a Riemann surface and \(S'\subset S\) another Riemann surface contained in \(S\). The author shows that there exists a unique (up to multiplication) measure \(\mu_S\) on the set of self avoiding loops on \(S\) which satisfies the following two conditions: (1) It is conformally invariant, that is, for any two domains \(D, D'\subset S\), which are conformally equivalent, the image of the measure \(\mu_S\) restricted to loops contained in \(D\) via any conformal map from \(D\) to \(D'\) is the same as \(\mu_S\) restricted to loops contained in \(D'\). (2) It is invariant under restriction, that is, \(\mu_S'\) is equal to \(\mu_S\) restricted to loops contained in \(S'\).
For the special case \(S=\mathbb C\), the author first shows that (2) and a weaker version of (1), namely its restriction to simply connected domains, imply the uniqueness. The measure \(\mu_{\mathbb C}\) is then constructed from the measure on Brownian loops defined in G. F. Lawler and W. Werner [Probab. Theory Relat. Fields 128, 565–588 (2004; Zbl 1049.60072)] by considering outer boundaries of Brownian loops. Finally, by some SLE\(_{8/3}\) considerations for \(\mu_{\mathbb C}\) restricted to annular domains, it is proved that \(\mu_{\mathbb C}\) satisfies (1). The construction is then lifted to Riemann surfaces. The author then studies properties of \(\mu_{\mathbb C}\) and discuses some consequences concerning outer and inner boundaries of Brownian loops and critical percolation cluster.
For the special case \(S=\mathbb C\), the author first shows that (2) and a weaker version of (1), namely its restriction to simply connected domains, imply the uniqueness. The measure \(\mu_{\mathbb C}\) is then constructed from the measure on Brownian loops defined in G. F. Lawler and W. Werner [Probab. Theory Relat. Fields 128, 565–588 (2004; Zbl 1049.60072)] by considering outer boundaries of Brownian loops. Finally, by some SLE\(_{8/3}\) considerations for \(\mu_{\mathbb C}\) restricted to annular domains, it is proved that \(\mu_{\mathbb C}\) satisfies (1). The construction is then lifted to Riemann surfaces. The author then studies properties of \(\mu_{\mathbb C}\) and discuses some consequences concerning outer and inner boundaries of Brownian loops and critical percolation cluster.
Reviewer: Jiří Černý (Zürich)
MSC:
| 60D05 | Geometric probability and stochastic geometry |
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 82B43 | Percolation |
| 30C99 | Geometric function theory |
| 60J65 | Brownian motion |
Keywords:
conformal invariance; self-avoiding loops; Brownian loops; critical percolation; Schram-Loewner evolutionCitations:
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