A. Robert, . Adams, J. John, and . Fournier, Sobolev spaces Academic press, 2003.

L. Valerian and A. , Conformal invariants: topics in geometric function theory, 2010.

J. Aru, T. Lupu, and A. Sepúlveda, The first passage sets of the 2D Gaussian free field. arXiv preprint, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01798812

J. Aru, T. Lupu, and A. Sepúlveda, Excursion decomposition of the 2D Gaussian free field, 2018.

J. Aru, The geometry of the Gaussian free field combined with SLE processes and the KPZ relation, 2015.
URL : https://hal.archives-ouvertes.fr/tel-01195774

J. Aru and A. Sepúlveda, Survey in preparation, 2018.

J. Aru and A. Sepúlveda, Two-valued local sets of the 2D continuum Gaussian free field: connectivity , labels, and induced metrics, 2018.
DOI : 10.1017/s1474748017000160

URL : http://arxiv.org/pdf/1603.03362

J. Aru, A. Sepúlveda, and W. Werner, On bounded-type thin local sets of the twodimensional Gaussian free field, Journal of the Institute of Mathematics of Jussieu, pp.1-28
DOI : 10.1017/s1474748017000160

URL : http://arxiv.org/pdf/1603.03362

J. R. Baxter and R. Van-severen-chacon, The equivalence of diffusions on networks to Brownian motion The random walk representation of classical spin systems and correlation inequalities, Contemporary Mathematics, pp.33-48123, 1982.

M. Biskup and O. Louidor, Conformal symmetries in the extremal process of two-dimensional discrete Gaussian free field. arXiv preprint arXiv:1410.4676, 2014. [Com13] Mark Comerford. The Carathéodory topology for multiply connected domains I, Open Mathematics, vol.11, issue.2, pp.322-340, 2013.

D. Chelkak and S. Smirnov, Discrete complex analysis on isoradial graphs, Advances in Mathematics, vol.228, issue.3, pp.1590-1630, 2011.
DOI : 10.1016/j.aim.2011.06.025

URL : https://doi.org/10.1016/j.aim.2011.06.025

J. Ding and L. Li, Chemical Distances for Percolation of Planar Gaussian Free Fields and Critical Random Walk Loop Soups, Communications in Mathematical Physics, vol.67, issue.4, pp.523-553, 2018.
DOI : 10.1007/978-3-319-44465-9_19

J. Dubédat, Commutation relations for Schramm-Loewner evolutions, Communications on Pure and Applied Mathematics, vol.129, issue.12, pp.1792-1847, 2007.
DOI : 10.1007/978-3-540-39982-7_2

J. Dubédat, SLE and the free field: Partition functions and couplings, Journal of the American Mathematical Society, vol.22, issue.4, pp.995-1054, 2009.
DOI : 10.1090/S0894-0347-09-00636-5

E. Dynkin, Markov processes as a tool in field theory, Journal of Functional Analysis, vol.50, issue.2, pp.167-187, 1983.
DOI : 10.1016/0022-1236(83)90066-6

URL : https://doi.org/10.1016/0022-1236(83)90066-6

E. Dynkin, Gaussian and non-Gaussian random fields associated with Markov processes, Journal of Functional Analysis, vol.55, issue.3, pp.344-376, 1984.
DOI : 10.1016/0022-1236(84)90004-1

URL : https://doi.org/10.1016/0022-1236(84)90004-1

E. Dynkin, Local Times and Quantum Fields, Seminar on Stochastic Processes of Progress in Probability and Statistics, pp.69-84, 1983.
DOI : 10.1007/978-1-4684-9169-2_5

N. Eisenbaum, Une version sans conditionnement du theoreme d???isomorphisme de Dynkin, Séminaire de Probabilités XXIX, pp.266-289, 1995.
DOI : 10.1007/BFb0084140

N. Enriquez and Y. Kifer, Markov chains on graphs and Brownian motion, Journal of Theoretical Probability, vol.14, issue.2, pp.495-510, 2001.
DOI : 10.1023/A:1011119932045

N. Eisenbaum, H. Kaspi, M. B. Marcus, J. Rosen, and Z. Shi, A Ray-Knight theorem for symmetric Markov processes. The Annals of Probability, pp.1781-1796, 2000.

P. Fitzsimmons and J. Rosen, Markovian loop soups: permanental processes and isomorphism theorems, Gaw96] Krzysztof Gawedzki. Lectures on conformal field theory, pp.1-30733, 1996.
DOI : 10.1214/EJP.v19-3255

URL : http://doi.org/10.1214/ejp.v19-3255

S. Janson, Bounds on the distributions of extremal values of a scanning process, Stochastic Processes and their Applications, pp.313-328, 1984.
DOI : 10.1016/0304-4149(84)90303-X

S. Janson, Gaussian Hilbert Spaces, volume 129 of Cambridge Tracts in Mathematics, 1997.

P. Koebe, ??ber die Konforme Abbildung Endlich- und Unendlich-Vielfach Zusammenh??ngender Symmetrischer Bereiche, Acta Mathematica, vol.43, issue.0, pp.263-287, 1922.
DOI : 10.1007/BF02401759

G. Lawler, Conformally invariant processes in the plane Number 114 Markov loops, determinants and Gaussian fields. arXiv preprint arXiv:1012.4797 Markov loops and renormalization. The Annals of Probability Markov paths, loops and fields, Lecture Notes in Mathematics, vol.38, issue.2026, pp.1280-1319, 2007.
DOI : 10.1090/surv/114

G. Lawler and V. Limic, Random walk: a modern introduction, 2010.
DOI : 10.1017/CBO9780511750854

URL : http://www.math.uchicago.edu/~lawler/srwbook.pdf

H. Lacoin, R. Rhodes, and V. Vargas, Large deviations for random surfaces: the hyperbolic nature of Liouville Field Theory, 2014.

G. Lawler, O. Schramm, and W. Werner, Conformal restriction: the chordal case, Journal of the American Mathematical Society, vol.16, issue.04, pp.917-955, 2003.
DOI : 10.1090/S0894-0347-03-00430-2

G. Lawler and J. Trujillo-ferreras, Random walk loop soup. Transactions of the, pp.767-787, 2007.

T. Lupu, Convergence of the two-dimensional random walk loop soup clusters to CLE. arXiv preprint, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01120001

T. Lupu, From loop clusters and random interlacements to the free field. The Annals of Probability, pp.2117-2146, 2016.
DOI : 10.1214/15-aop1019

T. Lupu, Loop percolation on discrete half-plane, Electronic Communications in Probability, vol.21, issue.0, p.2016
DOI : 10.1214/16-ECP4571

URL : http://doi.org/10.1214/16-ecp4571

G. Lawler, W. W. , T. Lupu, and W. Werner, The Brownian loop soup. Probability Theory and Related Fields The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph. Probability Theory and Related Fields, pp.565-588, 2004.
DOI : 10.1007/s00440-003-0319-6

B. Michael, J. Marcus, and . Rosen, Markov processes, Gaussian processes and local times, 2006.

J. Miller and S. Sheffield, The GFF and CLE(4), 2011. Slides, talks and private communications

J. Miller and S. Sheffield, Imaginary geometry I: interacting SLEs. Probability Theory and Related Fields, pp.553-705, 2016.
DOI : 10.1007/s00440-016-0698-0

URL : https://link.springer.com/content/pdf/10.1007%2Fs00440-016-0698-0.pdf

J. Miller and S. Sheffield, Imaginary geometry II: reversibility of SLE?(?1;?2) for ? ? (0, 4) The Annals of Probability, pp.1647-1722, 2016.

J. Miller and S. Sheffield, Imaginary geometry III: reversibility of \SLE_\kappa for \kappa \in (4,8), Annals of Mathematics, vol.184, issue.2, pp.455-486, 2016.
DOI : 10.4007/annals.2016.184.2.3

URL : http://arxiv.org/pdf/1201.1498

J. Miller and S. Sheffield, Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees. Probability Theory and Related Fields, pp.729-869, 2017.
DOI : 10.1007/s00440-017-0780-2

URL : https://link.springer.com/content/pdf/10.1007%2Fs00440-017-0780-2.pdf

L. Pitt, Positively correlated normal variables are associated. The Annals of Probability, pp.496-499, 1982.
DOI : 10.1214/aop/1176993872

URL : http://doi.org/10.1214/aop/1176993872

E. Peltola and H. Wu, Global multiple SLEs for ? ? 4 and connection probabilities for level lines of GFF. arXiv preprint

W. Qian and W. Werner, Decomposition of Brownian loop-soup clusters. arXiv preprint

A. Sepúlveda, On thin local sets of the Gaussian free field. arXiv preprint

A. Shamov, On Gaussian multiplicative chaos, Journal of Functional Analysis, vol.270, issue.9, pp.3224-3261, 2016.
DOI : 10.1016/j.jfa.2016.03.001

URL : http://arxiv.org/pdf/1407.4418.pdf

S. Sheffield, Local sets of the Gaussian free field: Slides and audio, 2005.

S. Sheffield, Gaussian free fields for mathematicians. Probability theory and related fields, pp.521-541, 2007.
DOI : 10.1007/s00440-006-0050-1

URL : http://arxiv.org/pdf/math/0312099

B. Simon, The P (?)2 Euclidean (Quantum) Field Theory. Princeton Series in Physics, 1974.

O. Schramm and S. Sheffield, Contour lines of the two-dimensional discrete Gaussian free field, Acta Mathematica, vol.20221, issue.1, 2009.
DOI : 10.1007/978-1-4419-9675-6_32

O. Schramm and S. Sheffield, A contour line of the continuum Gaussian free field. Probability Theory and Related Fields, pp.47-80, 2013.
DOI : 10.1007/s00440-012-0449-9

URL : http://arxiv.org/pdf/1008.2447

S. Sheffield and W. Werner, Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Annals of Mathematics, vol.176, issue.3, pp.1827-1917, 2012.
DOI : 10.4007/annals.2012.176.3.8

URL : http://annals.math.princeton.edu/wp-content/uploads/annals-v176-n3-p08-p.pdf

K. Symanzik, Euclidean Quantum Field Theory. I. Equations for a Scalar Model, Journal of Mathematical Physics, vol.12, issue.3, pp.510-525, 1966.
DOI : 10.1103/PhysRev.130.1605

K. Symanzik, Euclidean quantum field theory, Scuola intenazionale di Fisica, pp.152-223, 1969.

A. Sznitman, An isomorphism theorem for random interlacements, Electronic Communications in Probability, vol.17, issue.0, pp.1-9, 2012.
DOI : 10.1214/ECP.v17-1792

URL : http://doi.org/10.1214/ecp.v17-1792

A. Sznitman, Topics in occupation times and Gaussian free field. Zurich lectures in advanced mathematics, 2012.
DOI : 10.4171/109

A. Sznitman, On scaling limits and Brownian interlacements, Bulletin of the Brazilian Mathematical Society, New Series, vol.64, issue.12, pp.555-592, 2013.
DOI : 10.1002/cpa.20382

URL : http://arxiv.org/pdf/1209.4531.pdf

T. Van-de-brug, F. Camia, and M. Lis, Random walk loop soups and conformal loop ensembles . Probability Theory and Related Fields [vdBK96] Rob van den Berg and Harry Kesten. A note on disjoint-occurrence inequalities for marked Poisson point processes, Journal of Applied Probability, vol.166, issue.332, pp.553-584420, 1996.

W. Werner, Conformal restriction and related questions, Probability Surveys, vol.2, issue.0, pp.145-190, 2005.
DOI : 10.1214/154957805100000113

URL : https://hal.archives-ouvertes.fr/hal-00012645

W. Werner, Topics on the GFF and CLE(4), 2016.

W. Werner and H. Wu, From CLE($\kappa$) to SLE($\kappa,\rho$)'s, Electronic Journal of Probability, vol.18, issue.0, pp.1-20, 2013.
DOI : 10.1214/EJP.v18-2376

URL : http://doi.org/10.1214/ejp.v18-2376

M. Wang and H. Wu, Level lines of Gaussian Free Field I: Zero-boundary GFF, Stochastic Processes and their Applications, 2016.
DOI : 10.1016/j.spa.2016.07.009

URL : http://arxiv.org/pdf/1412.3839

. Department-of-mathematics and . Zürich, Rämistr. 101, 8092 Zürich, Switzerland E-mail address: juhan.aru@math.ethz.ch CNRS and LPSM, UMR 8001