Skip to main content
    • Aa
    • Aa


  • Juhan Aru (a1), Avelio Sepúlveda (a1) and Wendelin Werner (a1)

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble $\text{CLE}_{4}$ and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the $\text{CLE}_{4}$ carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of $\text{CLE}_{4}$ ) are in fact measurable functions of the GFF.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

J. Dubédat , Commutation relations for Schramm–Loewner evolutions, Commun. Pure Appl. Math. 60 (2007), 17921847.

Z.-X. He and O. Schramm , Fixed points, Koebe uniformization and circle packings, Ann. Math. 137 (1995), 369406.

K. Izyurov and K. Kytölä , Hadamard’s formula and couplings of SLEs with free field, Probab. Theory Related Fields 155 (2013), 3569.

J. P. Miller and S. Sheffield , Imaginary Geometry I. Interacting SLEs, Probab. Theory Related Fields 164 (2016), 553705.

J. P. Miller and S. Sheffield , Imaginary Geometry II. Reversibility of SLE𝜅(𝜌1; 𝜌2) for 𝜅 ∈ (0, 4), Ann. Probab. 44 (2016), 16471722.

Ş. Nacu and W. Werner , Random soups, carpets and fractal dimensions, J. Lond. Math. Soc. (2) 83 (2011), 789809.

S. C. Port and C. J. Stone , Brownian Motion and Classical Potential Theory (Academic Press, New York, 1978).

O. Schramm and S. Sheffield , Contour lines of the discrete two-dimensional Gaussian free field, Acta Math. 202 (2009), 21137.

S. Sheffield , Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab. 44 (2016), 34743545.

S. Sheffield and W. Werner , Conformal Loop Ensembles: the Markovian characterization and the loop-soup construction, Ann. Math. 176 (2012), 18271917.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 42 *
Loading metrics...

* Views captured on Cambridge Core between 27th April 2017 - 24th June 2017. This data will be updated every 24 hours.