Electronic Communications in Probability

Weighted graphs and complex Gaussian free fields

Gregory F. Lawler and Petr Panov

Full-text: Open access

Abstract

We prove a combinatorial statement about the distribution of directed currents in a complex “loop soup” and use it to give a new proof of the isomorphism, which relates loop measures and complex Gaussian free fields.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 38, 9 pp.

Dates
Received: 7 June 2018
Accepted: 2 April 2019
First available in Project Euclid: 22 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1561169054

Digital Object Identifier
doi:10.1214/19-ECP225

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces

Keywords
Gaussian free field random walk loop soup isomorphism theorem

Rights
Creative Commons Attribution 4.0 International License.

Citation

Lawler, Gregory F.; Panov, Petr. Weighted graphs and complex Gaussian free fields. Electron. Commun. Probab. 24 (2019), paper no. 38, 9 pp. doi:10.1214/19-ECP225. https://projecteuclid.org/euclid.ecp/1561169054


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References

  • [1] E. B. Dynkin (1983). Local times and quantum fields. Seminar on Stochastic Processes, 64-84. Birkhauser.
  • [2] A. Kassel, T. Lévy (2016). Covariant Symanzik identities. Preprint available: arXiv:1607.05201.
  • [3] Y. Le Jan (2008). Dynkin’s isomorphism without symmetry. Stochastic analysis in mathematical physics. ICM 2006 Satellite conference in Lisbon. 43-53 World Scientific.
  • [4] Y. Le Jan (2011). Markov Paths, Loops and Fields, Lecture Notes in Mathematics 2026, Springer-Verlag.
  • [5] G. F. Lawler (2018). Topics in loop measures and the loop-erased walk. Probability Surveys 15, 28-101.
  • [6] G. F. Lawler, V. Limic (2010). Random walk: a modern introduction, volume 123 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010.
  • [7] G. F. Lawler, J. Perlman (2015). Loop measures and the Gaussian free field, in Random Walks, Random Fields, and Disordered Systems, Lecture Notes in Mathematics 2144, M. Biskup, J. Černý, R. Kotecký, ed., Springer-Verlag, 211-235.
  • [8] G. F. Lawler, O. Schramm, W. Werner (2003). Conformal restriction: the chordal case (electronic). J. Am. Math. Soc. 16(4), 917-955.
  • [9] G. F. Lawler, J. A. Trujillo Ferreras (2007). Random walk loop soup. Trans. Amer. Math. Soc., 359(2): 767-787 (electronic).
  • [10] G. F. Lawler, W. Werner (2004). The Brownian loop soup. Probab. Theory Related Fields, 128(4): 565-588.