zbMATH — the first resource for mathematics

Markov loops and renormalization. (English) Zbl 1197.60075
From the introduction: The purpose of this paper is to explore some simple relations between Markov path and loop measures, spanning trees, determinants and Markov fields such as the free field. The main emphasis is put on the study of occupation fields defined by Poissonian ensembles of Markov loops. These were defined by G. F. Lawler and W. Werner [Probab. Theory Relat. Fields 128, No. 4, 565–588 (2004; Zbl 1049.60072)] for planar Brownian motion. We first present the results in the elementary framework of symmetric Markov chains on a finite space, proving also in passing several interesting results such as the relation between loop ensembles and spanning trees. We can show that the renormalized powers of the occupation field (i. e. the self intersection local times of the loop ensemble) converge in the two dimensional case and that they can be identified with higher even Wick powers of the free field when the intensity parameter is a half integer.

60J27 Continuous-time Markov processes on discrete state spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J45 Probabilistic potential theory
Full Text: DOI
[1] Dubédat, J. (2007). SLE and the free field: Partition functions and couplings. J. Amer. Math. Soc. 22 995-1054. · Zbl 1204.60079 · doi:10.1090/S0894-0347-09-00636-5
[2] Dynkin, E. B. (1984). Local times and quantum fields. In Seminar on Stochastic Processes , 1983 ( Gainesville , Fla. , 1983). Progress in Probability Statist. 7 69-83. Birkhäuser, Boston, MA. · Zbl 0554.60058
[3] Eisenbaum, N. and Kaspi, H. (2006). A characterization of the infinitely divisible squared Gaussian processes. Ann. Probab. 34 728-742. · Zbl 1102.60031 · doi:10.1214/009117905000000684
[4] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics 19 . de Gruyter, Berlin. · Zbl 0838.31001
[5] Gawedzki, K. Conformal field theory. Lecture notes. I.A.S., Princeton. · Zbl 0842.17031
[6] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3 . Oxford Univ. Press, New York. · Zbl 0771.60001
[7] Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47 655-693. · Zbl 0445.60058 · doi:10.1215/S0012-7094-80-04741-9
[8] Lawler, G. F. (1999). Loop-erased random walk. In Perplexing Problems in Probability. Progress in Probability 44 197-217. Birkhäuser, Boston, MA. · Zbl 0947.60055
[9] Lawler, G. F. and Werner, W. (2004). The Brownian loop soup. Probab. Theory Related Fields 128 565-588. · Zbl 1049.60072 · doi:10.1007/s00440-006-0319-6
[10] Lawler, G. F. and Trujillo Ferreras, J. A. (2007). Random walk loop soup. Trans. Amer. Math. Soc. 359 767-787 (electronic). · Zbl 1120.60037 · doi:10.1090/S0002-9947-06-03916-X
[11] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939-995. · Zbl 1126.82011 · doi:10.1214/aop/1079021469
[12] Le Jan, Y. (1978). Mesures associées à une forme de Dirichlet. Applications. Bull. Soc. Math. France 106 61-112. · Zbl 0393.31008 · numdam:BSMF_1978__106__61_0 · eudml:87337
[13] Le Jan, Y. (1988). On the Fock space representation of functionals of the occupation field and their renormalization. J. Funct. Anal. 80 88-108. · Zbl 0671.60064 · doi:10.1016/0022-1236(88)90067-5
[14] Le Jan, Y. (2008). Dynkin’s isomorphism without symmetry. In Stochastic Analysis in Mathematical Physics 43-53. World Sci. Publ., Hackensack, NJ. · Zbl 1167.60016
[15] Lyons, R. and Peres, Y. Probability on trees networks.
[16] Marcus, M. B. and Rosen, J. (1992). Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Probab. 20 1603-1684. · Zbl 0762.60068 · doi:10.1214/aop/1176989524
[17] Neveu, J. (1968). Processus Aléatoires Gaussiens. Séminaire de Mathématiques Supérieures ( Été , 1968) 34 Les Presses de l’Université de Montréal, Montreal, Que. · Zbl 0192.54701
[18] Qian, M. P. and Qian, M. (1982). Circulation for recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 59 203-210. · Zbl 0483.60060 · doi:10.1007/BF00531744
[19] Schramm, O. and Sheffield, S. (2006). Contour lines of the two dimensional discrete Gaussian free field. Available at . · Zbl 1210.60051 · arxiv.org
[20] Simon, B. (1979). Trace Ideals and Their Applications. London Math. Soc. Lect. Notes 35 . Cambridge, U.K. · Zbl 0423.47001
[21] Simon, B. (1974). The P ( \varphi ) 2 Euclidean ( Quantum ) Field Theory . Princeton Univ. Press, Princeton, NJ. · Zbl 1175.81146
[22] Symanzik, K. (1969). Euclidean quantum field theory. In Scuola intenazionale di Fisica “Enrico Fermi.” XLV Corso 152-223. Academic Press.
[23] Vere-Jones, D. (1988). A generalization of permanents and determinants. Linear Algebra Appl. 111 119-124. · Zbl 0665.15007 · doi:10.1016/0024-3795(88)90053-5
[24] Vere-Jones, D. (1997). Alpha-permanents and their applications to multivariate gamma, negative binomial and ordinary binomial distributions. New Zealand J. Math. 26 125-149. · Zbl 0879.15003
[25] Werner, W. (2008). The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc. 21 137-169 (electronic). · Zbl 1130.60016 · doi:10.1090/S0894-0347-07-00557-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.